Scientific Papers of Josiah Willard Gibbs, Volume 1/Chapter V

by Josiah Willard Gibbs

V.

ON THE VAPOR-DENSITIES OF PEROXIDE OF NITROGEN, FORMIC ACID, ACETIC ACID, AND PERCHLORIDE OF PHOSPHORUS.

[American Journal of Science, ser. 3, vol. xviii, Oct.–Nov. 1879.]

The relation between temperature, pressure, and volume, for the vapor of each of these substances differs widely from that expressed by the usual laws for the gaseous state, the laws known most widely by the names of Mariotte, Gay-Lussac, and Avogadro. The density of each vapor, in the sense in which the term is usually employed in chemical treatises, i.e., its density taken relatively to air of the same temperature and pressure,^[1] has not a constant value, but varies nearly in the ratio of one to two. And these variations are exhibited at pressures not exceeding that of the atmosphere and at temperatures comprised between zero and 200 or 300 of the centigrade scale.

Such anomalies have been explained by the supposition that the vapor consists of a mixture of two or three different kinds of gas or vapor, which have different densities. Thus it is supposed that the vapor of peroxide of nitrogen is a gas-mixture, the components of which are represented (in the newer chemical notation) by NO₂ and N₂O₄ respectively. The densities corresponding to these formulæ are 1.589 and 3.178. The density of the mixture should have a value intermediate between these numbers, which is substantially the case with the actual vapor. The case is similar with respect to the vapor of formic acid, which we may regard as a mixture of CH₂O₂ (density 1.589) and C₂H₄O₄ (density 3.178), and the vapor of acetic acid, which we may regard as a mixture of C₂H₄O₂ (density 2.073) and C₄H₈O₄ (density 4.146). In the case of perchloride of phosphorus, we must suppose the vapor to consist of three parts; PCl₅ (the proper perchloride, density 7.20), PCl₃ (the protochloride, density 4.98), and Cl₂ (chlorine, density 2.22). Since the chlorine and protochloride arise from the decomposition of the perchloride, there must be as many molecules of the type Cl₂ as of the type PCl₃. Now a gas-mixture containing an equal number of molecules of PCl₃ and Cl₂ will have the density ${\tfrac {1}{2}}$(4.98 + 2.22) or 3.60. It follows that, at least so far as the range of the possible values of its density is concerned, we may regard the vapor as a mixture in variable proportions of two kinds of gas having the densities 7.20 and 3.60 respectively. The observed values of the density accord with this supposition.

These hypotheses respecting the constitution of the vapors are corroborated, in the case of peroxide of nitrogen and perchloride of phosphorus, by other circumstances. The varying color of the first vapor may be accounted for by supposing that the molecules of the type N₂O₄ are colorless, while each molecule of the type NO₂ has a constant color. This supposition affords a simple relation between the density of the vapor and the depth of its color, which has been verified by experiment.^[2]

The vapor of the perchloride of phosphorus shows with increasing temperature in an increasing degree the characteristic color of chlorine. The amount of the color appears to be such as is required by the hypothesis respecting the constitution of the vapor on the very probable supposition that the perchloride proper is colorless, but the case hardly admits of such exact numerical determinations as are possible with respect to the peroxide of nitrogen.^[3] But since the products of dissociation are in this case dissimilar, they may be partially separated by diffusion through a neutral gas, the lighter chlorine diffusing more rapidly than the heavier protochloride. The fact of dissociation has in this way been proved by direct experiment.^[4]

In the case of acetic and formic acids, we have no other evidence than the variations of the densities in support of the hypothesis of the compound nature of the vapor, yet if these variations shall appear to follow the same law as those of the peroxide of nitrogen and the perchloride of phosphorus, it will be difficult to refer them to a different cause.

But however it may be with these acids, the peroxide of nitrogen and the perchloride of phosphorus evidently furnish us with the means of studying the laws of chemical equilibrium in gas-mixtures in which chemical change is possible and does in fact take place reversibly, with varying conditions of temperature and pressure. Or, if from any considerations we can deduce a general law determining the proportions of the component gases necessary for the equilibrium of such a mixture under any given conditions, these substances afford an appropriate test for such a law.

In a former paper^[5] by the present writer, equations were proposed to express the relation between the temperature, the pressure or the volume, and the quantities of the components in such a gas-mixture as we are considering a gas-mixtwe of convertible components in the language of that paper. Applied to the vapor of the peroxide of nitrogen, these equations led to a formula giving the density in terms of the temperature and pressure, which was shown to agree very closely with the experiments of Deville and Troost, and much less closely, but apparently within the limits of possible error, with the experiments of Playfair and Wanklyn. Since the publication of that paper, new determinations of the density have been published in different quarters, which render it possible to compare the equation with the results of experiment throughout a wider range of temperature and pressure. In the present paper, all experimental determinations of the density of this vapor which have come to the knowledge of the writer are cited, and compared with the values demanded by the formula, and a similar comparison of theory and experiment is made with respect to each of the other substances which have been mentioned.

The considerations from which these formulae were deduced may be briefly stated as follows. It will be observed that they are based rather upon an extension of generally acknowledged principles to a new class of cases than upon the introduction of any new principle.

The energy of a gas-mixture may be represented by an expression of the form

$m_{1}(c_{1}t+{\text{E}}_{1})+m_{2}(c_{2}t+{\text{E}}_{2})+{\text{etc.,}}$

with as many terms as there are different kinds of gas in the mixture, $m_{1},m_{2}$, etc. denoting the quantities (by weight) of the several component gases, $c_{1},c_{2}$, etc., their several specific heats at constant volume, ${\text{E}}_{1},{\text{E}}_{2}$, etc., other constants, and $t$ the absolute temperature. In like manner the entropy of the gas-mixture is expressed by

$m_{1}\left({\text{H}}_{1}+c_{1}\log _{\text{N}}t-a_{1}\log _{\text{N}}{\frac {m_{1}}{v}}\right)+m_{2}\left({\text{H}}_{2}+c_{2}\log _{\text{N}}t-a_{2}\log _{\text{N}}{\frac {m_{2}}{v}}\right)+{\text{etc.,}}$

where $v$ denotes the volume, and ${\text{H}}_{1},a_{1},{\text{H}}_{2},a_{2}$, etc. denote constants relating to the component gases, $a_{1},a_{2}$, etc. being inversely proportional to their several densities. The logarithms are Naperian. These expressions for energy and entropy will undoubtedly apply to mixtures of different gases, whatever their chemical relations may be (with such limitations and with such a degree of approximation as belong to other laws of the gaseous state), when no chemical action can take place under the conditions considered. If we assume that they will apply to such cases as we are now considering, although chemical action is possible, and suppose the equilibrium of the mixture with respect to chemical change to be determined by the condition that its entropy has the greatest value consistent with its energy and its volume, we may easily obtain an equation between $m_{1},m_{2}$, etc., $t$ and $v$.^[6]

The condition that the energy does not vary, gives

$(m_{1}c_{1}+m_{2}c_{2}+{\text{etc.}})dt+(c_{1}t+{\text{E}}_{1})dm_{1}+(c_{2}t+{\text{E}}_{2})dm_{2}+{\text{etc.}}=0.$

(1)

The condition that the entropy is a maximum implies that its variation vanishes, when the energy and volume are constant.

This gives

${\frac {m_{1}c_{1}+m_{2}c_{2}+{\text{etc.}}}{t}}dt+\left({\text{H}}_{1}-a_{1}+c_{1}\log _{\text{N}}t-a_{1}\log _{\text{N}}{\frac {m_{1}}{v}}\right)dm_{1}$ $+\left({\text{H}}_{2}-a_{2}+c_{2}\log _{\text{N}}t-a_{2}\log _{\text{N}}{\frac {m_{2}}{v}}\right)dm_{2}+{\text{etc.}}=0.$

(2)

Eliminating $dt$, we have

$\left({\text{H}}_{1}-a_{1}-c_{1}-{\frac {{\text{E}}_{1}}{t}}+c_{1}\log _{\text{N}}t-a_{1}\log _{\text{N}}{\frac {m_{1}}{v}}\right)dm_{1}$ $+\left({\text{H}}_{2}-a_{2}-c_{2}-{\frac {{\text{E}}_{2}}{t}}+c_{2}\log _{\text{N}}t-a_{2}\log _{\text{N}}{\frac {m_{2}}{v}}\right)dm_{2}+{\text{etc.}}=0.$

(3)

If the case is like that of the peroxide of nitrogen, this equation will have two terms, of which the second may refer to the denser component of the gas-mixture. We shall then have $a_{1}=2a_{2},$, and $dm_{1}=-dm_{2}$, and the equation will reduce to the form

$\log {\frac {m_{2}v}{m_{1}^{2}}}=-{\text{A}}-{\text{B}}\log t+{\frac {\text{C}}{t}},$

(4)

where common logarithms have been substituted for Naperian, and ${\text{A, B}}$ and ${\text{C}}$ are constants. If in place of the quantities of the components we introduce the partial pressures, $p_{1},p_{2}$, due to these components and measured in millimeters of mercury, by means of the relations

$m_{1}={\frac {p_{1}v}{a_{1}t}}$$m_{2}={\frac {p_{2}v}{a_{2}t}},$

where $a_{1}$ denotes a constant, we have

${\begin{aligned}\log {\frac {p_{2}}{p_{1}^{2}}}&=-({\text{A}}+\log 2a_{1})-(1+{\text{B}})\log t+{\frac {\text{C}}{t}}\\&=-{\text{A}}'-{\text{B}}'\log t+{\frac {\text{C}}{t}},\\\end{aligned}}$

(5)

where ${\text{A}}'$ and ${\text{B}}'$ are new constants. Now if we denote by $p$ the total pressure of the gas-mixture (in millimeters of mercury), by ${\text{D}}$ its density (relative to air of the same temperature and pressure), and by ${\text{D}}_{1}$ the theoretical density of the rarer component, we shall have

$p:p+p_{2}::{\text{D}}_{1}:{\text{D}}.$

This appears from the consideration that $p+p_{2}$ represents what the pressure would become, if without change of temperature or volume all the matter in the gas-mixture could take the form of the rarer component. Hence,

$p_{2}=p{\frac {{\text{D}}-{\text{D}}_{1}}{{\text{D}}_{1}}},$ $p_{1}=p-p_{2}=p{\frac {2{\text{D}}_{1}-{\text{D}}}{{\text{D}}_{1}}},$

By substitution in (5) we obtain

$\log {\frac {{\text{D}}_{1}({\text{D}}-{\text{D}}_{1})}{(2{\text{D}}_{1}-D)^{2}}}=-{\text{A}}'-{\text{B}}'\log t+{\frac {\text{C}}{t}}+\log p.$

(6)

By this formula, when the values of the constants are determined, we may calculate the density of the gas-mixture from its temperature and pressure. The value of ${\text{D}}_{1}$ may be obtained from the molecular formula of the rarer component. If we compare equations (3), (4) and (5), we see that

${\text{B}}'={\text{B}}+1,$${\text{B}}={\frac {c_{1}-c_{2}}{a_{2}}}\cdot$

Now $c_{1}-c_{2}$ is the difference of the specific heats at constant volume of NO₂ and N₂O₄. The general rule that the specific heat of a gas at constant volume and per unit of weight is independent of its condensation, would make $c_{1}=c_{2},{\text{B}}=0$, and ${\text{B}}'=1$. It may easily be shown, with respect to any of the substances considered in this paper,^[7] that unless the numerical value of ${\text{B}}'$ greatly exceeds unity, the term ${\text{B}}'\log t$ may be neglected without serious error, if its omission is compensated in the values given to ${\text{A}}'$ and ${\text{C}}$. We may therefore cancel this term, and then determine the remaining constants by comparison of the formula with the results of experiment.

In the case of a mixture of Cl₂, PCl₃ and PCl₅, equation (3) will have three terms distinguished by different suffixes. To fix our ideas, we may make these suffixes ₂, ₃ and ₅, referring to Cl₂, PCl₃ and PCl₅ respectively. Since the constants $a_{2},a_{3}$, and $a_{5}$ are inversely proportional to the densities of these gases,

$a_{2}dm_{2}=a_{3}dm_{3}=-a_{5}dm_{5},$

and we may substitute ${\frac {1}{a_{2}}},{\frac {1}{a_{3}}},{\frac {-1}{a_{5}}}$ for $dm_{2},dm_{3}$ and $dm_{5}$ in equation (3), which is thus reduced to the form

$\log {\frac {m_{5}v}{m_{2}m_{3}}}=-{\text{A}}-{\text{B}}\log t+{\frac {\text{C}}{t}}\cdot$

(7)

If we eliminate $m_{2},m_{3},m_{5}$ by means of the partial pressures $p_{2},p_{3},p_{6}$, we obtain

$\log {\frac {p_{5}}{4p_{2}p_{5}}}=-{\text{A}}'-{\text{B}}'\log t+{\frac {\text{C}}{t}},$

(8)

when ${\text{A}}',{\text{B}}'$, like ${\text{A, B}}$ and ${\text{C}}$, are constants. If the chlorine and the protochloride are in such proportions as arise from the decomposition of the perchloride, $p_{2}=p_{3}$ and $4p_{2}p_{3}=(p_{2}+p_{3})^{2}$. In this case, therefore, we have

$\log {\frac {p_{5}}{(p_{2}+p_{3})^{2}}}=-{\text{A}}'-{\text{B}}'\log t+{\frac {\text{C}}{t}}\cdot$

(9)

It will be seen that this equation is of the same form as equation (5), when $p_{5}$ in (9) is regarded as corresponding to $p_{2}$ in (5), and $p_{2}+p_{3}$ in (9), which represents the pressure due to the products of decomposition, is regarded as corresponding to $p_{1}$ in (5), which has the same signification. It follows that equation (5), as well as (6), which is derived from it, may be regarded as applying to the vapor of perchloride of phosphorus, when the values of the constants are properly determined. This result might have been anticipated, but the longer course which we have taken has given us the more general equations, (7) and (8), which will apply to cases in which there is an excess of chlorine or of the protochloride.

If the gas-mixture considered, in addition to the components capable of chemical action, contains a neutral gas, the expressions for the energy and entropy of the gas-mixture should properly each contain a term relating to this neutral gas. This would make it necessary to add $c_{n}m_{n}$ to the coefficient of $dt$ in (1), and ${\frac {c_{n}m_{n}}{t}}$ to the coefficient of $dt$ in (2), the suffix _n being used to mark the quantities relating to the neutral gas. But these quantities would disappear with the elimination of $dt$, and equation (3) and all the subsequent equations would require no modification, if only $p$ and ${\text{D}}$ are estimated (in accordance with usage) with exclusion of the pressure and weight due to the neutral gas. This result, which may be extended to any number of neutral gases, is simply an expression of Dalton's Law.

We now proceed to the comparison of the formulæ, especially of equation (6), with the results of experiment.

Table I.—Peroxide of Nitrogen.
Experiments at Atmospheric Pressure.
Mitscherlich,—R. Müller,—Deville and Troost.

Temperature		Pressure		Density calculated by eq. (10).		Density observed.					Excess of observed density.
							Deville & Troost					Deville & Troost
						M—h.	$\scriptstyle {\overbrace {\quad \quad \quad \quad \quad } }$				M—h.	$\scriptstyle {\overbrace {\quad \quad \quad \quad \quad \quad } }$
						M—r.	I.	II.	III.	M—r.	I.	II.	III.
183.2		(760)		1.592					1.57					-.022
154.0		(760)		1.597					1.58					-.017
151.8		(760)		1.598			1.50					-.10
135.0		(760)		1.607				1.60						-.007
121.8		(760)		1.622			1.64					+.02
121.5		(760)		1.622					1.62					-.002
111.3		(760)		1.641					1.65					+.009
100.25		760		1.677		1.72					-.04
100.1		(760)		1.676					1.68					+.004
100.0		(760)		1.677			1.71					+.03
90.0		(760)		1.728					1.72					-.008
84.4		(760)		1.768			1.83					+.06
80.6		(760)		1.801					1.80					-.001
79		748		1.814		1.84					+.03
77.4		(760)		1.833				1.85					+.02
70.0		(760)		1.920					1.92					.000
70		754.5		1.919		1.95					+.03
68.8		(760)		1.937			1.99					+.05
66.0		(760)		1.976				2.03					+.05
60.2		(760)		2.067					2.06					-.013
55.0		(760)		2.157				2.20					+.04
52		757		2.211		2.26					+.05
49.7		(760)		2.255			2.34					+.09
49.6		(760)		2.256					2.27					+.014
45.1		(760)		2.342				2.40					+.06
39.8		(760)		2.443					2.46					+.017
35.4		(760)		2.524					2.53					+.006
35.2		(760)		2.528			2.66					+.13
34.6		(760)		2.539				2.62					+.08
32		748		2.582		2.65					+.07
28.7		(760)		2.642			2.80					+.16
28		751		2.652		2.70					+.05
27.6		(760)		2.661				2.70					+.04
26.7		(760)		2.676					2.65					+.026

Peroxide of nitrogen.—If we take the constants of the equation for this substances from the paper already cited,^[8] we have

$\log {\frac {15.89({\text{D}}-1.589)}{(3.178-{\text{D}})^{2}}}={\frac {3118.6}{t_{\text{C}}+273}}+\log p-12.451,$

(10)

$t_{\text{C}}$ denoting the temperature on the centigrade scale. The numbers 3.178 and 1.589 represent the theoretical densities of N₂O₄ and NO₂ respectively. The two other constants were determined by the experiments of Deville and Troost.

The results of these and other experiments at atmospheric pressure, all made by Dumas' method, are exhibited in Table I. The first three columns give the temperature (centigrade), the pressure (in millimeters of mercury),^[9] and the density calculated from the temperature and pressure by equation (10). The subsequent columns give the densities observed by different authorities, and the excess of the observed over the calculated densities. In the first column of observed densities, we have one observation by Mitscherlich^[10] (at 100.25°) and five by R. Müller.^[11] The three remaining columns contain each the results of a series of experiments by Deville and Troost.^[12] In each series the experiments were made with increasing temperatures, and with the same vessel, without refilling. It should be observed that the results of the three series are not regarded by their distinguished authors as of equal weight. It is expressly stated that the numbers in the two earlier series, and especially in the first, may be less exact. The last series agrees very closely with the formula. It was from this that the constants of the formula were determined. The experiments of series I and II, and those of Mitscherlich and Muller, give somewhat larger values, with a single exception, as is best seen in the columns which give the excess of the observed density. The differences between the different columns are far too regular to be attributed to the accidental errors of the individual observations, except in the case of the experiment at 151.8°, where some accident has evidently occurred either in the experiment itself or in the reduction of the result. Setting this observation aside, we must look for some constant cause for the other discrepancies between the different series.

We can hardly attribute these discrepancies to difference in the material employed, or to air or other foreign substance imperfectly expelled from the flask. For impurities which increase the density would make the divergence between the different series greatest when the densities are the least, whereas the divergences seem to vanish as the density approaches the limiting value. (A similar objection would apply to the supposition of any error in the determination of the weight of the flask when filled with air alone.) But if we should attribute the divergences to an impurity which diminishes the density (as air), we should be driven to the conclusion that the first series of Deville and Troost gives the most correct results, and that all the best attested numbers at temperatures below 90° are considerably in the wrong. It does not seem possible to account for these discrepancies by any causes which would apply to cases of normal or constant density. They are illustrations of the general fact that when the density varies rapidly with the temperature, determinations of density for the same temperature and pressure by different observers, or different determinations by the same observer, exhibit discordances which are entirely of a different order of magnitude from those which occur with substances of normal or constant densities, or which occur with the same substance at temperatures at which the density approaches a constant value. In some cases such results may be accounted for by carelessness on the part of the observers, not controlled by a comparison of the result with a value already known. But such an explanation is inadequate to explain the general fact, and evidently inadmissible in the present case.

It is probable that these discrepancies are in part attributable to a circumstance which has been noticed by M. Wurtz, in his account of his experiments upon the vapor-density of bromhydrate of amylene, in the following words:—"Le temps pendant lequel la vapeur est maintenue à la temperature où l'on détermine la densité n'est pas sans influence sur les nombres obtenus. C'est ce qui result des deux expériences faites à 225 degrés avec des produits identiques. Dans la première, la vapeur a été portée rapidement à 225 degrés. Dans la seconde elle a été maintenue pendant dix minutes à cette température. On voit que les nombres trouvés pour les densités ont été fort differénts. (The numbers were 4.69 and 3.68 respectively.) Ce résultat ne doit point surprendre si l'on considere que le phénomène de decomposition de la vapeur doit absorber de la chaleur, et que les quantités de chaleur necessaires pour produire et la dilatation et la décomposition ne sauraient être fournies instantanément."^[13]

It is not difficult to form an estimate of the quantities of heat which come into play in such cases. With respect to peroxide of nitrogen, it was estimated in the paper already cited that the heat absorbed in the conversion of a unit of N₂O₄ into NO₂ under constant pressure is represented by 7181 $a_{2}$. (The heat is supposed to be measured in units of mechanical work.) Now the external work done by the conversion of a unit of N₂O₄ into NO₂ under constant pressure is $a_{2}t$. Therefore, the ratio of the heat absorbed to the external work done by the conversion of N₂O₄ into NO₂ is $7181\div t$, or 23 at the temperature of 40° centigrade. Let us next consider how much more rapidly this vapor expands with increase of temperature at constant pressure than air. From the necessary relation

$v={\frac {kmt}{p{\text{D}}}},$

where $m$ denotes the weight of the vapor, and $k$ a constant, we obtain

$\left({\frac {dv}{dt}}\right)_{p}={\frac {v}{t}}-{\frac {v}{\text{D}}}\left({\frac {d{\text{D}}}{dt}}\right)_{p},$

where the suffix _p indicates that the differential coefficients are for constant pressure. The last term of this expression evidently denotes the part of the expansion which is due to the conversion of N₂O₄ into NO₂, and the preceding term the expansion which would take place if there were no such conversion, and which is identical with the expansion of the same volume of air under the same circumstances. The ratio of the two terms is $-{\frac {t}{\text{D}}}\left({\frac {d{\text{D}}}{dt}}\right)_{p}$, the numerical value of which for the temperature of 40° is 2.42, as may be found by differentiating equation (10), or, with less precision, from the numbers in the third column of Table I. Let us now suppose that equal volumes of peroxide of nitrogen and of air at the temperature of 40° and the pressure of one atmosphere receive equal infinitesimal increments of temperature under constant pressure. The heat absorbed by the peroxide of nitrogen on account of the conversion of N₂O₄ into NO₂ is 23 times the external work due to the same cause, and this work is 2.42 times the external work done by the expansion of the air. But the heat absorbed by the air in expanding under constant pressure is well known to be 3.5 times the work done. Therefore the heat absorbed on account of the conversion of N₂O₄ into NO₂ is (23 x 2.42 ÷ 3.5 =) 15.9 times the heat absorbed by the air. To obtain the whole heat absorbed by the vapor we must add that which would be required if no conversion took place. At 40° the vapor of peroxide of nitrogen contains about 54 molecules of N₂O₄ to 46 of NO₂, as may easily be calculated from its density. The specific heat for constant pressure of a mixture in such proportions of gases of such molecular formulae, if no chemical action could take place, would be about twice that of the same volume of air. Adding this to the heat absorbed by the chemical action we obtain the final result,—that at 40° and the pressure of the atmosphere the specific heat of peroxide of nitrogen at constant pressure is about eighteen times that of the same volume of air.^[14]

But the greater amount of heat which is required to bring the vapor to the desired temperature is only one factor in the increased liability to error in cases of this kind. The expansion of peroxide of nitrogen for increase of temperature under constant pressure at 40° is 3.42 times that of air. If, then, in a determination of density, the vapor fails to reach the temperature of the bath, the error due to the difference of the temperature of the vapor and the bath, will be 3.42 times as great as would be caused by the same difference of temperatures in the case of any vapor or gas having a constant density. When we consider that we are liable not only to the same, but to a much greater difference of temperatures in a case like that of peroxide of nitrogen, when the exposure to the heat is of the same duration, it is evident that the common test of the exactness of a process for the determination of vapor-densities, by applying it to a case in which the density is nearly constant, is entirely insufficient.

That the experiments of the III^d series of Deville and Troost give numbers so regular and so much lower than the other experiments is probably to be attributed in part to the length of time of exposure to the heat of the experiment, which was half an hour in this series, for the other series, the time is not given.

Another point should be considered in this connection. During the heating of the vapor in the bath, it is not immaterial whether the flask is open or closed. This will appear, if we compare the values of $\left({\frac {d{\text{D}}}{dt}}\right)_{p}$ and $\left({\frac {d{\text{D}}}{dt}}\right)_{v},$ the differential coefficients of the density with respect to the temperature on the suppositions, respectively, of constant pressure, and of constant volume. For 40°, we have

$\left({\frac {d{\text{D}}}{dt}}\right)_{p}=.0189,$$\left({\frac {d{\text{D}}}{dt}}\right)_{v}=.0163,$

the first number being obtained immediately from equation (10) by differentiation, and the second by differentiation after substitution of ${\frac {kmt}{v{\text{D}}}}$ for $p$. The ratio of these numbers evidently gives the proportion in which the chemical change takes place under the two suppositions. This shows that only about six-sevenths of the heat required for the chemical change can be supplied before opening the flask, and the remainder of this heat as well as that required for expansion must be supplied after the opening. The errors due to this source may evidently be diminished by diminishing the intervals of temperature between the successive experiments in a series of this kind, and also by diminishing the opening made in the flask, which increases the time for which the flask may be left open without danger of the entrance of air. In the III^d series of experiments by Deville and Troost, the intervals of temperature did not exceed ten degrees (except after the density had nearly reached its limiting value), and the necit of the flask was drawn out into a very fine tube.

In Table II, which relates to experiments on the same substance at pressures less than that of the atmosphere, the principal series is that of Naumann,^[15] which commences a few degrees below the lowest temperatures of Deville and Troost, and extends to -6° centigrade, the pressures varying from 301 to 84 millimeters. These experiments were made by the method of Gay-Lussac. The numbers in the column of observed densities have been re-calculated from the more immediate results of the experiments, and are not in all cases identical with those given in Professor Naumann's paper. Every case of difference is marked with brackets. Instead of the numbers [2.66], [2.62], [2.85], [2.94], Naumann's paper has 2.57, 2.65, 2.84, 3.01, respectively. In some cases the temperatures and pressures of two experiments are so nearly the same that it would be allowable to average the results, at least in the column of excess of observed density. In such cases the numbers in this column have been united by a brace. The greatest difference between the observed and calculated densities is .16, which occurs at the least pressure, 84 millimeters. In this experiment the weight of the substance employed is also less than in any other experiment. Under such circumstances, the liability to error is of course greatly increased. The average difference between the observed and calculated densities is .063. Since these differences are almost uniformly positive and increase as the temperature diminishes, it is evident that they might be considerably diminished by slight changes in the constants of equation (10), without seriously impairing the agreement of that equation with the experiments of Deville and Troost. But it has not seemed necessary to re-calculate the formula, which, in its present form, will at least illustrate the degree of accuracy with which densities at low pressures and at temperatures below the boiling point of the liquid may be derived from experiments at atmospheric pressure above the boiling point. Moreover, the excess of observed density may be due in part to a circumstance mentioned by Professor Naumann, that the chemical action between the vapor and the mercury diminished the volume of the vapor, and thus increased the numbers obtained for the density.

Table II.—Peroxide of Nitrogen.
Experiments at less than Atmospheric Pressure.
Playfair and Wanklyn,—Troost,—Naumann.

Temperature		Pressure		Density calculated by eq. (10).		Density observed.				Excess of observed density.
						P. & W.	T.	N.		P. & W.	T.	N.
97.5		(301)		1.631		1.783				+.152
27		35		1.90			1.6				-.30
27		16		1.77			1.59				-.18
24.5		(323)		2.524		2.52				-.004
22.5		136.5		2.34				2.35				+.01
22.5		101		2.26				2.28				+.02
21.5		161		2.41				2.38				-.03	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$
20.8		153.5		2.41				2.46				+.05
20		301		2.59				2.70				+.11
18.5		136		2.43				2.45				+.02
18		279		2.61				2.71				+.10
17.5		172		2.51				2.52				+.01	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$
16.8		172		2.53				2.55				+.02
16.5		224		2.59				[2.66]				+.07	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$
16		228.5		2.61				[2.62]				+.01
14.5		175		2.58				2.63				+.05
11.3		(159)		2.620		2.645				+.025
11		190		2.66				2.76				+.10
10.5		163		2.64				2.73				+.09
4.2		(129)		2.710		2.588				-.122
4		172.5		2.77				2.85				+.08
2.5		145		2.76				[2.85]				+.09
1		138		2.78				2.84				+.06
-1		153		2.83				2.87				+.04
-3		84		2.76				2.92				+.16
-5		123		2.85				2.98				+.13	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$
-6		125.5		2.87				[2.94]				+.07

The same table includes two experiments of Troost,^[16] by Dumas' method, but at the very low pressures of 35^mm and 16^mm. In such experiments we cannot expect a close agreement with the formula, for the same error in the determination of the weight of the vapor, which would make a difference of '01 in the density in experiments at atmospheric pressure, would make a difference of .21 or .47 in the circumstances of these experiments. In fact, the numbers obtained differ considerably from those demanded by the formula.

There remain four experiments by Playfair and Wanklyn^[17] in which Dumas' method was varied by diluting the vapor with nitrogen. The numbers in the column of pressures represent the total pressure diminished by the pressure which the nitrogen alone would have exerted. They are not quite accurate, since the data given in the memoir cited only enable us to determine the ratios of the total and the partial pressures. The numbers here given are obtained by setting the total pressure, which was that of the atmosphere at the time of the experiment, equal to 760^mm. The effect of this inaccuracy upon the calculated densities would be small. Two of these observations agree closely with the formula; and two show considerable divergence, but in opposite directions, and these are the two in which the quantities of peroxide of nitrogen were the smallest. The differences appear to be attributable rather to the difficulty of a precise determination of the quantities of nitrogen and of vapor, than to any effect of the one upon the other.

Special interest attaches to experiments at the same or nearly the same temperature but different pressures. For with experiments at the same temperature, the constants of the formula which are determined by observation are reduced to one, so that the verification of the formula by experiment cannot possibly be regarded as a case of interpolation. It is not necessary that the temperatures should be exactly the same, for it will be conceded that the formula represents the actual function well enough to answer for adjusting slight differences of temperature ; but it is necessary that the range of pressures should be considerable in order that the differences of density should be large in proportion to the probable errors of observation. But the pressures must not be so low that accurate determinations become impossible.

In the experiments of Naumann we see some fair correspondences with the formula in respect to the influence of pressure, especially in the first four experiments of the list, where, if we average the results of the third and fourth experiments, as is evidently allowable, the observed values follow very closely the fluctuations of the calculated, extending from 2.26 to 2.41. In other cases the agreement is less satisfactory. The circumstance that the experiments at the two highest pressures (301 and 279^mm) give results exceeding the calculated values considerably more than any other experiments at adjacent temperatures may seem to indicate that the densities increase with the pressures more rapidly than the formula allows; but the differences are not too large to be ascribed to errors of observation, and the experiment at the lowest pressure (84^mm) also shows a large excess of observed density.

A much more critical test may be found in the comparison of Naumann's experiments with those of Deville and Troost, notwithstanding the interval of about 4° of temperature. The formula requires that a diminution of pressure from 760 to 101 millimeters shall reduce the density from 2.676 at 26.7 to 2.26 at 22.5, notwithstanding the effect of the change of temperature. Experiment gives a reduction of density from 2.65 to 2.28, which is about one-ninth less. This is, it will be observed, a deviation from the formula in the opposite direction from that which the experiments of Naumann alone, or a comparison of the experiments of Troost with those of Deville and Troost, seemed to indicate. The experiment here compared with Naumann's belongs to the III^d series of Deville and Troost. If instead of this experiment we should take an average of the experiments at lowest temperature in the II^d and III^d series, the agreement with the formula with respect to the effect of change of pressure would be almost perfect.

Formic acid.—In Table III, the determinations of Bineau are compared with the densities calculated by the formula

$\log {\frac {1.589({\text{D}}-1.589)}{(3.178-{\text{D}})^{2}}}={\frac {3800}{t_{\text{C}}+273}}+\log p-12.641.$

(11)

The observed densities are taken from the eighteenth volume of the third series of the Annales de Chimie et de Physique (1846), except in three cases, distinguished by parentheses, which are earlier determinations published in the nineteenth volume of the Comptes Rendus (1844). It may be added that the pressure (687) for the experiment at 108° is taken from Erdmann's Journal für praktische Chemie (vol. xl, p. 44), the impression being imperfect in the Annales, in the copies to which the writer has been able to refer, where the figures look much like 637. (The pressure 637 would make the calculated density 2.28.)

In the column which gives the excess of observed densities, the effect of nearness to the state of saturation is often very marked. Such cases are distinguished by an asterisk. The temperature of 99.5° is below the boiling point of formic acid, and the higher pressures employed at this temperature cannot be far from the pressure of saturated vapor. With respect to lower temperatures, we have the statement of Bineau that the pressure of saturated vapor is about 19^mm at 13°, 20.5^mm at 15°, 33.5^mm at 22°, and 53.5^mm at 32°. By interpolation between the logarithms of these pressures (in a single case, by extrapolation), we obtain the following result:—

Temperature, . . . . . . .	10.5	12.5	16	18.5	22
Pressure of sat. vapor, .	16.6	18.5	22	26.2	33.5
Pressure of experiment,	14.69	15.20	15.97	23.53	25.17

Table III.—Formic Acid.
Experiments of Bineau.

Temperature		Pressure		Density calculated by eq. (11).		Density observed.		Excess of observed density.
216.0		690		1.60		1.61		+.01
184.0		750		1.64		1.68		+.04
125.5		687		2.03		2.05		+.02
125.5		645		2.02		2.03		+.01
124.5		670		2.04		2.06		+.02
124.5		640		2.03		2.04		+.01
118.0		655		2.13		(2.14)		(+.01)
118.0		650		2.13		2.13		.00
117.5		688		2.15		2.13		-.02
115.5		649		2.17		2.20		+.03
115.5		640		2.16		2.16		.00
115		655		2.18		(2.13)		(-.05)
111.5		690		2.25		2.22		-.03
111.5		690		2.25		2.25		.00
115		608		2.22		(2.13)		(-.09)
108		[687]		2.30		2.31		+.01
105.0		691		2.35		2.35		.00
105.0		650		2.34		2.33		-.01
105.0		630		2.33		2.32		-.01
101.0		693		2.42		2.44		+.02
101.0		650		2.40		2.41		+.01
99.5		690		2.44		2.52		+.08*
99.5		684		2.44		2.49		+.05
99.5		676		2.44		2.46		+.02
99.5		662		2.43		2.44		+.01
99.5		641		2.42		2.42		.00
99.5		619		2.41		2.41		.00
99.5		602		2.41		2.40		-.01
99.5		557		2.39		2.34		-.05
34.5		28.94		2.82		2.77		-.05
31.5		3.04		2.40		2.60		+.20
30.5		8.83		2.67		2.69		+.02
30.0		18.28		2.81		2.76		-.05
29.0		27.40		2.88		2.83		-.05
24.5		17.39		2.88		2.86		-.02
22.0		25.17		2.95		3.05		+.10*
20.0		16.67		2.93		2.94		+.01
20.0		7.99		2.84		2.85		+.01
20.0		2.72		2.64		2.80		+.16
18.5		23.53		2.98		3.23		+.25*
16.0		15.97		2.97		3.13		+.16*
15.5		2.61		2.72		2.86		+.14
15.0		7.60		2.90		2.93		+.03
12.5		15.20		3.00		3.14		+.14*
11.0		7.26		2.95		3.02		+.07
10.5		14.69		3.01		3.23		+.22*

Whether the large excess of observed density in these cases represents a property of the vapor, or an incipient condensation on the walls of the vessel which contains it, as has been supposed by eminent physicists in similar cases, we need not here discuss.

If we reject these cases of nearly saturated vapor, as well as the three earlier determinations, there remain 25 experiments at pressures somewhat less than one atmosphere in which the maximum difference between the observed and calculated densities is .05, and the average difference .016; nine experiments at pressures ranging from 29^mm to 7^mm, in which the maximum difference is .07 and the average .035; and three experiments at pressures of about 3^mm, in which the average difference is .17. The extraordinary precision of the determinations at low pressures is doubtless due to the large scale on which the experiments were conducted. All the experiments at temperatures below 99° were made with a globe of the capacity of 5${\tfrac {1}{2}}$ liters with a stem of suitable length to hold the barometric column.

The agreement is certainly as good as could be desired, and shows the accuracy of which the method of observation is capable. But in no part of the thermometric scale do we find so great a range of pressures as might be desired, without using pressures too low for accurate results, or observations which are to be rejected for other reasons.

Acetic acid.—For this substance the densities have been calculated by the formula

$\log {\frac {2.073({\text{D}}-2.073)}{(4.146-{\text{D}})^{2}}}={\frac {3520}{t_{\text{C}}+273}}+\log p-11.349,$

(12)

the constants 3520 and 11.349 being derived from the determinations of Cahours and Bineau, which with those of Horstmann and Troost are given in Table IV. The experiments of Cahours and Horstmann were made under atmospheric pressure, those of Horstmann^[18] by the method of Bunsen, those of Cahours presumably by the method of Dumas. The numbers in the first column of the densities observed by Cahours are taken from the twentieth volume (1845) of the Comptes Rendus, except a few cases, distinguished by parentheses, which are taken from the preceding volume (1844).

The numbers in the second column are taken from his Leçons de chimie générale élémentaire, 1856. These numbers seem to be based in part upon new experiments and in part upon a revision of the observations recorded in the Comptes Rendus, the calculations being carried out to another figure of decimals. They are therefore entitled to a greater weight than the numbers of the preceding column.

The agreement of the formula with the numbers given in the Leçons de chimie is very good, the greatest divergences being .080 at 190° and .062 at 180°. But at 190° the table in the Comptes Rendus agrees precisely with the formula, and at 171 (the next experiment) it shows a divergence in the opposite direction. The next divergences in the order of magnitude are -.033, -.036, -.032

Table IV.—Acetic Acid.
Experiments of Cahours,—Horstmann,—Troost.

Temperature		Pressure		Density calculated by eq. (10).		Density observed.				Excess of observed density.
						Cahours $\scriptstyle {\overbrace {\quad \quad \quad \quad \quad } }$		Horstmann.		Cahours $\scriptstyle {\overbrace {\quad \quad \quad \quad \quad } }$		Horstmann.
						C. R.	Leçons.			C. R.	Leçons.
338		(760)		2.077		2.08				.00
336		(760)		2.077			2.082				+.005
327		(760)		2.078		2.08	2.085			.00	+.007
321		(760)		2.079		2.08	2.083			.00	+.004
308		(760)		2.081			2.085				+.004
300		(760)		2.082		2.08				.00
295		(760)		2.084			2.083				-.001
280		(760)		2.089		2.08				-.01
272		(760)		2.093			2.088				-.005
254.6		747.2		2.105				2.135				+.030
252		(760)		2.108			2.090				-.018
250		(760)		2.111		2.08				-.03
240		(760)		2.122			2.090				-.032
233.5		752.8		2.132				2.195				+.063
231		(760)		2.137		(2.12)	2.101			(-.02)	-.036
230		(760)		2.139		2.09				-.05
219		(760)		2.165		2.17	2.132			+.01	-.033
200		(760)		2.239		2.22	2.248			-.02	+.009
190		(760)		2.298		2.30	2.378			.00	+.080
181.7		749.7		2.359				2.419				+.060
180		(760)		2.376			2.438				+.062
171		(760)		2.466		2.42				-.05
170		(760)		2.477			2.480				+.003
165.0		754.1		2.534				2.647				+.113
162		(760)		2.575			2.583				+.008
160.3		751.6		2.594				2.649				+.055
160		(760)		2.601		2.48				-.12
152		(760)		2.716		(2.72)	2.727			(.00)	+.011
150		(760)		2.747		2.75				.00
145		(760)		2.826		(2.75)				(-.08)
140		(760)		2.910		2.90	2.907			-.01	-.003
134.3		748.8		3.001				3.108				+.107
131.3		754.1		3.055				3.070				+.107
130		(760)		3.082		3.12	3.105			+.04	+.023
128.6		752.9		3.103				3.079				-.024
125		(760)		3.168		3.20				+.03
124		(760)		3.185			3.194				+.009
						Bineau		Troost.	Bineau.		Troost.
132	757	3.05	(2.86)			(-.19)
130	59.7	2.31			2.12			-.19
130	30.6	2.21			2.10			-.11
129	633	3.03	(2.88)			(-.15)
36.5	11.32	3.63	3.62			-.01
35.0	11.19	3.65	3.64			-.01
30.0	6.03	3.61	3.60			-.01
28.0	10.03	3.75	3.75			.00
24.0	5.75	3.71	3.70			-.01
22.0	8.64	3.82	3.85			+.03
22	2.70	3.59	3.56			-.03
21.0	4.06	3.70	3.72			+.02
20.5	10.03	3.86	3.95			+.09
20.0	8.55	3.84	3.88			+.04
20.0	5.56	3.77	3.77			.00
19.0	4.00	3.73	3.75			+.02
19	2.60	3.65	3.66			+.01
12.0	5.23	3.88	3.92			+.04
12	2.44	3.77	3.80			+.03
11.5	3.76	3.84	3.88			+.04

at 219°, 231°, 240°, respectively. Here the table in the Comptes Rendus agrees substantially with that of the Leçons, but the experiments of Horstmann show a divergence in the opposite direction. In fact, the three columns of observed densities nowhere agree in the direction of their divergence from the formula.

The somewhat decided differences between the results of Horstmann and those of Cahours may be due in part to the different methods of observation, especially to the entirely different manner of applying the heat and measuring the temperature. But the higher values obtained by Horstmann cannot be accounted for by too short an exposure to the source of heat, for his experiments were made with decreasing temperatures.

The determinations of Bineau are taken from the same sources as those on formic acid, the earlier determinations being distinguished as before by parentheses. One of these (at 132) was made by the method of Dumas, the other by that of Gay-Lussac. The smallness of the observed densities appears due to the presence of water. (An acidimetric test gave 295 parts of acid in 306.) The other experiments were made with the same apparatus which was used with formic acid and show even greater regularity in their results than the experiments with that substance. Only in one case is the influence of proximity to saturation seen, viz., at 20.5 and 10.03^mm, the pressure of saturated vapor at this temperature being about 12.7^mm.^[19] In the remaining fifteen observations of this series, notwithstanding the very low pressures employed (from 2.44 to 11.32), the greatest difference between the observations and the formula is .04, and the average difference .02.

The two observations by Troost^[20] were made by the method of Dumas, but at pressures very low for this method. The results obtained differ considerably from the formula, but not so much as in the case of his experiments at low pressure with peroxide of nitrogen.

Table V contains the experiments of Naumann^[21] on acetic acid. These consist of ten series (distinguished by the letters A, B, C, etc.) of observations by Hoffmann's method.^[22] The temperatures of the observations in the different series are for the most part the same, so that for each temperature we have observations through a wide range of pressures. Within each compartment of the table are given

Table V.—Acetic Acid.
Experiments of Naumann.

			Temperature
			78°		100°		110°		120°		130°		140°		150°		160°		185°
A $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right.}$	Pressure.				393.5		411		432		455		477		498.5				565
	D. calc.				3.39		3.23		3.06		2.90		2.75		2.61				2.28
	D. Obs.				3.44		3.31		3.14		2.97		2.82		2.68				2.36
	Exc. of D. obs.				+.05		+.08		+.08		+.07		+.07		+.07				+.08
B $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right.}$	Pressure.		342.3	359.3	377.5	398.5	417.5	436.5		495
	D. calc.		3.35	3.18	3.02	2.85	2.70	2.57		2.26
	D. Obs.		3.37	3.22	3.06	2.89	2.75	2.63		2.31
	Exc. of D. obs.		+.02	+.04	+.04	+.04	+.05	+.06		+.05
C $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right.}$	Pressure.		258							382
	D. calc.		3.26							2.22
	D. Obs.		3.17							2.25
	Exc. of D. obs.		-.09							+.03
D $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right.}$	Pressure.		232		252	274	287.5	300		335
	D. calc.		3.23		2.87	2.72	2.58	2.46		2.21
	D. Obs.		3.12		2.94	2.68	2.54	2.44		2.23
	Exc. of D. obs.		-.11		+.07	-.04	-.05	-.02		+.02
E $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right.}$	Pressure.	164	186	197	209	221	232	243	253	269
	D. calc.	3.53	3.15	2.97	2.81	2.65	2.52	2.41	2.32	2.18
	D. Obs.	3.41	3.06	2.91	2.75	2.61	2.50	2.40	2.31	2.22
	Exc. of D. obs.	-.12	-.09	-.06	-.06	-.04	-.02	-.01	-.01	+.04
F $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right.}$	Pressure.	149	168			201
	D. calc.	3.50	3.12			2.62
	D. Obs.	3.34	3.01			2.56
	Exc. of D. obs.	-.16	-.11			-.06
G $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right.}$	Pressure.	137	156	166.5	180	188	199	208.2		230
	D. calc.	3.48	3.09	2.92	2.75	2.60	2.47	2.37		2.17
	D. Obs.	3.26	2.98	2.81	2.61	2.50	2.40	2.29		2.14
	Exc. of D. obs.	-.22	-.11	-.11	-.14	-.10	-.07	-.08		-.03
H $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right.}$	Pressure.	113	130	138.5	149	157.5	168.2	175		191.5
	D. calc.	3.42	3.03	2.85	2.69	2.55	2.43	2.33		2.15
	D. Obs.	3.25	2.94	2.78	2.60	2.47	2.32	2.26		2.13
	Exc. of D. obs.	-.17	-.09	-.07	-.09	-.08	-.11	-.07		-.02
J $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right.}$	Pressure.	80	92	98.5	106	112.5	117.3		129.2
	D. calc.	3.32	2.91	2.73	2.58	2.45	2.35		2.21
	D. Obs.	3.06	2.76	2.61	2.46	2.34	2.27		2.11
	Exc. of D. obs.	-.26	-.15	-.12	-.12	-.11	-.08		-.10
K $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right.}$	Pressure.	66	77.7	84	89.5	93	98	103		110.5
	D. calc.	3.26	2.85	2.68	2.53	2.40	2.31	2.24		2.12
	D. Obs.	3.04	2.66	2.49	2.37	2.32	2.24	2.16		2.11
	Exc. of D. obs.	-.22	-.19	-.19	-.16	-.08	-.07	-.08		-.01

in order the pressure of an experiment, the density calculated by equation (12), the observed density, and the excess of observed density, the temperature of the experiment being given at the head of the column. These experiments, taken by themselves, seem to show an effect of pressure upon the density about one third greater than is indicated by the formula. But the divergences (of which the greatest is .26 and the average .085) are not large in view of the fact that the experiments were undertaken rather with the desire of obtaining a great number of observations with moderate labor, than with the intention of attaining the greatest possible accuracy.

The quantity of acid diminishes somewhat regularly from .2084 grams in series A to .0185 in series K. The volume, which was 154^cc in the experiment at 185° in series A, diminishes in the successive series, and in the same series with diminishing temperature, to 69.6^cc in the experiment at 78° in series K. It is worthy of notice that the greatest deviations from the formula occur where the liability to error is most serious with respect to pressure (which was measured without a cathetometer), to volume, and to the quantity of acid.

Far more serious than the absolute amount of these divergences, is the regularity which they exhibit. But it must be remembered that the observations are by no means entirely independent, and many sources of possible error, such as the calibration of the tube and the determination of the quantity of acid, might affect the results with considerable regularity.

Only to a slight degree can the divergences from the formula be accounted for by an insufficient exposure to the temperature of the experiment. The observations, except those at 78°, were made with increasing temperatures, and the greatest divergences from the formula are not in the positive direction. Yet the positive divergences occur where we should most expect to find them, if they were due to this cause, viz., in the series in which the greatest quantities of acid were used, and in cases in which the temperature seems to have been raised at once an unusual number of degrees. (See especially the observation at 120° in series D, 'and in general the observations at 185°, which exhibit if not a positive at least a diminution of negative excess.) In the observations at 78°, which were the last of each series, and therefore followed a fall of temperature from 185°, we find in some cases, especially in series G, H, and J, a negative divergence much greater than in the other determinations of the same series, and which appears to be referable to this circumstance.

In Table VI are exhibited the results of experiments by Playfair and Wanklyn,^[23] in which the vapor of the acid was diluted with hydrogen or, in a single case (the experiment at 95.5°), by air. Columns I and II of the observed densities relate each to a series of observations by the method of Gay-Lussac, column III contains four independent determinations by the method of Dumas. The numbers in the column of pressures are, as in other similar cases, the partial pressures obtained by subtracting from the total pressure (which was never very much less than that of the atmosphere) that which would be exerted by the hydrogen or air alone.

The first observation of the first series gives the density 1.936, which is doubtless too small, since it is much less than the theoretical limit 2.073. Since the greater part of the measurements from which this number was calculated were also used in reducing the other observations of the series, the error probably affects the other observations, and in a somewhat increased degree. This will account only for a part of the difference between the observations and the formula. The remaining part of the differences in this series, and the somewhat smaller differences in the next, may be due to the fact that the experiments of both series were conducted with descending temperatures. Yet the experiments of the third column, which were made by Dumas' method, do not exhibit any preponderance of positive values for the excess of observed density, but rather the opposite.

Table VI.—Acetic Acid.
Experiments of Playfair and Wanklyn.

Temperature		Pressure		Density calculated by eq. (10).		Density observed.				Excess of observed density.
						I.	II.	III.		I.	II.	III.
212.5		322.8		2.124			2.060				-.064
194		326.0		2.168			2.055				-.113
186		254.4		2.173		1.936				-.237
182		319.4		2.213			2.108				-.105
166.5		289.5		2.293			2.350				+.057
163		245.8		2.290		2.017				-.273
132		227.5		2.628		2.292				-.336
130.5		285.7		2.729			2.426				-.303
119		269.0		2.914			2.623				-.291
116.5		211.3		2.876		2.371				-.505
95.5		(123.8)		3.105				2.594				-.511
86.5		(200.4)		3.432				3.172				-.260
79.9		(83.3)		3.297				3.340				+.043
62.5		(46.2)		3.473				3.950				+.477

On the whole, these experiments furnish no decisive indication of any influence of the hydrogen or air upon the vapor. They may be thought to corroborate slightly the tendency observed in the experiments of Naumann and Troost toward lower densities than the formula gives at very low pressures. Yet where the experiments of Naumann show the greatest deficiency in observed density (at 78 and 80^mm), an experiment of Playfair and Wanklyn, at almost precisely the same temperature and pressure, gives a trifling excess of observed density, and at a little lower temperature and pressure, where we should expect from the experiments of Naumann that the deficiency would be still greater, an experiment of Playfair and Wanklyn shows a great excess of density.

By combining the experiments of Cahours, Naumann and Troost, we may obtain observations of density at 130° for a very wide range of pressures. For one atmosphere, we may regard the formula as coinciding with the average of the numbers given by Cahours. For pressures between three-quarters and one-half of an atmosphere the experiments of Naumann show an excess of density; at pressures below half an atmosphere the experiments both of Naumann and of Troost show a deficiency of density as compared with the formula. For an indefinite diminution of pressure, there can be little doubt that the real density, like the value given by the formula, approaches the theoretical value 2.073. The greatest excess in numbers obtained by experiment is .07; the greatest deficiency is .19, which occurs at 59.7^mm; the next in order of magnitude is .11, which occurs more than once. These discrepancies are certainly such as may be accounted for by errors of observation. They do not appear to be greater than we might expect on the hypothesis of the entire correctness of the formula. On the other hand, the agreement is greater than we should expect, if we reject the theory on which the formula was obtained. It is about such as we might expect in a suitable formula of interpolation with three constants, which have been determined by the values of the density for one atmosphere, for half an atmosphere, and for infinitesimal pressures. But we must regard the actual formula, in its application to this single temperature, as having only two constants, of which one is determined so as to make the formula give the theoretical value for infinitesimal pressures, and the other so as to make it agree with the experiments of Cahours at the pressure of one atmosphere.

An entirely different method has been employed by Horstmann^[24] to determine the vapor-density of this substance. A current of dried air is forced through the liquid acid, which is heated to promote evaporation, and the mixture of air and vapor is cooled to any desired temperature, with deposition of the excess of acid, by passing upward through a spiral tube in a suitable bath. The acid is then separated from the air, and the quantity of each determined. It is assumed that the air is exactly saturated with vapor on leaving the coil, and that it has the temperature of the bath. If we know the pressure of saturated vapor for that temperature, and assume the validity of Dalton's law, it is easy to calculate the density of the vapor. For the pressure of the air is found by subtracting the pressure of the vapor from the total pressure (the experiments were so conducted that this was the same as the actual pressure of the atmosphere), and the ratio of the weights of the acid and the air obtained by analysis, divided by the ratio of their pressures, will give the ratio of their densities. The pressures of saturated vapor employed by Horstmann are those given by Landolt,^[25] and differ greatly from the determinations of Regnault, in some cases being nearly twice as great, a difference noticed but not explained by Landolt, who however gives determinations (previously unpublished) of Wüllner, which somewhat exceed his own. (On the other hand, the observations of Bineau substantially agree with those of Regnault.)

If we compare the observations of Horstmann with the values given by equation (12), on the basis of Landolt's pressures, we find a very marked disagreement, as may be seen by the following numbers, which relate to the highest temperatures of Horstmann's experiments, where the disagreement is least:—

Temperature	63.1	62.9	59.9	51.1	49.0	48.7	44.6	41.4
Pressure (Land.)	110.0	109.2	97.0	69.0	63.4	63.0	53.1	46.6
Density cale. eq. (12)	3.67	3.67	3.69	3.75	3.77	3.77	3.79	3.81
Density obs.	3.19	3.11	3.12	3.16	2.89	2.98	2.75	2.62

It will be observed that while the values obtained from equation (12) increase with diminishing temperatures, the values obtained from Horstmann's experiments diminish. This diminution continues as far as the experiments go, until finally at 12° or 15° the densities are only one half as great as those obtained by Bineau, by direct experiment at the same temperatures and at somewhat less pressures, in a series of observations which bear every mark of a very exceptional precision. (Compare Tables VII and IV.) The explanation of this disagreement is doubtless to be found in the values of the pressures employed in the calculations, and it will be interesting to see how the results may be modified by the adoption of different pressures.

In determinations of the pressure of saturated vapors, too great values are so much more easily accounted for than errors in the opposite direction, especially when the pressures are small, that especial interest attaches to the lowest figures which are supported by a competent authority. The experiments of Regnault^[26] were made with three different preparations of acetic acid, of which the second was once, and the third twice, purified by distillation over anhydrous phosphoric acid. Each distillation considerably diminished the pressure of the saturated vapor, the effect of the second distillation being about half that of the first. The numbers obtained with the third preparation are given in the following table with their logarithms, and the differences of the logarithms for one degree of temperature:—

Temperature	Pressure.	log. pressure.	diff. per 1°.
9.71	6.42	.8075
12.12	7.33	.8651	.0239
14.33	8.42	.9253	.0272
14.87	8.59	.9340	.0161
17.23	9.85	.9934	.0252
19.84	11.455	1.0590	.0251
22.37	13.15	1.1189	.0237
25.28	15.36	1.1864	.0232

The uniformity of the numbers in the last column shows the remarkable precision of the determinations. At the same time it is evident that the differences in these numbers are due principally to the errors of observation, so that numbers obtained by interpolation between the logarithms of the observed pressures will be somewhat better (on account of averaging of the errors) than the original determinations.

The values obtained by such an interpolation have been used for the comparison of Horstmann's experiments with the formula (12) which is given in Table VII. Unfortunately this comparison cannot be extended above 25°, which is the limit of Regnault's experiments. The first three columns of the table give the temperatures of Horstmann's experiments, the pressures corresponding to these temperatures according to the determinations of Landolt, and the density deduced from Horstmann's experiments by the use of these pressures. To

Table VII.—Acetic Acid.
Determinations of Vapor-density by Distillation.

Temperature		Pressure acc. to Landolt.			Density observed, Horstmann and Landolt. (10).		Pressure acc. to Regnault.			Density calc. from Regnault's pressures by eq. (12).		Density observed, Horstmann and Regnault.		Excess of observed density.
														I.	II.
25.0			23.5		2.42			15.13		3.86		3.80		-.06
23.8			22.4		2.23			14.19		3.86		3.56			-.30
22.6			21.6		2.29			13.31		3.87		3.76		-.11
21.5			20.4		2.24			12.54		3.87		3.68		-.19
20.4			19.2		2.05			11.81		3.88		3.37			-.51
20.2			19.0		2.28			11.68		3.88		3.75		-.13
20.0			18.9		2.13			11.56		3.88		3.52			-.36
17.4			16.8		2.09			9.95		3.89		3.56		-.33
15.6			15.6		1.98			8.96		3.90		3.48		-.42
15.3			15.3		1.95			8.81		3.90		3.42			-.48
15.3			15.3		1.85			8.81		3.90		3.24			-.66
14.7			15.1		1.78			8.54		3.91		3.18		-.73
12.7			13.7		1.96			7.60		3.91		3.56		-.35
12.4			13.5		1.89			7.46		3.92		3.45		-.47

these columns, which are taken from Horstmann's paper, are added the pressure derived from Regnault's observations by the logarithmic interpolation described above, the density calculated by equation (12) from these pressures and the temperatures of the first column, and the densities obtained by combining Horstmann's experiments with Regnault's pressures. This column is derived from the second, third and fourth, as follows. If $w$ and ${\text{W}}$ denote respectively the weights of vapor and of air which pass through the apparatus in the same time, ${\text{P}}$ the height of the barometer, and $p_{\text{L}}$ the pressure of saturated vapor as determined by Landolt, the densities obtained on the basis of Landolt's pressures, and given in the third column, are evidently represented by ${\frac {w({\text{P}}-p_{\text{L}})}{{\text{W}}p_{\text{L}}}}$. The numbers of the fifth column, which are represented in the same way by ${\frac {w({\text{P}}-p_{\text{R}})}{{\text{W}}p_{\text{R}}}}$, where $p_{\text{R}}$ denotes the pressure as determined by Regnault's experiments, have been calculated by the present writer by multiplying the numbers of the third column by ${\frac {p_{{\text{L}}({\text{P}}-p_{\text{E}})}}{p_{\text{R}}({\text{P}})-p_{\text{L}}}}$.

As the height of the barometer in Horstmann's experimente is not given, it has been necessary to assume ${\text{P}}=760$. The inaccuracy due to this circumstance is evidently trifling. The last two columns of the table, which relate to different series of experiments by Horstmann (a distinction not observed in other parts of the table), give the excess of the densities thus obtained from Horstmann's and Regnault's experiments above the values calculated from equation (12) with the use of Regnault's determinations of pressure.

The densities obtained by experiment are without exception less than those obtained from equation (12). At the highest temperatures, where the liability to error is the least, both in respect to the measurement of the pressure of saturated vapor and in respect to the analysis of the product of distillation, the results of experiment are most uniform, and most nearly approach the numbers required by the formula. At the lowest temperatures, the greatest observed density is about one-eleventh less than that required by the formula, the difference being about the same as between the highest and lowest observed values for the same temperature.

Since each successive purification of the substance employed by Regnault diminished the pressure of its vapor, it is not improbable that the pressures might have been still farther diminished by farther purification of the substance. The pressures which we have used are therefore liable to the suspicion of being too high, and it is quite possible that more accurate values of the pressure would still farther reduce the deficiency of observed density.

Perchloride of phosphorus.—For this substance, we have at atmospheric pressure a single determination of vapor-density by Mitscherlich,^[27] and a series of determinations by Cahours;^[28] at lower pressures we have determinations by Wurtz^[29] and by Troost and Hautefeuille.^[30] In the experiments of Wurtz the pressure was reduced by mixing the vapor with air. In Table VIII all these determinations are compared with the formula

$\log {\frac {3.6({\text{D}}-3.6)}{(7.2-{\text{D}})^{2}}}={\frac {5441}{t_{\text{C}}+273}}+\log p-14.353.$

(13)

The differences between the calculated and observed values are often large, in six cases exceeding .30; but they exhibit in general that irregularity which is characteristic of errors of observation. We should expect large errors in the observed densities, on account of the difficulty of obtaining the substance in a state of purity, and because the large value of the density renders it very sensitive to the effect of impurities which diminish the density,—also because the specific heat of the vapor is great, as shown by the numerator of the fraction in the second member of (13),^[31] and because the density varies very rapidly with the temperature as seen by the numbers in the third column of Table VIII.

Table VIII.—Perchloride of Phosphorus.
Experiments of Mitscherlich, Cahours, Wurtz, and Troost
and Hautefeuille.

Temperature		Pressure		Density calculated by eq. (13).		Density observed.			Excess of observed density.
						Mitscherlich.	Cahours.		Mitscherlich.	Cahours.
336		(760)		3.610			3.656			+.046
327		754		3.614			3.656			+.042
300		765		3.637			3.654			+.017
289		(760)		3.656			3.69			+.034
288		763		3.659			3.67			+.011
274		755		3.701			3.84			+.139
250		751		3.862			3.991			+.129
230		746		4.159			4.302			+.142
222		753		4.344		4.85			+.506
208		(760)		4.752			4.73			-.021
200		758		5.018			4.851			-.167
190		758		5.368			4.987			-.381
182		757		5.646			5.078			-.568
						Wurtz.	T. & H.	Wurtz.	T. & H.
178.5	227.2	5.053		5.150		+.097
167.6	221.8	5.456		5.235		+.012
154.7	221.8	5.926		5.619		-.041
150.1	225	6.086		5.886		-.200
148.6	244	6.169		5.964		-.205
145	391	6.45	6.55		+.10
145	311	6.37	6.70		+.33
145	307	6.36	6.33		-.03
145	307	6.36	6.33		-.03
144.7	247	6.287		6.14		-.147
137	281	6.53	6.48		-.05
137	269	6.51	6.54		+.03
137	243	6.48	6.46		-.02
137	234	6.47	6.42		-.05
137	148	6.31	6.47		+.16
129	191	6.59	6.18		-.41
129	170	6.56	6.63		+.07
129	165	6.55	6.31		-.24

But at the two lowest temperatures of Cahours' experiments, the differences of the observed and calculated densities (.381 and .568) are not only great, but exhibit, in connection with the adjacent numbers, a regularity which suggests a very different law from that of the formula. In fact, the densities obtained by Cahours at atmospheric pressure and those obtained by Troost and Hautefeuille at pressures a little less than one-third of an atmosphere seem to form a continuous series, notwithstanding the abrupt change of pressure. Yet it is difficult to admit that the density is independent of the pressure. So radical a difference between the behavior of this substance and that of the others which we have been considering requires unequivocal evidence. Now it is worthy of notice that the experiment at 182°, in which the greatest discrepancy is seen, is not given in the first record of the experiments, which was in the Comptes Rendus in 1845. It is given in the Annales de Chimie et de Physique in 1847, where it is called the first experiment. (The experiment at 336° is also omitted in the Comptes Rendus and that at 208° in the Annales, otherwise the lists are the same.) If it was the first experiment in point of time, which is apparently the meaning, it was made before the publication in the Comptes Rendus, and we can only account for its omission by supposing that it was a preliminary experiment, in which its distinguished author did not feel sufficient confidence to include it at first with his other determinations, although he afterwards concluded to insert it. If we reject this observation as doubtful, the disagreement between the formula and observation appears to be within the limits of possible error, but additional experiments will be necessary to confirm the formula.^[32]

Experiments have also been made by M. Wurtz in which the vapor of the perchloride of phosphorus was diluted with that of the protochloride.^[33] These experiments may be used to test equation (8), which, when the values of its constants are determined by equation (13), reduces to the form

$\log {\frac {p_{5}}{p_{2}p_{3}}}={\frac {5441}{t_{\text{C}}+273}}-13.751$

(14)

where $p_{5},p_{2}$, and $p_{3}$ denote the partial pressures due respectively to the PCl₅, the Cl₂, and the PCl₃, existing as such in the gas-mixture. Since these quantities cannot be the subjects of immediate observation, a farther transformation of the equation will be convenient. Let ${\text{M}}_{3},{\text{M}}_{2}$ denote the quantities of the protochloride and of chlorine of which the mixture may be formed, and ${\text{P}}_{3},{\text{P}}_{2}$ the pressure which would belong to each of these if existing by itself with the same volume and temperature. These quantities will be connected by the equations

${\text{P}}_{2}={\frac {kt{\text{M}}_{2}}{2.22v}},$${\text{P}}_{3}={\frac {kt{\text{M}}_{3}}{4.98v}},$

(15)

where $k$ denotes the same constant as on page 381. From the evident relations

${\text{P}}_{2}=p_{2}+p_{5},$${\text{P}}_{3}=p_{3}+p_{5},$$p=p_{2}+p_{3}+p_{5},$

we obtain

$p_{5}={\text{P}}_{2}+{\text{P}}_{3}-p,$$p_{2}=p-{\text{P}}_{3},$$p_{3}=p-{\text{P}}_{2},$

and by substitution of these values in equation (14),

$\log {\frac {{\text{P}}_{2}+{\text{P}}_{3}-p}{(p-{\text{P}}_{2})(p-{\text{P}}_{3})}}={\frac {5441}{t_{\text{C}}+273}}-13.751.$

(16)

In view of the relations (15), this may be regarded as an equation between the pressure, the temperature, the volume, and the quantities of protochloride of phosphorus and chlorine into which the gas-mixture is resolvable.

Table IX.—Perchloride and Protochloride of Phosphorus.
Experiments on the Mixed Vapors by Wurtz.

No. of exp.		$t_{\text{C}}$		$p$ (obs.)		$\pi$		$\delta$		${\text{P}}_{2}$		${\text{P}}_{3}$		$p$ calculated by eq. (16).		Excess of obs. value of $p$.
XII		173.29		756.1		423		6.68		392.4		725.5		760.7		-4.6
X		165.4		748.4		413		6.80		390.1		725.5		747.9		+.5
VII		176.24		751.0		411		6.88		392.7		732.7		773.1		-22.1
VIII		169.35		724.1		394		7.16		391.8		721.9		750.5		-26.4
V		175.26		743.3		343		7.03		334.9		735.2		764.4		-21.1
II		164.9		758.5		338		7.38		346.4		766.9		782.9		-24.4
XI		175.75		760.0		318		7.00		309.2		751.2		776.8		-16.8
IV		175.26		756.3		271		7.06		265.7		751.0		770.9		-14.6
IX		160.47		753.5		214		7.44		221.1		760.6		766.8		-13.3
I		165.4		760.0		194		7.25		195.3		761.3		768.5		-8.5
VI		170.34		751.2		174		8.30		200.6		777.8		787.6		-36.4
III		174.28		742.7		168		7.74		180.6		755.3		766.5		-23.8

It is in this form that we shall apply the equation to the experiments of M. Wurtz, the results of which are exhibited in Table IX. The first column gives the number distinguishing each experiment in the original memoir; the second, the temperature; the third, the observed pressure ($p$) of the mixture of PCl₅, PCl₃, and Cl₂, which is the barometric pressure corrected for the small quantity of air remaining in the flask; the fourth, the pressure $\pi$ due to the possible perchloride, found by subtracting the pressure due to the excess of protochloride (this pressure is calculated from the theoretical density of the protochloride) from the total pressure; the fifth, the density $\delta$ of the possible perchloride calculated from its pressure $\pi$ with the temperature and volume. The numbers of these five columns are taken from the memoir cited, except that the correction of the barometric pressures has been applied by the present writer in accordance with the data furnished in that memoir. The two next columns contain the values of ${\text{P}}_{2}$ and ${\text{P}}_{3}$. These would naturally be calculated from ${\text{M}}_{2}$ and ${\text{M}}_{3}$ by equations (15). But since the values of ${\text{M}}_{2}$ and ${\text{M}}_{3}$ have not been given explicitly, those of ${\text{P}}_{2}$ and ${\text{P}}_{3}$ have been calculated from the recorded values of $\pi$ and $\delta$. Since the weight of the possible perchloride is ${\frac {7.2}{2.22}}{\text{M}}_{2}$, we have

$\delta ={\frac {7.2{\text{M}}_{2}kt}{2.22v\pi }}={\frac {7.2}{\pi }}{\text{P}}_{3}\cdot$

Moreover,

$p-\pi ={\text{P}}_{3}-{\text{P}}_{2},$

since both members of the equation express the pressure due to the excess of the protochloride. The values of ${\text{P}}_{2}$ and ${\text{P}}_{3}$ were obtained by these equations.

The eighth column of the table gives the values of $p$ calculated from the preceding values of $t_{\text{C}},{\text{P}}_{2}$, and ${\text{P}}_{3}$, by equation (16); and the last column, the difference of the observed and calculated values of $p$. The average difference is 18^mm, or a little more than two per cent., the observed pressure being almost uniformly less than the calculated value. This deficiency of pressure is doubtless to be accounted for by a fact which MM. Troost and Hautefeuille have noticed in this connection. The protochloride of phosphorus deviates quite appreciably from the laws of Mariotte, Gay-Lussac, and Avogadro, the product of the volume and pressure of a given quantity of vapor at 180° and the pressure of one atmosphere being 1.548 per cent, less than at the same temperature and the pressure of one-half an atmosphere.^[34] Now we may assume as a general rule that when the product of volume and pressure of a gas is slightly less than the theoretical number (calculated by the laws of Mariotte, Gay-Lussac, and Avogadro) the difference for any same temperature is nearly proportional to the pressure.^[35] It is therefore probable that between 160° and 180°, at pressures of about one atmosphere, the product of volume and pressure for protochloride of phosphorus is somewhat more than three per cent, less than the theoretical number. The experiments of Wurtz, as exhibited in Table IX, show that the pressure, and therefore the product of volume and pressure (we may evidently give the volume any constant value as unity), in a mixture consisting principally of the protochloride is on the average a little more than two per cent, less than is demanded by theory, the differences being greater when the proportion of the protochloride is greater. The deviation from the calculated values is therefore in the same direction and about such in quantity as we should expect.^[36]

M. Wurtz has remarked that the average value of $\delta$ (the density of the possible perchloride) is nearly identical with the theoretical density of the perchloride, and appears inclined to attribute the variations from this value to the errors of experiment. Yet it appears very distinctly in Table IX, in which the experiments are arranged according to the value of $\pi$ (the pressure due to the possible perchloride), that $\delta$ increases as $\pi$ diminishes. The experiments of MM. Troost and Hautefeuille show that the coincidence remarked by M. Wurtz is due to the fact that on the average in these experiments the deficiency of the density of the possible perchloride (compared with the theoretical value) is counterbalanced by the excess of density of the protochloride. When $\pi >400$, the effect of the deficiency in the density of the possible perchloride distinctly preponderates; when $\pi <250$, the effect of the excess of density in the protochloride distinctly preponderates. But the magnitude of the differences concerned is not such as to invalidate the general conclusion established by the experiments of M. Wurtz, that the dissociation of the perchloride may be prevented (at least approximately) by mixing it with a large quantity of the protochloride.

Table for facilitating calculation.—The numerical solution of equations (10), (11), (12) and (13) for given values of $t$ and $p$ may be facilitated by the use of a table. If we set

$\Delta ={\frac {\text{D}}{{\text{D}}_{1}}},$

(17)

${\text{L}}=\log {\frac {1000{\text{D}}_{1}({\text{D}}-{\text{D}}_{1})}{(2{\text{D}}_{1}-{\text{D}})^{2}}}=\log {\frac {1000(\Delta -1)}{(2-\Delta )^{2}}}\cdot$

(18)

we have for peroxide of nitrogen,

${\text{L}}={\frac {3118.6}{t_{\text{C}}+273}}+\log p-9.541;$

(19)

for formic acid,

${\text{L}}={\frac {3800}{t_{\text{C}}+273}}+\log p-9.641;$

(20)

for acetic acid,

${\text{L}}={\frac {3520}{t_{\text{C}}+273}}+\log p-8.349;$

(21)

and for perchloride of phosphorus,

${\text{L}}={\frac {5441}{t_{\text{C}}+273}}+\log p-11.353.$

(22)

By these equations the values of ${\text{L}}$ are easily calculated. The values of ${\text{A}}$ may then be obtained by inspection (with interpolation when necessary) of the following table. From $\Delta$ the value of ${\text{D}}$ may be obtained by multiplying by ${\text{D}}_{1}$, viz., by 1.589 for peroxide of nitrogen or formic acid, by 2.073 for acetic acid, and by 3.6 for perchloride of phosphorus.^[37]

Table X.
For the solution of the equation: $\log {\frac {1000(\Delta -1)}{(2-\Delta )^{2}}}={\text{L}}.$

${\text{L}}$		$\Delta$		Diff.		${\text{L}}$			$\Delta$		Diff.		${\text{L}}$			$\Delta$		Diff.
.7		1.005					3.0		1.382					5.3		1.932
.8		1.006		1			3.1		1.421		39			5.4		1.939		7
.9		1.008		2			3.2		1.461		40			5.5		1.945		6
1.0		1.010		2			3.3		1.500		31			5.6		1.951		6
1.1		1.012		2			3.4		1.537		37			5.7		1.956		5
1.2		1.015		3			3.5		1.574		37			5.8		1.961		5
1.3		1.019		4			3.6		1.609		35			5.9		1.965		4
1.4		1.024		5			3.7		1.642		33			6.0		1.969		4
1.5		1.030		6			3.8		1.673		31			6.1		1.972		3
1.6		1.037		7			3.9		1.703		30			6.2		1.975		3
1.7		1.046		9			4.0		1.730		27			6.3		1.978		3
1.8		1.056		10			4.1		1.755		25			6.4		1.980		2
1.9		1.069		13			4.2		1.778		23			6.5		1.982		2
2.0		1.084		15			4.3		1.800		22			6.6		1.984		2
2.1		1.102		18			4.4		1.819		19			6.7		1.986		2
2.2		1.122		20			4.5		1.837		18			6.8		1.987		1
2.3		1.146		24			4.6		1.854		17			6.9		1.989		2
2.4		1.172		26			4.7		1.868		14			7.0		1.990		1
2.5		1.202		30			4.8		1.882		14			7.2		1.992
2.6		1.234		32			4.9		1.894		12			7.4		1.994
2.7		1.268		34			5.0		1.905		11			7.6		1.995
2.8		1.305		37			5.1		1.915		10			7.8		1.996
2.9		1.343		38			5.2		1.924		9			8.0		1.997
3.0		1.382		39			5.3		1.932		8			9.0		1.999

The constants of these equations are of course subject to correction by future experiments, which must also decide the more general question in what cases, and within what limits, and with what degree of approximation, the actual relations can be expressed by equations of such form. In the case of perchloride of phosphorus especially, the formula proposed requires confirmation.

1. The language of this paper will be conformed to this usage.
2. Salet, "Sur la coloration du peroxyde d'azote," Comptes Rendus, t. lxvii, p. 488.
3. H. Sainte-Claire Deville, "Sur les densites de vapeur," Comptes Rendus, t. lxii, p. 1157.
4. Wanklyn and Robinson, "On Diffusion of Vapours: a means of distinguishing between apparent and real Vapour-densities of Chemical Compounds," Proc. Roy. Soc., vol. xii, p. 507.
5. "On the Equilibrium of Heterogeneous Substances," this volume, page 55. The equations referred to are (313), (317), (319), and (320), on pages 171 and 172. The applicability of these equations to such cases as we are now considering is discussed under the heading "Gas-mixtures with Convertible Components," page 172.
6. For certain a priori considerations which give a degree of probability to these assumptions, the reader is referred to the paper already cited.
7. For the case of peroxide of nitrogen, see pp. 180, 181 in the paper cited above.
8. See equation (336) on page 177,—also the following equations in which the density is given in terms of the temperature and pressure. In comparing these equations, it must be observed that in (336) the pressures are measured in atmospheres, but in this paper in milimeters of mercury.
9. 760^mm has been assumed as the pressure of the atmosphere in all cases in which the precise pressure is not recorded in the published account of the experiments. The figures inserted in the columns of pressures are in such cases enclosed in parentheses. The same course has been followed in the subsequent tables. With respect to the principal series of observations by Deville and Troost (series III), it is stated that the barometer varied between 747 and 764 millimeters. A difference of 13 millimeters in the pressure would in no case cause a difference of .005 in the calculated densities. In this series, therefore, the errors due to this circumstance are not very serious.
10. Pogg. Ann., vol. xxix (1833), p. 220.
11. Lieb. Ann., vol. cxxii (1862), p. 15.
12. Comptes Rendus, vol. lxiv (1867), p. 237.
13. Comptes Rendus, t. lx, p. 730.
14. Similar calculations from less precise data for the bromhydrate of amylene at 225° seem to indicate a specific heat as much as forty times as great as that of the same volume of air.
15. Berichte der deutschen chemischen Gesellschaft, Jahrgang xi (1878), S. 2045.
16. Comptes Rendus, t. lxxxvi (1878), p. 1395.
17. Trans. Roy. Soc. Edinb., vol. xxii (1861), p. 463.
18. Lieb. Ann., suppl. vi, p. 65.
19. This number is obtained from data given by Bineau by the same kind of interpolation which was used for formic acid.
20. Comptes Rendus, vol. lxxxvi (1878), p. 1395.
21. Lieb. Ann., vol. civ, p. 325.
22. This is a modification of the method of Gay-Lussac, in which the heat is supplied by a vapor bath.
23. Trans. Roy. Soc. Edinb., vol. xxii, p. 455.
24. Berichte der deutschen chemischen Gesellschaft, Jahrg. iii (1870), S. 78; and Jahrg. xi (1878), S. 1287.
25. Lieb. Ann., suppl. vi (1868), p. 157.
26. Mém. Acad. Sciences, vol. xxvi, p. 758. The experiments date from 1844.
27. Pogg. Ann., vol. xxix (1833), p. 221.
28. Comptes Rendus, vol. xxi (1845), p. 625; and Annales de Chimie et de Physique, ser. 3, vol. xx (1847), p. 369.
29. Comptes Rendus, vol. lxxvi (1873), p. 601.
30. Ibid., vol. lxxxiii (1876), p. 977.
31. Compare Equilib. Het. Subs., this volume p. 180, and supra pp. 380–382.
32. Additional experiments on the density of this vapor have been made by M. Cahours, concerning which he says in 1866: "Les déterminations qui je viens d'effectuer à 170 et 172 degrés (ce corps bout vers 160 à 165 degrés) m'ont donné des nombres qui, bien que notablement plus forts que ceux que j'ai obtenus antérieurement à 182 et 185 degrés, sont encore bien éloignés de celui que correspond à volumes," Comptes Rendus, t. 63, p. 16. So far as the present writer has been able to ascertain, these determinations have not been published. The formula gives 6.025 for 170° and 5.973 for 172°, at atmospheric pressure. The number corresponding to four volumes is 7.20.
33. Comptes Rendus, vol. lxxvi (1873), p. 601.
34. Troost and Hautefeuille, Comptes Rendus, vol. lxxxiii (1876), p. 334.
35. Andrews, "On the Gaseous State of Matter," Phil. Trans., vol. clxvi (1876), p. 447.
36. The deviation of the protochloride of phosphorus from the laws of ideal gases shows the impossibility of any very close agreement between such equations as have been deduced in this paper and the results of experiment in the case of gas-mixtures in which this substance is one of the components. With respect to the question whether future experiments on the vapor of the perchloride (alone, or with an excess of chlorine or of the protochloride) will reduce the disagreement between the calculated and observed values to such magnitudes as occur in the case of the protochloride alone, it would be rash to attempt to anticipate the result of experiment.
37. The value of $\Delta$ diminished by unity expresses the ratio of the number of the molecules of the more complex type to the whole number of molecules. Thus, if $\Delta =1.20$, in the case of peroxide of nitrogen there are 20 molecules of the type N₂O₄ to 80 of the type NO₂, or in the case of perchloride of phosphorus there are 20 molecules of the type PCl₅ to 40 of the type PC_l3 and 40 of the type Cl₂. A consideration of the varying values of $\Delta$ is therefore more instructive than that of the values of ${\text{D}}$, and it would in some respects be better to make the comparison of theory and experiment with respect to the values of $\Delta$.