S/s was 1.40 for so many gases, instead of being less than 4⁄3. Somewhat later L. Boltzmann suggested that a diatomic molecule regarded as a rigid dumb-bell or figure of rotation, might have only five effective degrees of freedom, since the energy of rotation about the axis of symmetry could not be altered by collisions between the molecules. The theoretical value of the ratio S/s in this case would be the required 7⁄5. For a rigid molecule on this theory the smallest value possible would be 4⁄3. Since much smaller values are found for more complex molecules, we may suppose that, in these cases, the energy of rotation of a polyatomic molecule may be greater than its energy of translation, or else that heat is expended in splitting up molecular aggregates, and increasing energy of vibration. A hypothesis doubtfully attributed to Maxwell is that each additional atom in the molecule is equivalent to two extra degrees of freedom. From an m-atomic molecule we should then have S/s = 1 + 2/(2m + 1). This gives a series of ratios 5⁄3, 7⁄5, 9⁄7, 11⁄9, &c., for 1, 2, 3, 4, &c., atoms in the molecule, values which fall within the limits of experimental error in many cases. It is not at all clear, however, that energy of vibration should bear a constant ratio to that of translation, although this would probably be the case for rotation. For the simpler gases, which are highly diathermanous and radiate badly even at high temperature, the energy of vibration is probably very small, except under the special conditions which produce luminosity in flames and electric discharges. For such gases, assuming a constant ratio of rotation to translation, the specific heat at low pressures would be very nearly constant. For more complex molecules the radiative and absorptive powers are known to be much greater. The energy of vibration may be appreciable at ordinary temperatures, and would probably increase more rapidly than that of translation with rise of temperature, especially near a point of dissociation. This would account for an increase of S, and a diminution of the ratio S/s, with rise of temperature which apparently occurs in many vapours. The experimental evidence, however, is somewhat conflicting, and further investigations are very desirable on the variation of specific heat with temperature. Given the specific heat as a function of the temperature, its variation with pressure may be determined from the characteristic equation of the gas. The direct methods of measuring the ratio S/s, by the velocity of sound and by adiabatic expansion, are sufficiently described in many text-books.
§ 19. Atomic and Molecular Heats.—The ideal atomic heat is the thermal capacity of a gramme-atom in the ideal state of monatomic gas at constant volume. This would be nearly three calories. For a diatomic gas, the molecular heat would be nearly five calories, or the atomic heat of a gas in the diatomic state would be 2.5. Estimated at constant pressure the atomic heat would be 3.5. Some authors adopt 2.5 and some 3.5 for the ideal atomic heat. The atomic heat of a metal in the solid state is in most cases larger than six calories at ordinary temperatures. Considering the wide variations in the physical condition and melting points, the comparatively close agreement of the atomic heats of the metals at ordinary temperatures, known as Dulong and Petit's Law, is very remarkable. The specific heats as a rule increase with rise of temperature, in some cases, e.g. iron and nickel, very rapidly. According to W. A. Tilden (Phil. Trans., 1900), the atomic heats of pure nickel and cobalt, as determined from experiments at the boiling-points of O2, and CO2, diminish so rapidly at temperatures below 0° C. as to suggest that they would reach the value 2.42 at the absolute zero. This is the value of the minimum of atomic heat calculated by Perry from diatomic hydrogen, but the observations themselves might be equally well represented by taking the imaginary limit 3, since the quantity actually observed is the mean specific heat between 0° and −182.5° C. Subsequent experiments on other metals at low temperatures did not indicate a similar diminution of specific heat, so that it may be doubted whether the atomic heats really approach the ideal value at the absolute zero. No doubt there must be approximate relations between the atomic and molecular heats of similar elements and compounds, but considering the great variations of specific heat with temperature and physical state, in alloys, mixtures or solutions, and in allotropic or other modifications, it would be idle to expect that the specific heat of a compound could be accurately deduced by any simple additive process from that of its constituents.
Authorities.—Joule's Scientific Papers (London, 1890); Ames and Griffiths, Reports to the International Congress (Paris, 1900), “On the Mechanical Equivalent of Heat,” and “On the Specific Heat of Water”; Griffiths, Thermal Measurement of Energy (Cambridge, 1901); Callendar and Barnes, Phil. Trans. A, 1901, “On the Variation of the Specific Heat of Water”; for combustion methods, see article Thermochemistry, and treatises by Thomsen, Pattison-Muir and Berthelot; see also articles Thermodynamics and Vaporization.
(H. L. C.)
CALOVIUS, ABRAHAM (1612–1686), German Lutheran divine, was born at Mohrungen in east Prussia, on the 16th of April 1612. After studying at Königsberg, in 1650 he was appointed professor of theology at Wittenberg, where he afterwards became general superintendent and primarius. He died on the 25th of February 1686. Calovius was the most noteworthy of the champions of Lutheran orthodoxy in the 17th century. He strongly opposed the Catholics, Calvinists and Socinians, attacked in particular the reconciliation policy or “syncretism” of Georg Calixtus (cf. the Consensus repetitus fidei vere lutheranae, 1665), and as a writer of polemics he had few equals. His chief dogmatic work, Systema locorum theologicorum (12 vols. 1655–1677), represents the climax of Lutheran scholasticism. In his Biblia Illustrata (4 vols.), written from the point of view of a very strict belief in inspiration, his object is to refute the statements made by Hugo Grotius in his Commentaries. His Historia Syncretistica (1682) was suppressed.
CALPURNIUS, TITUS, Roman bucolic poet, surnamed Siculus from his birthplace or from his imitation of the style of the Sicilian Theocritus, most probably flourished during the reign of Nero. Eleven eclogues have been handed down to us under his name, of which the last four, from metrical considerations and express MS. testimony, are now generally attributed to Nemesianus (q.v.), who lived in the time of the emperor Carus and his sons (latter half of the 3rd century A.D.). Hardly anything is known of the life of Calpurnius; we gather from the poems themselves (in which he is obviously represented by “Corydon”) that he was in poor circumstances and was on the point of emigrating to Spain, when “Meliboeus” came to his aid. Through his influence Calpurnius apparently secured a post at Rome. The time at which Calpurnius lived has been much discussed, but all the indications seem to point to the time of Nero. The emperor is described as a handsome youth, like Mars and Apollo, whose accession marks the beginning of a new golden age, prognosticated by the appearance of a comet, doubtless the same that appeared some time before the death of Claudius; he exhibits splendid games in the amphitheatre (probably the wooden amphitheatre erected by Nero in 57); and in the words
maternis causam qui vicit Iulis (i. 45),
there is a reference to the speech delivered in Greek by Nero on behalf of the Ilienses (Suetonius, Nero, 7; Tacitus, Annals, xii. 58), from whom the Julii derived their family. Meliboeus, the poet's patron, has been variously identified with Columella, Seneca the philosopher, and C. Calpurnius Piso. Although the sphere of Meliboeus's literary activity (as indicated in iv. 53) suits none of these, what is known of Calpurnius Piso fits in well with what is said of Meliboeus by the poet, who speaks of his generosity, his intimacy with the emperor, and his interest in tragic poetry. His claim is further supported by the poem De Laude Pisonis (ed. C. F. Weber, 1859) which has come down to us without the name of the author, but which there is considerable reason for attributing to Calpurnius. The poem exhibits a striking similarity with the eclogues in metre, language and subject-matter. The author of the Laus is young, of respectable family and desirous of gaining the favour of Piso as his Maecenas. Further, the similarity between the two names can hardly be accidental; it is suggested that the poet may have been adopted by the courtier, or that he was the son of a freedman of Piso. The attitude of the author of the Laus towards the subject of the panegyric seems to show less intimacy than the relations between Corydon and Meliboeus in the eclogues, and there is internal evidence that the Laus was written during the reign of Claudius (Teuffel-Schwabe, Hist, of Rom. Lit. § 306, 6).
Mention may here be made of the fragments of two short hexameter poems in an Einsiedeln MS., obviously belonging to the time of Nero, which if not written by Calpurnius, were imitated from him.
- Iulis for in ulnis according to the best MS. tradition.
- According to Dr R. Garnett (and Mr Greswell, as stated in Conington's Virgil, i. p. 123, note) the emperor referred to is the younger Gordian (A.D. 238). His arguments in favour of this will be found in the article on Calpurnius by him in the 9th edition of the Encyclopaedia Britannica and in the Journal of Philology, xvi., 1888; see in answer J. P. Postgate, “The Comet of Calpurnius Siculus” in Classical Review, June 1902. Dean Merivale (Hist. of the Romans under the Empire, ch. 60) and Pompei, “Intorno al Tempo del Poeta Calpurnio” in Atti del Istituto Veneto, v. 6 (1880), identify the amphitheatre with the Colosseum (Flavian amphitheatre) and assign Calpurnius to the reign of Domitian.
- It has been variously ascribed to Virgil, Ovid, Lucan, Statius and Saleius Bassus.