358
M. CLAPEYRON ON THE MOTIVE POWER OF HEAT.
but the temperature remaining constant during the variation of the volume, we have
|
![{\displaystyle vdp+pdv=0,\;\mathrm {whence} \;dp=-{\frac {p}{v}}dv,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/097472b71c3b438a32b4bb8ead650255cdd882ee) | |
and consequently
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![{\displaystyle dQ=\left({\frac {dQ}{dv}}-{\frac {p}{v}}{\frac {dQ}{dp}}\right)dv.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/635b4aef12a6a4a07705ffe078f70d1a4f3052ed) | |
If we divide the effect produced by this value of
, we shall have
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![{\displaystyle {\frac {R\;dt}{v{\frac {dQ}{dv}}-p{\frac {dQ}{dp}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44c7eba3e72cc85c16c5905ca4e4be2d025d3179) | |
for the expression of the maximum effect which can be developed by the passage of a quantity of heat equal to unity, from a body maintained at the temperature
to a body maintained at the temperature
.
We have shown that this quantity of action developed is independent of the agent which has served to transmit the heat; it is therefore the same for all the gases, and is equally independent of the ponderable quantity of the body employed: but there is nothing that proves it to be independent of the temperature;
ought therefore to be equal to an unknown function of
, which is the same for all the gases.
Now by the equation
,
is itself the function of the product
; the partial differential equation is therefore
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![{\displaystyle v{\frac {dQ}{dv}}-p{\frac {dQ}{dp}}=F(p.v),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/baa1b9d9b46f22c8bcbe78a285d4cfe606ae2427) | |
having for its integral
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![{\displaystyle Q=f(p.v)-F(p.v)\log[(hyp)p].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/608f71286bdffbf81dcc5402b10718a3fe97ea63) | |
No change is effected in the generality of this formula by substituting for these two arbitrary functions of the product
, the functions
and
of the temperature, multiplied by the coefficient
; we shall thus have
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![{\displaystyle Q=R(B-C\log p).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1700bad093c0ed8b427d5de2744b2b07b576a78c) | |
That this value of
satisfies all the conditions to which it is subject may be easily verified; in fact we have
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![{\displaystyle {\begin{aligned}&{\frac {dQ}{dv}}=R\left({\frac {dB}{dt}}{\frac {p}{R}}-\log p{\frac {dC}{dt}}{\frac {p}{R}}\right)\\&{\frac {dQ}{dp}}=R\left({\frac {dB}{dt}}{\frac {v}{R}}-\log p{\frac {dC}{dt}}{\frac {v}{r}}-C{\frac {1}{p}}\right);\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/071bc8578d7708b81565ca102f1d2fee672ef9ee) | |