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Dependence of Heat Capacities on temperature pressure and volume

Dependence of Heat Capacities on temperature pressure and volume

We have defined thus far two quantities which are functions of the state of the system, namely, E and H, also Cv and Cp. If we tend to confine ourselves to pure substances, then these quantities are functions of any 2 of the 3 variables Pressure, temperature, and volume. In dealing with these variables, it is found that E and Cv are most conveniently expressed in terms of T and V, while for H and Cp the best choice is T and P. If we start now with the fact that

E = f (T,V)  then

 dE = (dE/dT)V dT + ( dE/dV  )T dV     …………………………………………(1)

 as  ( dE/dT )v = Cv

 A corresponding expression for  ( dE/dV  )T dV cannot be obtained without the second law of thermodynamics. However, this relationship is equal with

( dE/dV  )T = T (dP/dT)v – P……………………………………………………(2)

Substituting these values of partial derivatives into the equation, (1) we get

dE = Cv dT + [T (dP/dT)v - P] dv ………………………………………………..(3)

The first term on the right in Eq (3) gives the effect on E of temperature change at constant volume, while the second gives the effect of volume change at a constant temperature.

  Considering  H as a function of P and T we get an equation 

dH = (dH/dT)p dT + (dH/dP)T dP   …………………...………………………….(4)

but

Cp = (dH/dT)p

 While

 (dH/dP)T = [V-T (dV/dT)p]  …………..…………………………………………(5)

Consequently,

 dH = CP dT + [ V – T(dV/dT)p ]dP   ………………….(6)

The first term in Eq. (6) gives the variation of H with the temperature at constant pressure, while the second gives the effect of pressure at a constant temperature.

The effect of volume change on Cv can be deduced by differentiation of Eq.

 ( dE/dT )v = Cv with respect to volume at constant temperature and utilization of Eq. (2) The result is (7)

(dCv/dV)T =  T( d2p/dT2)V   ………………………………………….(7)

similarly, the effect of pressure on Cp follows the differential eq  Cp = (dH/dT)p dT

  with respect to P at constant T and use of eq (5) we get this eq.  (8)

(dCP/dP)T =-T(d2V/dT2)P ………………………………………………………..(8)


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