# Specific heat at constant pressure

1. ### rabbahs

16
Dear Forum members,

I have a bit confusion about the "Specific Heat at constant pressure".

Normally it only varies with the temperature (As given by many book at their Appendixs). But these values are only given at 1 atm pressure and with a wide range of temperature. Most of the books also specify the polynomial related to the specific heat (that only change with temperature, because pressure is held fixed at 1 atm).

My question is that what happen to Cp when the pressure increase from 1 atm to 5 atm at constant temperature. is Cp increase with increase or decrease of pressure ??

I know that its sounds bit odd that asking for Cp (which is indeed sp. heat at CONSTANT PRESSURE)

is there any polynomial which describe the change of Cp with both pressure and temperature ???

Please also view the attached file which clearly shows that Cp is changing with pressure.

I want to know that polynomial having both temperature and pressure.

Thanks alot

2. ### Mapes

2,532
I don't know if this will get you anywhere, but the change in $C_P$ with pressure is a classic problem in thermo (I remember getting it on an exam before I was familiar with it):

$$\left(\frac{\partial C_P}{\partial P}\right)_T=\frac{\partial}{\partial P}\left[\left(T\frac{\partial S}{\partial T}\right)_P\right]_T=T\frac{\partial}{\partial P}\left[\left(\frac{\partial S}{\partial T}\right)_P\right]_T=T\frac{\partial}{\partial T}\left[\left(\frac{\partial S}{\partial P}\right)_T\right]_P$$

Then we use a Maxwell relation to get

$$T\frac{\partial}{\partial T}\left[\left(-\frac{\partial V}{\partial T}\right)_P\right]_P=-T\left(\frac{\partial^2 V}{\partial T^2}\right)_P$$

Thus, the change you're looking for is related to the second derivative of volume with temperature. This is zero for an ideal gas, but it may get you somewhere if you have a constitutive equation for a real gas.

3. ### rabbahs

16
thanks Mapes, i will look it to these equations and let you know

4. ### rabbahs

16
could you kindly give me the reference of the book in which these relations related to specific heat are discussed ??

5. ### Mapes

2,532
Actually, nearly every thermodynamics book will define specific heat and explain Maxwell relations. Zemansky is pretty good.

16
thanks