# Specific Heat Capacity for Gas

So I have a question regarding the specific heat capacities in thermodynamics. In general the specific heat capacities for a gas (or gas mixture in thermo-chemical equilibrium) can be expressed as,

## c_p = \left(\frac{\partial h}{\partial T}\right)_p \qquad \text{and} \qquad c_v= \left(\frac{\partial e}{\partial T}\right)_v ##
.
Additionally, from the state postulate of thermodynamics one can write state relationships as,

## h = h\left(p,v\right) \qquad \text{and} \qquad e = e\left(p,v\right) \qquad \text{and} \qquad T=T\left(p,v\right) ##

Now I know that the specific heats are a defined thermodynamic property and not a state variable, however, would the above relationships imply,

## c_p =c_p\left(p,v\right) \qquad \text{and} \qquad c_v= c_v\left(p,v\right) ##

?? I have never come across such a relationship (obviously not explicit) in a textbook, or even seen a surface plot to indicate this relationship. Any help would be greatly appreciated.
Note: I am aware of the reciprocity relations and Maxwell relations, but I am trying to reduce the specific heats to functional relationships of density and pressure without the temperature appearing. These relationships will be formed numerically with Cantera, but I wan't to make sure my thought process is on the right track.

Thanks,

-Mr. Cosmos

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## Answers and Replies

Chestermiller
Mentor
So I have a question regarding the specific heat capacities in thermodynamics. In general the specific heat capacities for a gas (or gas mixture in thermo-chemical equilibrium) can be expressed as,

## c_p = \left(\frac{\partial h}{\partial T}\right)_p \qquad \text{and} \qquad c_v= \left(\frac{\partial e}{\partial T}\right)_v ##
.
Additionally, from the state postulate of thermodynamics one can write state relationships as,

## h = h\left(p,v\right) \qquad \text{and} \qquad e = e\left(p,v\right) \qquad \text{and} \qquad T=T\left(p,v\right) ##

Now I know that the specific heats are a defined thermodynamic property and not a state variable,
The specific heats certainly are state variables.

however, would the above relationships imply,

## c_p =c_p\left(p,v\right) \qquad \text{and} \qquad c_v= c_v\left(p,v\right) ##
Yes.
?? I have never come across such a relationship (obviously not explicit) in a textbook, or even seen a surface plot to indicate this relationship. Any help would be greatly appreciated.
Note: I am aware of the reciprocity relations and Maxwell relations, but I am trying to reduce the specific heats to functional relationships of density and pressure without the temperature appearing. These relationships will be formed numerically with Cantera, but I wan't to make sure my thought process is on the right track.

Thanks,

Mr. Cosmos
If the heat capacities are expressed as ##C_v=C_v(p,T)## and ##C_p(p,T)##, then, from the equation of state, T=T(p,v), we have ##C_v=C_v(p,T(p,v))=C_v(p,v)## and ##C_p=C_p(p,T(p,v))=C_p(p,v)##

Thanks for the quick reply. I guess my confusion was with the appropriate definitions of the heat capacities being state variables. In my textbook the heat capacities are declared as non-state variables, and the same is said here,
https://www.grc.nasa.gov/www/k-12/airplane/specheat.html
However, since reading your response I have found other sources that say that they are indeed state variables. Interesting discussion.

Thanks,

-Mr. Cosmos

Chestermiller
Mentor
Thanks for the quick reply. I guess my confusion was with the appropriate definitions of the heat capacities being state variables. In my textbook the heat capacities are declared as non-state variables, and the same is said here,
https://www.grc.nasa.gov/www/k-12/airplane/specheat.html
However, since reading your response I have found other sources that say that they are indeed state variables. Interesting discussion.

Thanks,

-Mr. Cosmos
There is one way of knowing whether a variable is a state variable or not. If you tell me the temperature and pressure of the material and I can tell you a unique value for the variable in question (e.g., heat capacity), then the variable is a state variable. Heat capacities satisfy this requirement.