Specific Heat Capacity for Gas

In summary, the conversation discusses the specific heat capacities in thermodynamics and how they can be expressed as partial derivatives of enthalpy and internal energy with respect to temperature. It also mentions the state postulate of thermodynamics and how the specific heat capacities are defined as state variables. The question is whether the relationships between specific heat capacities and pressure and volume imply that the heat capacities are also functions of pressure and volume alone, without temperature appearing. The expert confirms that this is the case and provides sources that discuss the state variable nature of heat capacities.
  • #1
Mr. Cosmos
9
1
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
 
Last edited:
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  • #2
Mr. Cosmos said:
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)##
 
  • #3
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
 
  • #4
Mr. Cosmos said:
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.
 

Related to Specific Heat Capacity for Gas

What is specific heat capacity for gas?

Specific heat capacity for gas is the amount of heat energy required to raise the temperature of one unit mass of gas by one degree Celsius.

How is specific heat capacity for gas measured?

Specific heat capacity for gas is measured by conducting experiments where a known amount of heat energy is added to a specific amount of gas, and the resulting change in temperature is recorded. The ratio of heat energy to temperature change gives the specific heat capacity value.

What factors affect the specific heat capacity for gas?

The specific heat capacity for gas can be affected by several factors, including the type of gas, the pressure and volume of the gas, and the temperature at which the measurement is taken.

Why is specific heat capacity for gas important?

Specific heat capacity for gas is important because it helps us understand how gases respond to changes in temperature and how much energy is required to produce those changes. This information is crucial for various industrial and scientific applications, such as in the design of engines and power plants.

How does specific heat capacity for gas differ from that of solids and liquids?

Unlike solids and liquids, gases have variable specific heat capacities that can change with temperature and pressure. This is because gases have more mobility and their particles are farther apart, making it easier for them to absorb heat energy and change temperature. Solids and liquids, on the other hand, have fixed specific heat capacities that are usually lower than that of gases.

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