Molar Heat capacities

Question

Molar Heat capacity is defined as the amount of heat required to raise the temperature of one mole of a substance by unit amount. As my textbook mentions the molar heat capacity of an ideal gas depends only upon temperature.(that too can also be neglected being almost negligible). So, it is a property of the material to a large extent. That means it must be independent of pressure and volume. Next, it defines Cv and Cp i. e molar heat capacity at constant volume and pressure. According to me Cv at all volumes=Cp at all pressures=molar heat capacity of the substance since Cv and Cp both are essentially heat capacities and they depend on what the substance is. But this is not the case. Do that implies heat capacity has some sort of dependency on pressure and volume? If that is the case then why do Cv values of same substance at different volume are equal since the formula of Cv has nothing to do with volume.(Depends upon degree of freedom which must be same at all volumes)

Answer 1

Assume the gas to be ideal and monotomic.

In theory the specific heat capacity of a mole of gas can have any value between $-\infty$ and $+\infty$ depending on the heat transfer in/out of the gas and the work done by/on the gas.

The defining equation for specific heat capacity is $Q=n\,C\,\Delta T$ where $Q$ is the heat transferred, $n$ is the number of moles of gas, $C$ is specific heat capacity of the gas and $\Delta T$ is the change in temperature of the gas.

The first law of thermodynamics $\Delta U = Q -W$ where $\Delta U$ is the change in internal energy of the gas, $Q$ is the heat transferred into the gas and $W$ is the work done by the gas shows that you can change the internal energy of the gas, which is dependent on the temperature of the gas, by an infinite number of permutations of $Q$ and $W$ thus allowing there to be an infinite number of specific heat capacities of a gas.

In practice for convenience the specific heat capacity is defined; at constant volume ie $W=0$ and then $\Delta U = Q$ and at constant pressure ie $W=P\Delta V$ where $\Delta V$ is the change in volume of the gas and then $\Delta U = Q-P\Delta V$.

As an example one could compress a gas in a chamber which allows no heat in or out of the chamber.
Work is done on the gas and so its temperature rises with no transfer of heat and the specific heat capacity of the gas is zero, $0=n\,C\, \Delta T$.

Another example might be of a gas which is being heated at a certain rate and the gas is doing work at the same rate ie $Q=W$.
There is no change in the internal energy of the gas $\Delta U = Q-W =0$ and so the gas does not change its temperature so the gas has an infinite specific heat capacity, $Q = n\,C\, 0$.

why do Cv values of same substance at different volume are equal since the formula of Cv has nothing to do with volume

Because if all else is equal (temperature and pressure) the number of moles of gas depends on the volume of the gas so if you have twice the volume of gas you have twice the number of moles but the value of the specific heat capacity of the gas has not changed even though twice the amount of heat is required to change the temperature of the gas by the same amount.

Answer 2

You were correct in saying that the heat capacities at constant volume and constant pressure are supposed to be physical properties of the material. However, in thermodynamics, the heat added to the system Q depends on the path of the process, so how can heat capacity depend on path if it is also supposed to be independent of path? In thermodynamics, we overcome this dilemma by modifying the definition of the heat capacity slightly. Instead of relating it to the heat Q, we relate it to the internal energy U and the enthalpy H, both of which are properties of the material (state properties), and not of the process path. So now $$C_v=\frac{1}{n}\left(\frac{\partial U}{\partial T}\right)_V$$and$$C_p=\frac{1}{n}\left(\frac{\partial H}{\partial T}\right)_P$$ These new definitions reduce to the definition we learned earlier (in terms of Q) if the process path is at constant volume ($C_v$) or constant pressure ($C_p$). However, they are also much more general and apply to any arbitrary process paths.

To state this all more categorically, these definitions of both heat capacities apply at all pressures and volumes.

For the special case of an ideal gas, $$dH=nC_pdT=dU+d(PV)=nC_vdT+nRdT=n(C_v+R)dT$$Therefore, $$C_p=C_v+R$$So, $C_v$ and $C_p$ are not even the same for this simplest of cases. Only for the case of an incompressible fluid are $C_v$ and $C_p$ equal.