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I have commonly seen that the work for an isentropic compression is written as: $$ W_{in} = c_{v}(T_1 - T_2) $$ What confuses me is that since it is a compression (volume is changing) why do we use the constant volume specific heat?

Thank you kindly for your time and help.

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Start with the first law.

$$dU = \delta q + \delta w$$

Isentropic sets $dS = \delta q /T = 0$, giving $dU = \delta w$. For an ideal gas with a constant specific heat, $dU = C_v dT$. This gives the equation being used.

$$\Delta U = C_v \Delta T = w $$

If temperature decreases, work is negative, meaning it is done by the gas. Alternatively, if temperature increases, work is done on the gas.

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The fundamental relation for any closed system can be written as

$$dU=\left(\frac{PC_V}{T\alpha K}\right)dT+\left(T-\frac{P}{\alpha K}\right)dS,$$

where $U$ is energy, $P$ is pressure, $C_V$ is the constant-volume heat capacity, $T$ is temperature, $\alpha$ is the thermal expansion coefficient, $K$ is the bulk modulus, and $S$ is entropy. If heating is absent (an adiabatic step, for example), then $dU$ is just the infinitesimal work $w$. If the step is isentropic, then $dS=0$:

$$w=\left(\frac{PC_V}{T\alpha K}\right)dT.$$

Now, the ideal gas is unique in that its reciprocal thermal expansion is its temperature and its stiffness (bulk modulus) is its pressure: $\alpha=1/T$, $P=K$:

$$w=C_V dT,$$ which we can integrate under the assumption of a constant heat capacity to give the work done $$W=C_V\Delta T,$$ the relation you asked about. This always holds. Unfortunately, the presence of the constant-volume heat capacity as a coefficient has confused many into concluding—not unreasonably, but incorrectly—that this relation only holds for constant-volume processes.

In fact, it didn't even matter that an isentropic process was specified; since $T-\frac{P}{\alpha K}=0$ for an ideal gas, the $dS$ term would have disappeared anyway. Indeed, every differential term except $dT$ is identically zero for an ideal gas! But this simplification often leaves new thermodynamics practitioners uncertain and frustrated because the only remaining material property has "constant-volume" in its name.

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Your problem is that much of the terminology of thermodynamics was defined before the underlying physical concepts were properly understood, which leaves us stuck with a lot of confusing names for things now.

In particular, you're worried about the applicability of the quantity commonly known as the "specific heat capacity at constant volume". The first thing you need to know is that that's a misleading name for the quantity, for two reasons. The first reason is that "heat capacity" is an oxymoron: "capacity" signifies storage, but we now know (originally from Joule, 1845, Lond. Edinb. Dublin Philos. Mag. J. Sci. 27(179):205-207) that, when it's being stored, energy doesn't have an identity as "heat" or "work", it's just energy; it only develops an identity as "heat" or "work" when it's being transferred. The second reason is that, as you've noticed, it's important even when the volume is not constant.

Therefore, a better name for the quantity commonly known as the "specific heat capacity at constant volume" would, following Stedman (1963, Educ. Train. 5(3): 127-128), be "specific internal energy capacity". It is, as that name suggests, the quotient of the change in specific internal energy of a body by the change in the temperature of that body.

Now let's think about the situation you're trying to analyse: an isentropic non-flow process, in which some material is compressed. In a non-flow process (unless something exotic is going on), internal energy is the only form in which the material can store energy; and "isentropic" implies that no heat is transferred, so the only way in energy can get in or out is as work. Hence, the specific work done is equal to the change in specific internal energy, which is (by definition) equal to the product of the specific internal energy capacity and the change in temperature, as your first equation says.

Now you may be wondering how the specific internal energy capacity came by the misleading name "specific heat capacity at constant volume" in the first place. Well, think about an isochoric (i.e. constant volume) non-flow process. As before, in a non-flow process (unless something exotic is going on), internal energy is the only form in which the material can store energy; and if the volume is constant in a non-flow process (again, unless something exotic is going on), then no work is done, so the only way energy can get in or out is as heat. Therefore, the specific input of heat is equal to the change in specific internal energy, which is (by definition) equal to the product of the specific internal energy capacity and the change in temperature, i.e. in an isochoric non-flow process, the quotient of the specific input of heat by the change in temperature is equal to the specific internal energy capacity. The quantity that Stedman and I suggest should be called "specific internal energy capacity" was first discussed in the context of measurements of the specific input of heat required to bring about a change in temperature in an isochoric process, before anyone knew about the interchangeability of heat and work or the idea of internal energy, and hence it got the name "specific heat capacity at constant volume" with which we seem to be stuck today.

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