Molar Specific Heat Definitions for Gases

Question

In most textbooks, the molar specific heat of a gas is defined for gases at constant volume and constant pressure as follows :

1. $C_v = \frac{Q}{n\Delta T} = \frac{\Delta U}{n \Delta T}$

2. $C_p = \frac{Q}{n \Delta T}$

But these definitions can also be related by $C_p = C_v + R$, with $R$, being the ideal gas constant.

But it seems that using definition $(1)$ in the first law of thermodynamics leads to contradictions, for example

$$\Delta U = Q - W$$ $$\implies C_v\cdot n \cdot\Delta T = (C_v\cdot n \cdot \Delta T) - W $$

$$\implies W = 0 \ \ (\forall\ C_V, n, \Delta T)$$

Which is obviously not true

Another seeming contradiction can be derived as follows:

$$C_p = C_v + R$$ $$\implies \frac{Q}{n \Delta T} = \frac{Q}{n \Delta T} + R$$ $$\implies R = 0$$

Which again is obviously not true. So it seems one can not just take the definitions of $C_v$ and $C_p$ at face value and use them in equations, there have to be certain conditions where I can or cannot use them.

Textbooks such as Fundamentals of Physics, and University Physics, give very little explanation why the definitions of molar specific heats of gases differ under constant volume and constant pressure, for example why $C_p \neq \frac{Q}{n \Delta T}$, is not explained in much detail if at all in either textbook.

So my question is why do the definitions of molar specific heats of gases differ under constant volume and constant pressure?

And why can I not take the definitions of $C_v$ and $C_p$ at face value and use them in equations?

Answer 1

When you learned this material as a beginning physics student, they taught you that $nC_p\Delta T=Q$, and the focus was typically on a solid or liquid (both of which are nearly incompressible). However, this was only introductory, and was not the correct and precise definition necessary to use in thermodynamics. In thermodynamics, it is recognized that heat capacity is really a physical property of the material, and has nothing to do with any specific process (which in thermo is characterized by W and Q). The new more precise definitions of heat capacities employed in thermodynamics involve the internal energy and the enthalpy of the material (and can apply to any material, including an ideal gas): $$nC_v=\left(\frac{\partial U}{\partial T}\right)_V$$ $$nC_p=\left(\frac{\partial H}{\partial T}\right)_p$$ If the first equation is combined with the first law, then, in heating tests at constant volume, the amount of heat added Q to the system can be used to experimentally measure Cv directly, since the amount of work is zero. If the second equation is combined with the first law, then, in heating tests at constant pressure, the amount of heat added to the system can be used to experimentally measure Cp directly, since, in this case, $W = p\Delta V$. So the subscripts v and p are used to refer to the conditions required to measure these heat capacities directly by determining the heat Q added is such special tests. However, once these measurements have established the values of the two heat capacities, they can be used for all differential changes in state to determine the partial derivatives of U and H with respect to temperature.

Answer 2

The expressions are specifically for the cases "gas heated in a constant volume" ($C_V$) and "gas allowed to expand so that the pressure stays constant" ($C_p$). So you can't substitute one expression into the other because the values for $Q$ would be different.

The $C_V$ case is easiest to understand: you add heat, increase the kinetic energy of the molecules, therefore increase the temperature. The ideal-gas law $pV=nRT$ prescribes that the pressure must increase, too.

If you heat the gas at constant pressure, the gas must expand its volume by an amount $\Delta V=nR\Delta T/p$, thereby doing a work $\Delta W=p\Delta V=nR\Delta T$. This extra work explains the relation $C_p-C_V=R$.