# The heat capacity of an ideal gas at a constant volume

On the graph:

1. blue line is the heat capacity of a classical monatomic ideal gas at constant volume;
2. the red line is the heat capacity of a classical monatomic ideal gas at a constant volume, taking into account the Earth's gravity field.

How to explain on a qualitative level why the red graph differs from the blue one in this way?

• The heat capacity at constant volume of a monoatomic ideal gas is (3/2)kN, not (5/2)kN, and it does not vary with altitude. Oct 6, 2021 at 22:54

Qualitatively, we can justify the endpoints by noting that gases spread out entropically as a result of their thermal energy; take away this thermal energy, and ideal gases stratify perfectly according to their density. (Real gases condense and then freeze.)

Thus, the monatomic ideal gas at high temperatures fills its container easily and thus maintains the familiar constant-volume heat capacity of $$\frac{3}{2}kN$$. The monatomic ideal gas at low temperatures in a gravity field mostly occupies only a lower portion of its container and thus assumes the behavior of constant-pressure confinement, including the constant-pressure heat capacity $$\frac{5}{2}kN$$. (The corresponding pressure depends on the height of the region we choose to look at.)

We can think of the presence of gravity as partially diverting any input heating of the system toward an increase in the gravitational potential energy; consequently, the heat capacity (i.e., the energy required to raise the temperature of a system solely by heating it) is larger.

For more discussion, see, for example, Landsberg, "Entropy of a column of gas under gravity," Am J Phys 62 (1994).

• What has this got to do with the heat capacity at constant volume? Oct 7, 2021 at 15:06
• The heat capacities we typically work with ignore gravity. The effect of gravity is typically negligible except for ideal gases (and certain other systems) at low temperature, as discussed in the paper. In this case, the gas doesn’t really fill its container and therefore isn’t really at constant volume. Oct 7, 2021 at 16:08
• But then the heat capacity varies locally, right? So this is an "apparent" heat capacity? Oct 7, 2021 at 16:44
• To my knowledge, it applies to the entire container of gas of height $H$ in a gravity field $g$. In the equation shown in the question, for example, the location (e.g., $z$) does not appear. Of course, $g$ varies somewhat with elevation on Earth. The linked paper was unknown to me before yesterday but is very straightforward and interesting because of its comparisons to the gravity-free classical ideal gas. Oct 7, 2021 at 19:51
• Thanks. I can see what's happening here now. When the gas is heated, not only does its temperature increase, but its potential energy also increases. So more heat is required to raise its temperature a specified amount. So what they calling the heat capacity at constant volume is really the "apparent heat capacity" calculated from $\frac{1}{M}\frac{dQ}{dT}$, where M is the total mass of gas in the container. Oct 7, 2021 at 23:23