A question about ideal gas heat capacities

I was asked why the heat capacity in problem 38 is Cp = 5/2R, rather than 7/2R. Here is a short explanation of heat capacities for ideal gases:

Generally, an ideal gas is something that obeys PV=RT and that has U(T) only (i.e., U is not a function of pressure or volume). Saying that something is an ideal gas does not immediately specify the heat capacities, although it does say that they are independent of P or V since U(T) and H(T) only.

In some specific instances, however, we can use some approximations to the heat capacities for the ideal gas if we know about the molecular structure:

Cp=5/2R for a monoatomic system
Cp=7/2R for a diatomic system

Thus it depends on the kind of ideal gas that you have (monoatomic or diatomic). So in problem 38, it is likely that the problem refers to an ideal gas with a monoatomic molecular structure.

Why should diatomic gases have a higher heat capacity? Recall that the heat capacity measures how much the internal energy or enthalpy changes with temperature. In a diatomic ideal gas, energy can be stored in the bond that connects the two atoms, in addition to the kinetic energy of the molecule. By stretching and compressing like a spring, the bond can contain potential energy. Therefore, the heat capacity is higher than for a monatomic gas because there is this additional place to store energy as the temperature increases.

In other more complicated systems, the heat capacity for an ideal gas can be a function of temperature, which is usually well-fitted by the form that we've been using in class. Note, however, that still for an ideal gas, these forms do not have a P or V dependence. This is why they are still called "ideal gas heat capacities". The way in which we figure out what the properties of real gases are, for example the enthalpy, is to first figure out what the ideal gas enthalpy would be at the given temperature and pressure using these heat capacity expressions, and then to add back in the part that might need to account for deviations from ideal gas behavior:

Delta H = Delta Hig + Delta Hresidual

where the Delta Hig comes from the ideal gas heat capacity integral and Delta Hresidual comes form the expressions developed recently in class that use PVT data.

MSS