Question

Could you decide between two structures from the Cp and Cv ratios?

Answer 1

The heat capacity ratio, δ = Cp/CV, is just as useful in understanding the structure of gaseous molecules and is more accessible experimentally.

Heat-Capacity Ratios for Gases

Cp = Cv + R

The ratio,

Cp = R. Cv

is related to the ability of the gas to do expansion work. Heat capacity at constant volume, Cv can be described using the equipartition theory, which states that each mode of motion will contribute to a molecule or atom’s energy.

E = E(translational) + E(rotational) + E(vibrational)

Setting up a Cartesian coordinate system, translational motion can occur in any of the three directions: x, y, or z. Thus for a monatomic gas energy can be represented as 3(RT/2); it is clear that no vibrational or rotational motions contribute. Rotational motion contributes to the energy of diatomic and polyatomic molecules; they are easily accessible at room temperature therefore will significantly contribute to . Vibrations can be separated into two categories: bending and stretching, where the number of modes can be described as 3N-5 for linear, and 3N-6 for nonlinear molecules. Vibrational levels are not as accessible as rotational ones are at room temperature, so it is valid to consider them, at most, only partially active; the extent depends on certain properties of the molecule. Stretching modes tend to have very high frequencies giving way to a small contribution to heat capacity ratios. It should be noted that electronic transitions will be ignored since most molecules are in their electronic ground state at room temperature. Applying classic statistical mechanics to the equipartition theory, an expression for the energy contribution of one mole of a gas from each mode of motion is given as RT/2. Since heat capacity varies with temperature the following relationship is given:

Cv = (E/T)v. (E is the internal energy)

Using these relationships, a theoretical value of  can be derived. In contrast, experimental values will be calculated through an adiabatic expansion method initiated by Clement and Desormes. Discrepancies between the experimental values and those predicted by the equipartition theory will be examined.