Degrees of Freedom and Molar Specific Heats — Question 4

Question 4

A measured heat-capacity ratio constrains an effective mode count

Use classical equipartition: each active independent quadratic energy term contributes $\tfrac12k_BT$ per molecule. Let $f$ count these terms, so for constant $f$, $C_V=fR/2$, $C_p=C_V+R$ and $\gamma=C_p/C_V$. Take $R=8.314$ J/(mol K). A gas has measured $\gamma=1.400$ with a hard uncertainty interval $1.390\le\gamma\le1.410$. Assume the stated ideal-gas relation between capacities.

1. Infer central $f$ and the exact corresponding interval of effective $f$.

2. Can $f\approx5$ alone prove that the sample is a pure rigid diatomic gas? Give a classical mixture of $f=3$ and $f=6$ species with the same central effective count.

Question 4 - Solution

A measured heat-capacity ratio constrains an effective mode count

(a) Method and calculation

$$\gamma=1+\frac2f\ \Rightarrow\ f=\frac2{\gamma-1}.$$$$f_0=5.000,\quad\frac2{.410}\le f\le\frac2{.390}\ \Rightarrow\ 4.87805\le f\le5.12821.$$

(b) Interpretation and further calculation

$$f_{\mathrm{mix}}=(1-x)(3)+x(6)=5\ \Rightarrow\ x=2/3.$$$$C_{V,\mathrm{mix}}=\tfrac52R,\quad\gamma_{\mathrm{mix}}=\tfrac75.$$A heat-capacity measurement constrains an energy response. Mixtures and partially active internal modes can share that response, so molecular identity is not uniquely determined.

Physical check

The inverse relationship makes symmetric uncertainty in $\gamma$ slightly asymmetric in $f$.