A Hint of Quantum Theory — Question 5

Question 5

A quantum vibration smoothly changes an ideal-gas heat-capacity ratio

A diatomic ideal gas has classical translations and rotations contributing $5R/2$ per mole to $C_V$. One vibration has molar excitation energy $U_v=R\Theta/(e^{\Theta/T}-1)$ with fixed $\Theta$. Use $C_p=C_V+R$.

1. Derive the vibrational molar capacity and evaluate $C_v^{\mathrm{vib}}/R$ at $T=\Theta$ and $T=\Theta/5$.

2. Find the total gas $\gamma$ at each temperature and its limits for frozen and fully classical vibration.

Question 5 - Solution

A quantum vibration smoothly changes an ideal-gas heat-capacity ratio

(a) Method and calculation

$$x=\Theta/T,\quad C_v^{\mathrm{vib}}=\frac{dU_v}{dT}=R\frac{x^2e^x}{(e^x-1)^2}.$$$$C_v^{\mathrm{vib}}(\Theta)/R\approx.920674,\quad C_v^{\mathrm{vib}}(\Theta/5)/R\approx.170742.$$

(b) Interpretation and further calculation

$$\gamma=1+\frac1{2.50+C_v^{\mathrm{vib}}/R}.$$$$\gamma(\Theta)\approx1.29234,\quad\gamma(\Theta/5)\approx1.37443.$$$$T/\Theta\to0:\ \gamma\to7/5;\qquad T/\Theta\to\infty:\ \gamma\to9/7.$$This model makes the transition continuous rather than assigning a sudden integer jump in active terms.

Physical check

The stated rotational contribution is held classical throughout; actual rotational freeze-out would need its own model.