A Hint of Quantum Theory — Question 6

Question 6

A fixed zero-point energy changes the energy reference but not heat capacity

A harmonic mode has levels $E_j=(j+\tfrac12)\Delta$, with $\Theta=\Delta/k_B=1000$ K independent of temperature and volume. Its mean energy is $\overline E=\Delta/2+\Delta/(e^{\Theta/T}-1)$. There are .100 mol of independent modes; use $R=8.314$ J/(mol K).

1. Find the total zero-point contribution and the heat absorbed when the modes warm at fixed volume from 500 K to 1000 K.

2. Show explicitly why including or subtracting zero-point energy gives the same heat capacity and heat input for this process.

Question 6 - Solution

A fixed zero-point energy changes the energy reference but not heat capacity

(a) Method and calculation

$$U_0=\tfrac12nR\Theta=415.700\,\mathrm J.$$$$Q=\Delta U=nR\Theta\left[\frac1{e^1-1}-\frac1{e^2-1}\right]\approx353.727\,\mathrm J.$$

(b) Interpretation and further calculation

$$\frac{dU_0}{dT}=\frac d{dT}(\tfrac12nR\Theta)=0\quad(\Theta\text{ fixed}).$$$$U^{\prime}(T)=U(T)-U_0\ \Rightarrow\ \frac{dU^{\prime}}{dT}=\frac{dU}{dT},\quad U_f^{\prime}-U_i^{\prime}=U_f-U_i.$$A temperature-independent offset cannot be treated as extra heat absorbed during this temperature change.

Physical check

The conclusion relies on a fixed level spacing; a changing mechanical parameter can make the ground energy relevant to work.