The Zeroth Law of Thermodynamics — Question 7

Question 7

Thermal equilibrium does not imply mechanical equilibrium

Two ideal gas chambers share a rigid conducting wall and have reached one temperature $T=300\,\mathrm K$. Use the supplied model $pV=nRT$, with $R=8.00\,\mathrm{J\,mol^{-1}K^{-1}}$. Chamber A has $n_A=1.0\,\mathrm{mol},V_A=.010\,\mathrm{m^3}$; B has $n_B=2.0\,\mathrm{mol},V_B=.010\,\mathrm{m^3}$.

1. Find both pressures and the force difference on a wall area $.020\,\mathrm{m^2}$. Can the temperatures be equal while the pressures differ?

2. The wall is replaced by a frictionless movable conducting piston at the same temperature. With total volume fixed, find the final chamber volumes when pressures are equal.

Question 7 - Solution

Thermal equilibrium does not imply mechanical equilibrium

(a) Method and calculation

$$p_A=\frac{1(8)(300)}{.010}=240\,\mathrm{kPa},\quad p_B=480\,\mathrm{kPa}.$$$$F_{\mathrm{net}}=(p_B-p_A)A=240000(.020)=4800\,\mathrm N\quad\text{toward A}.$$The rigid support balances this force; thermal equilibrium alone does not equalize pressure.

(b) Interpretation and further calculation

At equal final $p$ and $T$,$$\frac{V_B}{V_A}=\frac{n_B}{n_A}=2,\qquad V_A+V_B=.020\,\mathrm{m^3}.$$$$V_A=.006667\,\mathrm{m^3},\qquad V_B=.013333\,\mathrm{m^3}.$$Mechanical equilibrium supplies an additional condition.

Physical check

The ideal-gas relation is supplied here; equal temperature is compatible with different density and pressure.