PHYS6562 F4 Daily Quaetion

Ergodicity

The system of classical particles bouncing back and forth is not ergodic since each particles can only occupy certain close volume in phase space, with fixed \(x,y\) position, \(P_x, P_y\) momenta and \(\pm p_z\). Therefore, the given volume in phase space of the ensemble will not evolve with time.

Heat engine

(a) True.
(b) True, since the work in done on the gas.
(c) False, it’s a refrigerator.
(d) True.
\[W = (P_0)(3V_0) + \int_{4V_0}^{V_0} \frac{4P_0V_0}{V} dV = 3P_0V_0 - 4P_0V_0\log4\] (e) True, a part does no work so \(\Delta U = Q\). \[\Delta U = \frac{3}{2} nRT_c - \frac{3}{2} nRT_h = -\frac{3}{2} (3P_0V_0) = -\frac{9}{2}P_0V_0\] where n is the number of particles in mole and \(R = 8.317\) is the ideal gas constant. (f) True. The whole cycle \(\Delta U = 0\), the heat input and the net work done onto the gas must equal the heat output. \[Q_h + W = Q_c\]