Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 2.7-2

SOME FORMAL RELATIONSHIPS, AND SAMPLE SYSTEMS

Physics — Statistical Mechanics · mechanical-equilibrium-two-systems

Problem

Two particular systems have the following equations of state: 1T(1)=32RN(1)U(1),P(1)T(1)=RN(1)V(1)\frac{1}{T^{(1)}} = \frac{3}{2}R\frac{N^{(1)}}{U^{(1)}}, \quad \frac{P^{(1)}}{T^{(1)}} = R\frac{N^{(1)}}{V^{(1)}} and 1T(2)=52RN(2)U(2),P(2)T(2)=RN(2)V(2)\frac{1}{T^{(2)}} = \frac{5}{2}R\frac{N^{(2)}}{U^{(2)}}, \quad \frac{P^{(2)}}{T^{(2)}} = R\frac{N^{(2)}}{V^{(2)}} The mole number of the first system is N(1)=0.5N^{(1)} = 0.5 and that of the second is N(2)=0.75N^{(2)} = 0.75. The two systems are contained in a closed cylinder, separated by a fixed, adiabatic, and impermeable piston. The initial temperatures are T(1)=200T^{(1)} = 200 K and T(2)=300T^{(2)} = 300 K, and the total volume is 20 liters. The ``setscrew'' which prevents the motion of the piston is then removed, and simultaneously the adiabatic insulation of the piston is stripped off, so that the piston becomes moveable, diathermal, and impermeable. What is the energy, volume, pressure, and temperature of each subsystem when equilibrium is established? It is sufficient to take R=8.3R = 8.3 J/mole K and to assume the external pressure to be zero. Answer: U(1)=1700U^{(1)} = 1700 J

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