Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 6.2-3

THE EXTREMUM PRINCIPLE IN THE LEGENDRE TRANSFORMED REPRESENTATIONS

Physics — Statistical Mechanics · helmholtz-potential-mixture-piston-work

Problem

Two subsystems are contained within a cylinder and are separated by an internal piston. Each subsystem is a mixture of one mole of helium gas and one mole of neon gas (each to be considered as a monatomic ideal gas). The piston is in the center of the cylinder, each subsystem occupying a volume of 10 liters. The walls of the cylinder are diathermal, and the system is in contact with a thermal reservoir at a temperature of

100^\circ

C. The piston is permeable to helium but impermeable to neon. Recalling (from Problem 5.3-10) that the Helmholtz potential of a mixture of simple ideal gases is the sum of the individual Helmholtz potentials (each expressed as a function of temperature and volume), show that in the present case

F = N\frac{T}{T_0}f_0 - \frac{3}{2}NRT\ln\frac{T}{T_0} - N_1RT\ln\!\left(\frac{V}{V_0}\frac{N_0}{N_1}\right) - N_2^{(1)}RT\ln\frac{V^{(1)}N_0}{V_0 N_2^{(1)}} - N_2^{(2)}RT\ln\frac{V^{(2)}N_0}{V_0 N_2^{(2)}}

where

T_0

f_0

V_0

, and

N_0

are attributes of a standard state (recall Problem 5.3-1),

N

is the total mole number,

N_2^{(1)}

is the mole number of neon (component 2) in subsystem 1, and

V^{(1)}

and

V^{(2)}

are the volumes of subsystems 1 and 2, respectively. How much work is required to push the piston to such a position that the volumes of the subsystems are 5 liters and 15 liters? Carry out the calculation both by calculating the change in

F

and by a direct integration (as in Problem 6.2-1). Answer:

\text{work} = RT\ln(\tfrac{4}{3}) = 893\text{ J}

Related theory

Work And Kinetic Energy Work Energy Theorem

Browse the problem library