Question

Consider n moles of ideal gas kept in a heat-isolated cylinder (all processes are adiabatic) with a piston at external pressure p₀, and at temperature T₀. The external pressure is suddenly changed to p=2p₀, and we wait for the system to equilibrate. The volume and the temperature of the ideal gas after equilibration is V and T, respectively.

a) Calculate the amount w of work produced on the system in terms of p, p₀, V, T₀, and n. Using the first law of thermodynamics, find a relation between V and T in terms of the inputs.

b) Using the relation between the external and internal pressure at equillibrium, combined with the equation of state for ideal gas, find another relation between V and T.

c) Considering the derived relations as a system of two equations on V and T, solve it, and find the final values of V and T (i.e., after the equilibrium is reached).

d) Using the relation between the pressure and the temperature for adiabatic processes in ideal gases, find the final temperature T_rev if a slow (reversible) process is applied to increase the external pressure from p₀ to p=2p₀.

e) Compare T with T_rev , i.e., find out whether T<T_rev or it is the other way around. Provide a physical interpretation.

Answer 1

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