A rigid box containing one mole of air at temperature T0 (in...

A rigid box containing one mole of air at temperature $T_0$ (in K ) is initially in thermal contact with an "infinite' heat-capacity reservoir" at the same temperature $T_0$. The box is removed from the reservoir and a cyclic engine is used to take some heat from the reservoir and put some into the air in the box. What is the minimum amount of work from $T_0$ to $T_1$ ? Express $W$ in terms of $T_0, T_1$ and the gas constant $R$, and state units. Ignore vibrational degrees-of-freedom in the air molecules and the heat capacity of the container. Would inclusion of vibrational degrees-offreedom increase or reduce the value of $W$ ?

Key Concepts

Reversible Processes and Minimum Work

In thermodynamics, the minimum work required to change the state of a system is achieved when the process is carried out reversibly. In such a process, the work done is exactly equal to the difference in a thermodynamic potential – for constant volume processes, this is the Helmholtz free energy. This concept shows that instead of doing extra work, one should follow a path where irreversibilities are eliminated, so that the work invested is as low as possible.

Helmholtz Free Energy in Constant?Volume Systems

For a system at constant volume, the change in the Helmholtz free energy (?F) represents the maximum useful work obtainable (or the minimum work required) in a process where temperature is held constant at the boundary (here, the reservoir’s temperature T0). When heating an ideal gas at constant volume from an initial temperature T0 to a final temperature T1, the minimum work required is given by ?F = ?U ? T0?S. For one mole of a diatomic gas with only translational and rotational degrees of freedom, its constant-volume specific heat (C_v) is (5/2)R, and the corresponding expression becomes W_min = (5/2)R[(T1 ? T0) ? T0 ln(T1/T0)] with units of joules.

Degrees of Freedom and Heat Capacity

The internal energy and entropy changes in a gas depend on its heat capacity, which in turn depends on the number of degrees of freedom available to the molecules. When vibrational degrees of freedom are included, the heat capacity increases because there are additional modes in which energy can be stored. As the minimum work required in a reversible process is directly related to the heat capacity (via ?U and ?S), including vibrational degrees of freedom will increase the free energy change and, hence, increase the minimum work required to heat the gas from T0 to T1.

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