Two vessels A and B each contain N molecules of the same perfect monatomic gas. Initially the tw

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Two vessels A and B each contain N molecules of the same...

Two vessels $\mathrm{A}$ and $\mathrm{B}$ each contain $\boldsymbol{N}$ molecules of the same perfect monatomic gas. Initially the two vessels are thermally isolated from each other, the two gases being at the same pressure $P$, and at temperatures $T_{\mathrm{A}}$ and $T_{\mathrm{B}}$ respectively. The two vessels are now brought into thermal contact, the pressure of the gases being kept constant at the value $P$. Find the change in entropy of the system after equilibrium is established, and show that this change is non-negative.

Key Concepts

Entropy Change

Entropy is a measure of the degree of disorder in a system, and a change in entropy quantifies the transformation in that disorder. In thermodynamic processes, especially those that are irreversible, the entropy of an isolated system tends to increase. Calculating entropy change involves integrating the infinitesimal changes (dQ/T) over the process path, and the second law of thermodynamics guarantees that the total entropy change for an irreversible process is non-negative.

Second Law of Thermodynamics

This law states that in any natural thermodynamic process, the total entropy of an isolated system will tend to increase over time, reaching a maximum at equilibrium. It forms the basis for understanding why processes are irreversible and why, when two bodies at different temperatures interact, heat flows from the hotter to the cooler body, resulting in an overall increase in entropy.

Perfect Monatomic Gas

A perfect (ideal) monatomic gas is a simplified model where the gas molecules are considered as non-interacting point particles with translational kinetic energy only. This model allows for the use of simple expressions for thermodynamic properties such as internal energy and heat capacities, which are essential for calculating entropy change in processes involving such gases.

Constant Pressure Process

A process at constant pressure implies that the external pressure remains unchanged during the process. Under constant pressure conditions, the heat transfer into or out of a system is related to enthalpy change rather than internal energy change alone. This affects how temperature, volume, and other state variables change, and is directly applied when computing the entropy change of processes carried out under isobaric conditions.

Thermal Equilibrium

Thermal equilibrium is reached when two systems in thermal contact come to a common temperature, ceasing net heat exchange. In the context of the problem, when the two gases are allowed to exchange heat while maintaining constant pressure, they eventually settle at a common equilibrium temperature. This final state is crucial for determining the overall change in entropy of the combined system.

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