(a) A system, maintained at constant volume, is brought in...

(a) A system, maintained at constant volume, is brought in contact with a thermal reservoir at temperature $T_f$. If the initial temperature of the system is $T_i$, calculate $\Delta S$, change in the total entropy of the system + reservoir. You may assume that $c_v$, the specific heat of the system, is independent of temperature. (b) Assume now that the change in system temperature is brought about through successive contacts with $N$ reservoirs at temperature $T_1+$ $\Delta T, T_{\mathrm{i}}+2 \Delta T, \ldots, T_{\mathrm{f}}-\Delta T, T_{\mathrm{f}}$, where $N \Delta T=T_{\mathrm{f}}-T_{\mathrm{i}}$. Show that in the limit $N \rightarrow \infty, \Delta T \rightarrow 0$ with $N \Delta T=T_{\mathrm{f}}-T_{\mathrm{i}}$ fixed, the change in entropy of the system + reservoir is zero. (c) Comment on the difference between (a) and (b) in the light of the second law of thermodynamics.

(a) A system, maintained at constant volume, is brought in contact with a thermal reservoir at temperature $T_f$. If the initial temperature of the system is $T_i$, calculate $\Delta S$, change in the total entropy of the system + reservoir. You may assume that $c_v$, the specific heat of the system, is independent of temperature.
(b) Assume now that the change in system temperature is brought about through successive contacts with $N$ reservoirs at temperature $T_1+$ $\Delta T, T_{\mathrm{i}}+2 \Delta T, \ldots, T_{\mathrm{f}}-\Delta T, T_{\mathrm{f}}$, where $N \Delta T=T_{\mathrm{f}}-T_{\mathrm{i}}$. Show that in the limit $N \rightarrow \infty, \Delta T \rightarrow 0$ with $N \Delta T=T_{\mathrm{f}}-T_{\mathrm{i}}$ fixed, the change in entropy of the system + reservoir is zero.
(c) Comment on the difference between (a) and (b) in the light of the second law of thermodynamics.

Key Concepts

Entropy Change

Entropy change is a measure of the irreversibility of a process and indicates the degree of dispersal of energy. In thermodynamics, the total entropy change of a system and its surroundings is calculated by summing the contributions of heat transfers divided by temperature. For processes occurring at a constant volume, the calculation simplifies because work done by expansion or compression is absent, and any energy exchange is purely in the form of heat.

Heat Transfer at Constant Volume

In a constant volume process, the system does no work because there is no change in volume. Thus, any change in the internal energy of the system is solely due to heat transfer. This simplifies the analysis, as the temperature changes in the system are directly related to the amount of heat absorbed or released by the system.

Thermal Reservoirs

A thermal reservoir is an idealized body with an infinite heat capacity that can absorb or supply finite amounts of heat without undergoing any change in temperature. They are used in thermodynamic analyses to define controlled temperature environments for systems. In the context of the problem, the reservoir’s fixed temperature dictates the manner in which heat is transferred and influences the entropy change of the surroundings.

Quasi-static (Reversible) Process

A quasi-static process is one that occurs infinitely slowly so that the system remains in thermodynamic equilibrium at all stages. In such processes, the incremental changes in state variables are so small that the reverse process can be achieved without any net entropy production. The second scenario described involves multiple reservoirs to create an effectively continuous temperature change, approximating a reversible process where the total entropy change of the system plus surroundings becomes zero.

Second Law of Thermodynamics

The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and will usually increase in the case of irreversible processes. For irreversible processes, such as those involving finite temperature differences between the system and its surroundings, there is a net increase in total entropy. In contrast, reversible processes, as approximated by the limit of infinitely small temperature differences, result in zero net change in the total entropy, aligning with the ideal reversible limit dictated by the second law.

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