20.63. An object of mass m1 , specific heat capacity c1 ,...

20.63. An object of mass $m_{1}$ , specific heat capacity $c_{1}$ , and temperature $T_{1}$ is placed in contact with a second object of mass $m_{2},$ specific heat capacity $c_{2},$ and temperature $T_{2}>T_{1} .$ As a result, the temperature of the first object increases to $T$ and the temperature of the second object decreases to $T^{\prime} .$ (a) Show that the entropy increase of the system is $$ \Delta S=m_{1} c_{1} \ln \frac{T}{T_{1}}+m_{2} c_{2} \ln \frac{T^{\prime}}{T_{2}} $$ and show that energy conservation requires that $$ m_{1} c_{1}\left(T-T_{1}\right)=m_{2} c_{2}\left(T_{2}-T^{\prime}\right) $$ (b) Show that the entropy change $\Delta S$ , considered as a function of $T,$ is a maximum if $T=T,$ which is just the condition of thermodynamic equilibrium. (c) Discuss the result of part (b) in terms of the idea of entropy as a measure of disorder.

20.63. An object of mass $m_{1}$ , specific heat capacity $c_{1}$ , and temperature $T_{1}$ is placed in contact with a second object of mass $m_{2},$ specific heat capacity $c_{2},$ and temperature $T_{2}>T_{1} .$ As a result, the temperature of the first object increases to $T$ and the temperature of the second object decreases to $T^{\prime} .$ (a) Show that the entropy increase of the system is
$$
\Delta S=m_{1} c_{1} \ln \frac{T}{T_{1}}+m_{2} c_{2} \ln \frac{T^{\prime}}{T_{2}}
$$
and show that energy conservation requires that
$$
m_{1} c_{1}\left(T-T_{1}\right)=m_{2} c_{2}\left(T_{2}-T^{\prime}\right)
$$
(b) Show that the entropy change $\Delta S$ , considered as a function of $T,$ is a maximum if $T=T,$ which is just the condition of thermodynamic equilibrium. (c) Discuss the result of part (b) in terms of the idea of entropy as a measure of disorder.

Key Concepts

Thermodynamic Equilibrium

Thermodynamic equilibrium is a state in which a system’s macroscopic properties, such as temperature and pressure, are uniform and unchanging in time. At equilibrium, there are no net flows of matter or energy, and the entropy of an isolated system reaches a maximum, indicating that the system is in its most probable and disordered state.

Entropy Maximization Principle

The entropy maximization principle states that isolated systems evolve towards a state that maximizes their entropy. This principle explains why processes occur spontaneously in a certain direction and underpins the condition for thermodynamic equilibrium, where entropy is at a maximum and no further net change occurs in temperature or other thermodynamic variables.

Energy Conservation

Energy conservation, encapsulated in the first law of thermodynamics, states that energy cannot be created or destroyed but only converted from one form to another. In thermal processes, this principle ensures that the heat lost by one part of a system is exactly gained by another, maintaining the total energy balance.

Entropy

Entropy is a thermodynamic quantity that measures the degree of disorder or randomness in a system. It is central to the second law of thermodynamics, which states that the total entropy of an isolated system can never decrease over time. The concept is used to quantify how energy disperses in a process and to determine the direction of spontaneous processes.

Specific Heat Capacity

Specific heat capacity is a property of a material that describes the amount of heat required to change the temperature of a unit mass of the substance by one degree. It plays a key role in calculating energy changes and temperature variations in thermal processes and is crucial for quantifying how different objects respond to heat exchange.

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