A body of constant heat capacity Cp and a temperature Ti is...

A body of constant heat capacity $C_p$ and a temperature $T_i$ is put into contact with a reservoir at temperature $T_f$. Equilibrium between the body and the reservoir is established at constant pressure. Determine the total entropy change and prove that it is positive for either sign of $\left(T_{\mathrm{f}}-T_{\mathrm{i}}\right) / T_{\mathrm{f}}$. You may regard $\left|T_{\mathrm{f}}-T_{\mathrm{i}}\right| / T_{\mathrm{f}}<1$.

Key Concepts

Entropy

Entropy is a state function that quantifies the amount of disorder in a system. It plays a central role in the second law of thermodynamics, which states that the total entropy of an isolated system always increases for spontaneous processes. Calculating entropy changes helps to determine the direction of thermodynamic processes and their reversibility.

Heat Capacity

Heat capacity is the property of a substance that measures the amount of heat required to change its temperature by a given amount. When a system’s heat capacity is constant, it simplifies the analysis of thermal processes, as it permits straightforward integration over temperature changes when determining entropy changes.

Constant Pressure Process

A constant pressure process maintains the system’s pressure throughout the transformation, which is significant in many thermodynamic analyses. Under constant pressure, certain thermodynamic relations, such as those involving enthalpy and heat capacity at constant pressure, can be directly applied to calculate changes in energy and entropy.

Second Law of Thermodynamics

The second law of thermodynamics stipulates that for any spontaneous process, the overall entropy of the system and its surroundings increases. This principle is critical in assessing whether a given process is spontaneous, as it ensures that the total entropy change remains positive regardless of the direction of temperature change.

Entropy Change Calculation

The calculation of entropy change in a process involves integrating the ratio of heat transferred to temperature. For a body with constant heat capacity, this leads to an expression involving the natural logarithm of the ratio of final to initial temperatures. For a reservoir, the change is computed differently, often using a linear approximation due to its large size. Combining these calculations demonstrates that the total entropy change of the system plus reservoir is always positive, in accordance with the second law.

Transcript

Please give Ace some feedback