The temperature dependence of the heat capacity of a substance is commonly written in the form \(C_{P, \mathrm{~m}}=a+b T+\) \(c / T^{2}\), with \(a, b\), and \(c\) constants. Obtain an expression for the entropy change when the substance is heated from \(T_{1}\) to \(T_{2}\). Evaluate this change for graphite, for which \(a=16.86 \mathrm{~J} \cdot \mathrm{K}^{-1} \cdot \mathrm{mol}^{-1}, b=4.77 \mathrm{~mJ} \cdot \mathrm{K}^{-2} \cdot \mathrm{mol}^{-1}\), and \(c=-8.54 \times 10^{5} \mathrm{~J} \cdot \mathrm{K} \cdot \mathrm{mol}^{-1}\), heated from \(298 \mathrm{~K}\) to 400 . \(\mathrm{K}\). What is the percentage error in assuming that the heat capacity is constant with its mean value in this range?

Heat capacity of a substance is a measure of how much heat energy it takes to raise the temperature of that substance by a given amount. However, it's important to understand that this heat capacity can itself change as the temperature changes.

In thermodynamics, we often encounter the formula for heat capacity dependent on temperature as follows:
\[ C_{P, \mathrm{m}}=a+bT+\frac{c}{T^2} \]
where \(a\), \(b\), and \(c\) are constants that represent the substance's specific heat properties. This relationship reflects the reality that substances may require more or less energy to heat up as they get warmer or cooler.

Understanding how heat capacity changes with temperature is crucial when calculating properties like entropy, which relies on integrating the heat capacity over a temperature range. Misjudging this relationship can lead to inaccuracies in thermodynamic calculations, such as those for entropy change.

Entropy, a fundamental concept in thermodynamics, can be thought of as a measure of disorder or randomness in a system. Calculating the change in entropy (\(\Delta S\)) when a substance is heated involves integrating the heat capacity over a temperature range.

When temperature dependence is considered, the entropy change is given by the integral: \[ \Delta S = \int_{T_1}^{T_2} \frac{C_{P,\mathrm{m}}}{T}dT \]
This integral involves breaking down the heat capacity function into components that individually depend on temperature, such as the constants \(a\), \(b\), and \(c\), differentially affecting the evaluation depending on how they are paired with the temperature variable, either as a multiple of \(T\) or an inverse power of \(T\).

Actually integrating these components correctly is necessary for an accurate calculation of the entropy change. Errors or oversimplifications in this process can lead to incorrect entropy values, which may significantly impact the analysis of thermodynamic processes.

In practice, to simplify calculations, the heat capacity of a substance is sometimes assumed to be constant over a temperature range. However, as we've seen, heat capacity can depend on temperature.

When performing thermodynamic calculations, assuming a constant heat capacity (\(\overline{C_{P,\mathrm{m}}}\)) can introduce a percentage error when compared to the accurate variable heat capacity method. The percentage error quantifies the discrepancy and is calculated as:
\[ \text{Percentage Error} = \left|\frac{\Delta S_{constant} - \Delta S_{actual}}{\Delta S_{actual}} \right| \times 100\% \]
where \(\Delta S_{constant}\) is the entropy change using the mean heat capacity and \(\Delta S_{actual}\) is the entropy change with the temperature-dependent heat capacity. This metric helps to evaluate the trade-off between computational simplicity and accuracy. In academic and industrial applications, understanding and minimizing this error is essential to ensure reliability of the thermodynamic analyses.

Enthalpy is another thermodynamic property closely related to heat capacity and entropy. It represents the total heat content of a system and is often used to calculate the heat exchange in chemical reactions and phase changes.

In thermodynamic cycles, the changes in enthalpy (\(\Delta H\)) can be related to the heat capacity at constant pressure (\(C_{P,\mathrm{m}}\)) and temperature change. The dependency of heat capacity on temperature becomes crucial when calculating enthalpy changes for processes that occur over a wide range of temperatures.

Both enthalpy and entropy are state functions, meaning they are determined by the state of the system and not by how the system reached that state. A precise understanding of heat capacity's temperature dependence is instrumental in properly calculating changes in both entropy and enthalpy, ensuring accurate predictions of energy transfers and equilibrium states in various thermodynamic scenarios.