Given that \(s_{m}^{\circ}=29.79 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\) for bismuth at \(100 \mathrm{~K}\) and the following tabulated heat capacities data (D.G. Archer, \(J\). Chem. Eng. Data 40 , 1015 (1995)), compute the standard molar entropy of bismuth at \(200 \mathrm{~K}\). $$ \begin{array}{lccccccl} \text { TIK } & 100 & 120 & 140 & 150 & 160 & 180 & 200 \\ C_{\mathrm{Am}} /\left(\mathrm{K} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right) & 23.00 & 23.74 & 24.25 & 24.44 & 24.61 & 24.89 & 25.11 \end{array} $$ Compare the value to the value that would be obtained by taking the heat capacity to be constant at \(24.44 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\) over this range.

Entropy is a measure of the disorder or randomness in a system. The standard molar entropy, denoted as \( s_m^\circ \), quantifies this disorder for a substance at a standard state.

When calculating entropy, one must understand that entropy is a state function, meaning it depends only on the initial and final states of a system and not on the path taken. For a substance such as bismuth, we can start with a known standard molar entropy at a certain temperature and calculate its entropy at another temperature by integrating the heat capacity with respect to temperature.

To perform this entropy calculation using numerical data from a table, we apply numerical integration techniques like the trapezoidal rule, which approximates the area under the heat capacity versus temperature curve. This rule is particularly useful when dealing with tabulated heat capacities at discrete intervals, giving a practical way to estimate the change in entropy.

Numerical integration is a method used to approximate the integral or total area under a curve. This is especially useful when dealing with complex or discrete data where analytical integration is not feasible.

In the context of thermodynamics, the trapezoidal rule is often used to estimate the integration of heat capacity over a range of temperatures. This rule involves splitting the temperature interval into smaller segments, calculating the average heat capacity for each segment, and then computing an area (like that of a trapezoid), which represents the change in entropy.

By summing up the areas of all trapezoids, we gain an approximation of the integral. This approach is key to calculating entropy changes from empirical data, and its accuracy improves with smaller temperature intervals and a higher density of heat capacity values.

Heat capacity \( (C_p) \) is an intrinsic property of a substance that describes how much heat energy is required to raise the temperature of a given amount of the substance by one degree. It is often expressed per mole for a substance, which is why we refer to the molar heat capacity \( (C_{p,m}) \).

For entropy calculations, we are concerned with how the heat capacity changes with temperature, since the change in entropy is directly tied to how a substance absorbs heat. As shown in the table for bismuth, heat capacity varies with temperature, suggesting that more or less heat may be required at different temperatures to change the entropy of the system.

While it's possible to assume a constant heat capacity for simplicity, it leads to an approximation that may not accurately reflect the true change in entropy. This is why the given exercise compares the results of a constant heat capacity assumption to a calculation using variable values of heat capacity.

A state function is a property of a system that only depends on the current state of the system, not on how it arrived at that state. Entropy is an example of a state function, as is energy, temperature, and pressure.

For state functions, it doesn't matter what path or process the system takes as it changes from one state to another; the difference in the state function will always be the same. This concept is crucial in thermodynamics because it allows us to use tabulated values, like standard molar entropies at specific temperatures, and paths of integration, like the trapezoidal method, to find entropy changes without needing to consider the exact pathway the system took between the states.

When we are comparing different methods of entropy calculation, understanding entropy as a state function ensures that the only variation in our calculated results comes from the methodology or data used, not from the physical properties of the process itself.