The heat capacity of chloroform (trichloromethane, \(\mathrm{CHCl}_{3}\) ) in the range \(240 \mathrm{~K}\) to \(330 \mathrm{~K}\) is given by \(\mathrm{C}_{\mathrm{p}, \mathrm{m}} /\left(\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right)=91.47+7.5 \times 10^{-2}(T / \mathrm{K})\). In a particular experiment, \(1.00 \mathrm{~mol} \mathrm{CHCl}_{3}\) is heated from \(273 \mathrm{~K}\) to \(300 \mathrm{~K}\). Calculate the change in molar entropy of the sample.

Heat capacity is an intrinsic property of matter that measures the amount of heat required to change the temperature of a substance by a certain amount. When heat is added to or removed from a substance, its temperature changes according to the substance's heat capacity. The higher the heat capacity, the more heat is needed to change the temperature of the substance.

Different substances have different heat capacities due to their unique molecular structures and bonding. The heat capacity can be affected by the temperature, pressure, and phase of the substance. It's therefore an essential factor in understanding thermal systems and their behavior in various conditions.

In our chloroform example, knowing its heat capacity helps us determine the amount of energy required to heat a mole of chloroform from one temperature to another. The heat capacity, which can vary with temperature, provides a bridge between thermal energy and temperature change in a substance.

The molar heat capacity of a substance is the heat capacity per mole of that substance. It is often represented by the symbol \(C_{p,m}\) when measured at constant pressure, and is crucial for quantifying heat transfer in chemical reactions and physical changes.

The molar heat capacity formula for chloroform given in the textbook exercise, \(C_{p,m} = 91.47 + 7.5 \times 10^{-2}(T/K)\), shows that chloroform's heat capacity is dependent on temperature (\(T\)). This formula helps you calculate how much energy, in Joules, is needed to raise the temperature of one mole of chloroform by one Kelvin.

Understanding and using the correct molar heat capacity formula is vital because it allows for precise calculations of temperature-related entropy changes, which is a cornerstone for thermodynamics studies.

Entropy is a measure of the disorder or randomness within a system, and is one of the central concepts of thermodynamics. One important aspect of entropy is that it's a state function. This means that entropy is determined by the current state of the system, not by the path the system took to reach that state.

Entropy as a state function is analogous to elevation in geography; no matter how you climb a hill, the elevation at the top is the same. Similarly, for any thermodynamic system, the change in entropy between two states is independent of the transformation process. This characteristic allows us to calculate the change in molarity entropy (\(\Delta S_{m}\)) by integrating the heat capacity over a temperature range, assuming a reversible path, as done in the chloroform heating example.

Thermodynamic integration is a mathematical technique used to calculate changes in functions of state, such as entropy, over a specific process. When dealing with heat capacities and changes in temperatures, integration allows us to find the total entropy change for a process, even when the heat capacity varies with temperature.

In our exercise, the provided heat capacity formula for chloroform requires integration over the temperature range of interest to determine the change in entropy. The process involves setting up a definite integral that incorporates the variable molar heat capacity with respect to temperature. This approach uses calculus to embody the nature of entropy as a state function, yielding a precise value for the entropy change that results from heating the chloroform sample from one temperature to another.

Thermodynamic integration is essential in physical chemistry because it provides an exact quantitative answer to changes in state functions, which is fundamental for predicting system behavior and conducting thermodynamic analysis.