(a) What sign for \(\Delta S\) do you expect when the volume of 0.200 mol of an ideal gas at \(27^{\circ} \mathrm{C}\) is increased isothermally from an initial volume of \(10.0 \mathrm{~L} ?(\mathbf{b})\) If the final volume is 18.5 L, calculate the entropy change for the process. (c) Do you need to specify the temperature to calculate the entropy change? Explain.

An isothermal process is characterized by its constant temperature throughout the entire operation. This temperature stability is key in certain thermodynamic evaluations, most notably those involving ideal gases. When a gas undergoes such a process, it can either expand or compress isothermally. In both cases, since the temperature remains unchanged, the internal energy of an ideal gas also remains constant.

During an isothermal expansion, a gas does work on its surroundings because it increases in volume. This applies to our example where the volume of an ideal gas expands from 10.0 L to 18.5 L. Since this process is done without a change in temperature, we can deduce that the heat energy must have flowed into the gas, allowing the expansion to be an isothermal one. Understanding this concept is vital when calculating the entropy change for the process.

The ideal gas law is a fundamental relation in chemistry and physics, represented by the equation \( PV=nRT \), where \( P \) stands for pressure, \( V \) is the volume, \( n \) signifies the amount of substance (moles), \( R \) is the gas constant, and \( T \) denotes the temperature. This equation describes the state of an ideal gas—a theoretical gas in which there are no intermolecular attractions and whose particles occupy negligible space.

The utility of the ideal gas law is extensive as it can predict the behavior of a gas under different conditions, as long as it behaves ideally, which most gases approximately do under normal conditions. It is crucial in deriving other relationships, such as the one used for entropy change during isothermal processes. The equality between the internal energy variations by isothermal expansion or compression and heat exchange with the surroundings guides us to assess entropy without the need to know the specific temperature, assuming ideal gas behavior.

The concept of entropy is central to the second law of thermodynamics and provides essential insight into the spontaneity of processes. Entropy, represented by \( S \), is a measure of a system's disorder or randomness. Spontaneous processes tend to lead to an increase in the total entropy of the universe, which includes the entropies of the system and its surroundings.

In our textbook example, the expansion of a gas without energy input (isothermal process) results in an increase in entropy since the gas particles have more space to disperse, hence more ways to arrange themselves. This means there's an increase in the number of possible microstates for the system, which corresponds to a larger entropy value. Calculating the entropy change can thus predict the spontaneity of a process, where a positive entropy change suggests a spontaneous process under constant temperature and pressure conditions. The direct connection between entropy change and spontaneity is a fundamental principle in thermodynamics that governs a multitude of natural and industrial processes.