Question

(a) How can we calculate \(\Delta S\) foran isothermal process? (b) Does \(\Delta S\) for a process depend on the path taken from the initial to the final state of the system? Explain.

Short answer

(a) To calculate ΔS for an isothermal process, we use the formula ΔS = \( \int_{i}^{f} \frac{dQ_{rev}}{T} \), where ΔS is the change in entropy, dQ_rev is the infinitesimal amount of reversible heat, and T is the constant temperature. This formula is valid only for reversible processes. (b) ΔS for a process does not depend on the path taken from the initial to the final state of the system because entropy is a state function, determined only by the current state of the system and not the path taken to reach that state.

Step-by-step solution

(a) Calculating the Change in Entropy (ΔS) for an Isothermal Process

: To calculate the change in entropy (ΔS) for an isothermal process, we can use the following formula: ΔS = \( \int_{i}^{f} \frac{dQ_{rev}}{T} \) where ΔS is the change in entropy, dQ_rev is the infinitesimal amount of reversible heat and T is the constant temperature of the process. Note that the integration goes from the initial state (i) to the final state (f). It is important to remember that this formula is only valid for a reversible process. If the process is irreversible, the entropy change of the system can still be calculated by first finding a reversible path between the same initial and final states and then using the formula above.

(b) Dependency of ΔS on the Path Taken from Initial to Final State

: The change in entropy (ΔS) for a process depends only on the initial and final states of the system, and not on the path taken between them. This is because entropy is a state function, meaning its value is determined by the current state of the system and not by the path taken to reach that state. To understand this, we should take a closer look at the formula for calculating entropy change: ΔS = \( \int_{i}^{f} \frac{dQ_{rev}}{T} \) Notice that the integral depends only on the initial (i) and final (f) states of the system and the temperature (T). There is no term for the specific path taken. Therefore, the change in entropy (ΔS) for a process does not depend on the path taken from the initial to the final state of the system.