Question

Indicate whether each statement is true or false. (a) $\mathrm{\Delta }S$ for an isothermal process depends on both the temperature and the amount of heat reversibly transferred. (b) $\mathrm{\Delta }S$ is a state function. (c) The second law of thermodynamics says that the entropy of the system increases for all spontaneous processes.

Short answer

(a) True: $\mathrm{\Delta }S$ for an isothermal process does depend on both the temperature and the amount of heat reversibly transferred ($\mathrm{\Delta }S=\frac{Q_{reversible}}{T}$). (b) True: $\mathrm{\Delta }S$ is a state function, as it depends only on the initial and final states and not on the path taken. (c) False: The second law of thermodynamics states that the total entropy (system and surroundings) should increase or remain the same, not just the entropy of the system.

Step-by-step solution

Recall the formula for entropy change

For an isothermal reversible process, the entropy change of a system is given by: $$\mathrm{\Delta }S=\frac{Q_{reversible}}{T}$$ here, $\mathrm{\Delta }S$ represents the change in entropy, $Q_{reversible}$ is the amount of heat transferred reversibly, and $T$ is the constant temperature.

Analyze the statement

Since the entropy change ($\mathrm{\Delta }S$) in our formula depends on both the amount of reversible heat transferred ($Q_{reversible}$) and the constant temperature ($T$), the statement (a) is indeed true. Statement (b): $\mathrm{\Delta }S$ is a state function.

Define a state function

A state function is a property that only depends on the current state of a system and is independent of the path taken to reach that state.

Analyze if entropy is a state function

Entropy is a state function, as it does not depend on the path taken to reach the current state of the system. The change in entropy ($\mathrm{\Delta }S$) only depends on the initial and final states, and not on the process or path taken. Therefore, statement (b) is true. Statement (c): The second law of thermodynamics says that the entropy of the system increases for all spontaneous processes.

Recall the second law of thermodynamics

The second law of thermodynamics states that in any spontaneous process, the total entropy of a system and its surroundings always increases or remains the same. Mathematically, for a spontaneous process, $\mathrm{\Delta }{S}_{total}\ge 0$.

Analyze the statement

The statement (c) is not entirely accurate because the second law of thermodynamics states that the total entropy (system and surroundings) should increase or remain the same, not just the entropy of the system. Therefore, statement (c) is false. To summarize, Statement (a): True Statement (b): True Statement (c): False

Key concepts

Isothermal Process

An isothermal process is one where the temperature of the system remains constant throughout. In thermodynamics, this constant temperature allows us to evaluate how other properties of the system, such as volume and pressure in an ideal gas, might change. One helpful analogy for understanding isothermal processes involves envisioning a balloon that expands or contracts slowly enough that the heat transfer between the balloon and its surroundings keeps the temperature inside the balloon steady.

During an isothermal expansion or compression of an ideal gas, the amount of work done by or on the gas is related to the heat exchanged with the surroundings. By the definition of an isothermal process, the formula for entropy change, ewline ewline is essential. It's worth noting the universality of this equation for reversible processes, which highlights that even though the temperature is constant, the entropy can still change depending on the amount of heat transferred reversibly, which is in line with the solution presented in statement (a).

In simple terms, if you're sitting in a room at a comfortable, steady temperature and studying, when you work on your exercises, the effort (or 'heat') you 'transfer' to the task—your study—can increase with the difficulty (or 'entropy'), though the room's temperature—your learning environment—remains the same.

State Function

A state function in thermodynamics refers to a property whose value is determined exclusively by the current state of a system, regardless of how that state was achieved. Some examples of state functions include internal energy, enthalpy, and indeed, entropy. A helpful way to remember this is to think of your geographical location as a 'state function'—no matter whether you've taken a direct flight or a meandering road trip, your destination doesn't care; it's the location itself that matters.

Entropy, denoted as ewline , is particularly precious because it gives us a quantitative measure of disorder or randomness in a system. It's not concerned with the process—rapid heat exchange, slow heating, a series of complex reactions—it's the result that counts, the initial and final states. Thus, confirming statement (b) as true aligns with the idea of a state function. This concept is critical to understand, as it underlines fundamental laws of thermodynamics and tells us entropy change is intrinsic to the system's properties, bypassing the details of the transition.

Second Law of Thermodynamics

The second law of thermodynamics is a fundamental principle that dictates the behavior of energy and entropy in natural processes. You can think of it as the universe's way of saying, 'There are rules to this game.' One of these rules is that in a closed system, entropy—often interpreted as disorder—will either increase or remain the same; it never decreases on its own.

This law explains natural occurrences like why heat flows from hot to cold, not the other way around, and why highly ordered energy (like electricity) tends to become less ordered energy (like heat) over time. A common misconception, as shown in statement (c), is that it might lead someone to believe that the law says 'the entropy of the system increases for all spontaneous processes.' The clarification is that it pertains to the sum of the entropy changes in the system and its surroundings, ewline. Therefore, a process can have a decrease in the system's entropy, provided there is a greater increase in the entropy of the surroundings, thus sticking to the law that ewline totalewline. As a result, it is crucial to remember that the universe's entropy is the ultimate measure, which is why the statement (c) is false when considering the system in isolation.