Question

(a) Express the second law of thermodynamics in words. (b) If the entropy of the system increases during a reversible process, what can you say about the entropy change of the surroundings? (c) In a certain spontaneous process the system undergoes an entropy change, \(\Delta S=42 \mathrm{~J} / \mathrm{K}\). What can you conclude about \(\Delta S_{\text {surr }} ?\)

Short answer

(a) The second law of thermodynamics states that the total entropy of a closed system and its surroundings can only increase or remain the same, but never decrease. (b) In a reversible process, if the entropy of the system increases, the entropy change of the surroundings must decrease by the same amount, making \(\Delta S_{\text{surr}} < 0\). (c) For a spontaneous process with \(\Delta S_{\text {sys}} = 42 \mathrm{~J/K}\), the entropy change of the surroundings, \(\Delta S_{\text{surr}}\), must be positive and larger than \(42 \mathrm{~J/K}\).

Step-by-step solution

(a) Second Law of Thermodynamics in Words

The second law of thermodynamics states that in any natural process, the total entropy of a closed system and its surroundings can only increase or remain the same, but it can never decrease. In simpler terms, it means that any process tends to move from a state of order towards a state of disorder.

(b) Entropy change of the surroundings during a reversible process

In a reversible process, the system and its surroundings undergo infinitesimally small changes such that the total entropy change, \(\Delta S_{\text {total}}\), is zero. This means that if the entropy of the system, \(\Delta S_{\text {sys}}\), increases during a reversible process (i.e., \(\Delta S_{\text {sys}} > 0\)), the entropy change of the surroundings, \(\Delta S_{\text {surr}}\), must decrease by the same amount, making \(\Delta S_{\text {surr}} < 0\). This ensures that the total entropy change remains zero.

(c) Entropy change of the surroundings during a spontaneous process

We are given the entropy change of the system, \(\Delta S_{\text {sys}} = 42 \mathrm{~J/K}\). For a spontaneous process, the second law of thermodynamics states that the total entropy change must be positive, i.e., \(\Delta S_{\text {total}} = \Delta S_{\text {sys}} + \Delta S_{\text {surr}} > 0\). Since we know that \(\Delta S_{\text {sys}} = 42 \mathrm{~J/K}\), we can conclude that the entropy change of the surroundings, \(\Delta S_{\text {surr}}\), must be positive and larger than \(42 \mathrm{~J/K}\), in order for the total entropy change to be positive.

Key concepts

Entropy

Imagine a child's playroom at the end of a busy day; toys are scattered in a disorderly fashion, compared to the neat and tidy state earlier. In thermodynamics, this idea relates to the concept of entropy, a measurement of disorder or randomness in a system. The second law of thermodynamics tells us that entropy in a closed system tends to increase over time, much like the playroom getting progressively messier. Mathematically, entropy is represented by the symbol 'S'.

This increase of disorder is not just about mess and clutter; on a molecular level, it means that particles will spread out and energy will disperse until there's an even distribution — like a perfume diffusing through the air until its scent is equally present everywhere. By understanding entropy, students get a grasp on why certain processes occur, like ice melting into water, and others, like spontaneously refreezing, do not happen on their own.

Reversible Process

To continue with our analogy, if that busy day in the playroom was instead meticulously controlled - each toy being moved in such a way it could be exactly returned to its original position - we would have what thermodynamics calls a reversible process. Imagine being able to reverse the day smoothly, with each action and movement perfectly undone, leading the room back to its initial orderly state.

In the real world, truly reversible processes are an idealization because they require changes to occur infinitely slowly and without any loss of energy to friction or other dissipative forces. But why is this concept important? Because these ideal reversible processes define a limit on the maximum efficiency that can be achieved in any energy conversion system, like an engine or a refrigerator. Understanding reversible processes helps students appreciate the energy efficiently limits and why no real process can be 100% efficient.

Spontaneous Process

A spontaneous process is the thermodynamic equivalent of letting gravity do the work - when you let go of a book, it falls. It doesn't need additional prompting or energy. In a similar way, a spontaneous process in thermodynamics is a reaction that, once started, proceeds on its own without any input of external energy. It's like the playroom becoming messier without anyone actively contributing to the chaos.

Now the key point here is not speed; 'spontaneous' doesn't necessarily mean 'instantaneous'. Some processes, like the rusting of iron, are spontaneous but can take a long time to become noticeable. What distinguishes a spontaneous process is the total increase in entropy (including both the system and the surroundings). If after you've tidied the playroom the toys can scatter themselves (increasing entropy) without your help, the process could be considered spontaneous. This core concept is crucial for students to recognize that nature favors processes that contribute to the overall disorder.

Total Entropy Change

Understanding the total entropy change of a system and its surroundings is akin to monitoring the overall tidiness of two interconnected playrooms. If you tidy up one room (system) by moving the toys to the other (surroundings), you haven't actually decreased the total mess, just displaced it. In thermodynamics terms, this displacement can't lead to a total entropy decrease for a natural process. The sum of the entropy change in the system and the entropy change in the surroundings is what we call the total entropy change, represented by \(\Delta S_{\text{total}}\).

Drawing from the second law of thermodynamics, if the total entropy change is positive, the process can proceed spontaneously without any additional input. By learning about total entropy change, students get to understand how energy transformations in a real-world scenario, like in an engine, can never be perfectly efficient because some energy always spreads out, increasing the total entropy of the universe.