Problem 26 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) Does the entropy of the surroundings increase for spontaneous processes? (b) In a particular spontaneous process the entropy of the system decreases. What can you conclude about the sign and magnitude of \(\Delta S_{\text {surr }} ?\) (c) During a certain reversible process, the surroundings undergo an entropy change, \(\Delta S_{\text {surt }}=-78 \mathrm{~J} / \mathrm{K} .\) What is the entropy change of the system for this process?

Step-by-Step Solution

Answer

(a) For spontaneous processes, the overall entropy change in the universe must be positive. If the entropy of the system decreases but the entropy of the surroundings increases by an even larger amount, the total entropy change will still be positive, and the process will be spontaneous. (b) Since the given process is spontaneous and the entropy of the system decreases, the entropy change of the surroundings (∆S_surr) must be positive and greater in magnitude than the decrease in entropy of the system. (c) In a reversible process, the total entropy change in the universe is zero. Given ∆S_surr = -78 J/K, we can conclude that the entropy change of the system, ∆S_sys, is positive and equal in magnitude: \(\Delta S_{\text {sys}} = 78 \mathrm{~J} / \mathrm{K}\).

Step 1: Part (a) - Entropy of Surroundings in Spontaneous Processes

For spontaneous processes, the overall entropy change in the universe must be positive, meaning the total entropy should increase for the process to be spontaneous. This consists of both the entropy change in the system (∆S_sys) and the entropy change in the surroundings (∆S_surr). Thus, if the entropy of the system increases and the surroundings remain unchanged, the total entropy change will be positive and the process will be spontaneous. Similarly, if the entropy of the system decreases but the entropy of the surroundings increases by an even larger amount, then the total entropy change will still be positive, and the process will be spontaneous.

Step 2: Part (b) - Sign and Magnitude of ∆S_surr

Since the given process is spontaneous and the entropy of the system decreases, the entropy change of the surroundings (∆S_surr) must be positive and greater in magnitude than the decrease in entropy of the system. By doing so, the overall entropy change in the universe (∆S_univ = ∆S_sys + ∆S_surr) will be positive, and the process will be spontaneous.

Step 3: Part (c) - Entropy change of the system during a reversible process

In a reversible process, the total entropy change in the universe is zero. This means that the entropy change in the system (∆S_sys) is equal in magnitude and opposite in sign to the entropy change in the surroundings (∆S_surr). Since ∆S_surr is given as -78 J/K, we can conclude that the entropy change of the system, ∆S_sys, is positive and equal in magnitude: \[\Delta S_{\text {sys}} = 78 \mathrm{~J} / \mathrm{K}\]

Key Concepts

Second Law of Thermodynamics

The Second Law of Thermodynamics is a fundamental principle that gives us deep insights into how energy and entropy behave in natural processes. A key aspect of this law is that it describes the direction of spontaneous processes. According to the Second Law, for a process to be spontaneous, the total entropy of the universe must increase.
This total entropy is the sum of the entropy change in the system (\(\Delta S_{sys}\)) and the entropy change in the surroundings (\(\Delta S_{surr}\)). In mathematical terms, it can be expressed as:

• \(\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr} > 0\) for spontaneous processes.

This means that while individual parts of a process may see a decrease in entropy, the sum of all changes in the entropy of the system and its surroundings must still result in a net increase.

Entropy Change in the System

Entropy change within a system (\(\Delta S_{sys}\)) can occur in various ways during a spontaneous process. The system itself may see an increase or decrease in entropy, depending on what is happening internally.
For instance, if the system undergoes a process where it becomes more ordered, such as freezing, \(\Delta S_{sys}\) will be negative. Conversely, if the system becomes more disordered, such as during melting or vaporization, \(\Delta S_{sys}\) will be positive.
In interactions where the system gives off heat to the surroundings, like during exothermic reactions, the system's entropy might decrease. However, this is balanced by an equal or greater increase in the entropy of the surroundings, ensuring the overall entropy of the universe still rises.

Entropy Change in the Surroundings

The surroundings interact with the system and also experience a change in entropy (\(\Delta S_{surr}\)), crucial for determining the spontaneity of a process. When the system releases heat, like in exothermic reactions, the surroundings absorb this heat and their entropy increases.
Conversely, if the system absorbs heat, like in endothermic reactions, the surroundings lose heat and their entropy decreases. In the context of spontaneous processes, if the system's entropy decreases, it is necessary for the entropy change in the surroundings to increase by a larger amount.
This ensures that the total entropy change of the universe remains positive, in alignment with the Second Law of Thermodynamics. During reversible processes, \(\Delta S_{surr}\) is equal and opposite to \(\Delta S_{sys}\), resulting in a net change of zero for the universe's entropy.