Problem 14 - Chemistry The Science in Context

Question

Which of the following combinations of entropy changes for a process are mathematically possible? a. \(\Delta S_{\text {sys }}<0, \Delta S_{\text {surr }}>0, \Delta S_{\text {univ }}>0\) b. \(\Delta S_{\text {rys }}<0, \Delta S_{\text {surr }}<0, \Delta S_{\text {univ }}>0\) c. \(\Delta S_{\text {sys }}<0, \Delta S_{\text {surr }}>0, \Delta S_{\text {univ }}<0\)

Step-by-Step Solution

Answer

a) ΔS_sys < 0, ΔS_surr > 0, and ΔS_univ > 0 b) ΔS_sys < 0, ΔS_surr < 0, and ΔS_univ > 0 c) ΔS_sys < 0, ΔS_surr > 0, and ΔS_univ < 0 Answer: Combinations (a) and (b) are mathematically possible, while combination (c) is not mathematically possible.

Step 1: According to the second law of thermodynamics, the entropy of an isolated system should always increase. Hence, for a process to be possible, the entropy change of the universe \((\Delta S_{\text {univ}})\) should be greater than or equal to zero. #Step_2: Determine the mathematical possibility of combination (a)#

In combination a, we have \(\Delta S_{\text {sys}} < 0\), \(\Delta S_{\text {surr}} > 0\), and \(\Delta S_{\text {univ}} > 0\). Since the entropy change of the universe is positive \((\Delta S_{\text {univ}} > 0)\), this combination is mathematically possible according to the second law of thermodynamics. #Step_3: Determine the mathematical possibility of combination (b)#

Step 2: In combination b, we have \(\Delta S_{\text {sys}} 0\). We can observe that both the system and surroundings have negative entropy changes which make this combination unlikely. However, the entropy change of the universe is positive \((\Delta S_{\text {univ}} > 0)\), which satisfies the second law of thermodynamics. Therefore, this combination is also mathematically possible. #Step_4: Determine the mathematical possibility of combination (c)#

In combination c, we have \(\Delta S_{\text {sys}} < 0\), \(\Delta S_{\text {surr}} > 0\), and \(\Delta S_{\text {univ}} < 0\). In this scenario, the entropy of the universe decreases \((\Delta S_{\text {univ}} < 0)\). This violates the second law of thermodynamics, as the entropy of the universe should either increase or remain constant. Hence, this combination is not mathematically possible. In conclusion, combinations (a) and (b) are mathematically possible, while combination (c) is not mathematically possible.

Key Concepts

Second Law of Thermodynamics

The Second Law of Thermodynamics is a fundamental principle that describes the behavior of energy and entropy in physical processes. It states that in an isolated system, the total entropy can never decrease over time. Entropy, which can be thought of as a measure of disorder or randomness, must either increase or stay the same.
This law doesn't just apply to physical systems that are perfectly isolated. It's also a guide for understanding many real-world processes, like why heat flows from hot to cold or why certain chemical reactions occur spontaneously.
It's important to remember these key ideas:

• Entropy tends to increase, signaling that things naturally progress toward disorganization.
• Energy conversions are not 100% efficient; there will always be some energy "lost" as waste heat, increasing entropy.
• Reversible processes are idealized situations where total entropy remains constant, though in practice, they don't occur naturally.

Entropy Change of Universe

The entropy change of the universe (\(\Delta S_{univ}\)) combines both the entropy changes of a system and its surroundings. According to the second law, this overall change must be greater than or equal to zero for a process to be feasible.
When examining the universe during a process, it's crucial to add up individual contributions:

• \(\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr}\)
• A positive \(\Delta S_{univ}\) indicates irreversible, naturally occurring processes.
• Zero \(\Delta S_{univ}\) characterizes reversible processes, which are idealized and don’t occur spontaneously.
• Negative \(\Delta S_{univ}\) would denote a process that violates the second law – these are impossible in nature.

Comprehending this concept helps explain why certain processes don't happen instantaneously or spontaneously and solidifies the importance of accounting for the whole system when studying thermodynamics.

Entropy Change of System and Surroundings

Analyzing the entropy change of both a system and its surroundings is crucial in understanding whether a process can feasibly occur. Each part of the entropy equation plays a distinct role:

• \(\Delta S_{sys}\): This represents the change in entropy of the system directly involved in the process. For example, a hot cup of coffee cooling down.
• \(\Delta S_{surr}\): This represents the change in entropy of the surroundings, which encompasses everything outside the system. Using the coffee example, this includes the room's air absorbing heat.

A feasible process requires that the total entropy change of the universe (system plus surroundings) is non-negative.
Sometimes, a decrease in the system's entropy is allowed provided the surroundings increase in entropy enough to ensure the total \(\Delta S_{univ}\) is positive.

• This allows for situations where order or energy is localized at the expense of greater disorder elsewhere.
• Even processes that appear orderly at first glance can contribute to increased universal entropy.

Understanding these interactions is key for predicting and explaining the direction and possibility of processes in thermodynamics.