Problem 13 - Chemistry The Science in Context

Question

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

Step-by-Step Solution

Answer

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

Step 1: Check combination a

In the first case, we have: \(\Delta S_{sys} > 0\), \(\Delta S_{surr} > 0\), and \(\Delta S_{univ} > 0\) According to the second law of thermodynamics, we know that \(\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr}\) Since both the system and surrounding entropy changes are positive, their sum will also be positive. Therefore, this combination is mathematically possible.

Step 2: Check combination b

In the second case, we have: \(\Delta S_{sys} > 0\), \(\Delta S_{surr} < 0\), and \(\Delta S_{univ} > 0\) Again, according to the second law of thermodynamics, we know that \(\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr}\) Since the system entropy change is positive but the surrounding entropy change is negative, we can't determine whether the total entropy change is positive or negative without more information. However, it is mathematically possible for this combination to occur if the positive entropy change in the system is greater than the negative entropy change in the surroundings.

Step 3: Check combination c

In the third case, we have: \(\Delta S_{sys} > 0\), \(\Delta S_{surr} > 0\), and \(\Delta S_{univ} < 0\) According to the second law of thermodynamics, we know that \(\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr}\) Since both the system and surrounding entropy changes are positive, their sum will also be positive, making the combination impossible. This combination contradicts the second law of thermodynamics and is therefore not mathematically possible. In conclusion, combinations a and b are mathematically possible, while combination c is not.

Key Concepts

Second Law of Thermodynamics

The Second Law of Thermodynamics is a fundamental principle that provides insight into the direction of processes in nature. It states that the total entropy of an isolated system can never decrease over time. Entropy can be understood as a measure of disorder or randomness.
For any spontaneous process, the total change in the entropy of the universe, which includes both the system and the surroundings, will always be positive or zero.
This law highlights the fact that energy transactions are not 100% efficient and that some energy is always converted into a less useful form, typically heat, increasing the disorder.

• It tells us about the irreversibility of natural processes, meaning once a process happens, it cannot be undone completely without changing something else.
• The second law is crucial for understanding why energy tends to spread out and why systems naturally progress towards thermodynamic equilibrium, the state of maximum entropy.

The second law provides a way to calculate the possible entropy changes in a process, forming the backbone of thermodynamic analysis.

Entropy Changes

When discussing entropy changes, we look at how the order and disorder in a system evolve. Entropy changes can happen both in the system and in its surroundings. Entropy changes (\(\Delta S\)) are crucial in evaluating the feasibility of a process under thermodynamic laws.

• For a process where \(\Delta S_{sys} > 0\) and\(\Delta S_{surr} > 0\), the sum \(\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr} > 0\), which indicates that the process is possible and spontaneous.
• If \(\Delta S_{sys} > 0\) and \(\Delta S_{surr} < 0\), the total entropy \(\Delta S_{univ}\) can still be positive provided the system’s entropy increase overrides the decrease in the surroundings.
• Conversely, if \(\Delta S_{sys} > 0\), \(\Delta S_{surr} > 0\), but \(\Delta S_{univ} < 0\), it violates the second law, as both terms being positive can't logically combine to yield a negative total, showing the process is impossible.

These changes are evaluated in thermodynamic analyses to determine whether a reaction or process can occur spontaneously.

Mathematical Possibility in Thermodynamics

In thermodynamics, determining the mathematical possibility of a process involves analyzing how the changes in entropy align with the laws that govern energy and matter behavior.

• The entropy changes in processes must be assessed to ensure they respect the second law of thermodynamics. This means evaluating if the total entropy of the universe increases. If it doesn't, then the process is not feasible.
• For example, if both the system and surroundings experience an entropy increase, their sum, \(\Delta S_{univ}\) , must also be positive, supporting mathematical possibility.
• If one experiences an increase and the other a decrease, it demands careful calculation to see if the resultant entropy change is non-negative.

This mathematical scrutiny helps ensure that theoretical thermodynamic calculations predict processes that can physically happen, guiding scientific and engineering practices.