Question

A closed system undergoes a process in which work is done on the system and the heat transfer \(Q\) occurs only at temperature \(T_{\mathrm{b}}\). For each case, determine whether the entropy change of the system is positive, negative, zero, or indeterminate. (a) internally reversible process, \(Q>0\). (b) internally reversible process, \(Q=0\). (c) internally reversible process, \(Q<0\). (d) internal irreversibilities present, \(Q>0\). (e) internal irreversibilities present, \(Q=0\). (f) internal irreversibilities present, \(Q<0\).

Short answer

(a) positive, (b) zero, (c) negative, (d) indeterminate, (e) positive, (f) indeterminate.

Step-by-step solution

Understanding Entropy Change

The entropy change of a system is a measure of disorder or randomness. For a closed system undergoing a process, the entropy change can be influenced by heat transfer (Q) and whether the process is internally reversible or has internal irreversibilities.

Case (a): Internally Reversible Process, Q > 0

For an internally reversible process with heat transfer Q > 0, the entropy change (ΔS) of the system is positive. The system absorbs heat, resulting in an increase in entropy.

Case (b): Internally Reversible Process, Q = 0

For an internally reversible process with no heat transfer Q = 0, the entropy change (ΔS) of the system is zero. Since there is no entropy generation due to heat transfer, the entropy remains unchanged.

Case (c): Internally Reversible Process, Q < 0

For an internally reversible process with heat transfer Q < 0, the entropy change (ΔS) of the system is negative. The system releases heat, resulting in a decrease in entropy.

Case (d): Internal Irreversibilities Present, Q > 0

For a process with internal irreversibilities and heat transfer Q > 0, the entropy change (ΔS) of the system is indeterminate. Internal irreversibilities can generate entropy, and the total entropy change depends on an additional component from these irreversibilities.

Case (e): Internal Irreversibilities Present, Q = 0

For a process with internal irreversibilities and no heat transfer Q = 0, the entropy change (ΔS) is positive. Internal irreversibilities generate entropy even in the absence of heat transfer.

Case (f): Internal Irreversibilities Present, Q < 0

For a process with internal irreversibilities and heat transfer Q < 0, the entropy change (ΔS) of the system is indeterminate. The entropy reduction due to heat loss may be countered by entropy generation due to irreversibilities.

Key concepts

entropy

Entropy is a fundamental concept in thermodynamics that measures the amount of disorder or randomness in a system. It's denoted by the symbol \( S \). When heat is added to a system, the molecules gain energy and move more randomly, increasing the entropy. Conversely, removing heat decreases the entropy as the system becomes more ordered. The change in entropy, \( \text{Δ}S \), is given by the equation \( \text{Δ}S = \frac{Q}{T} \) for a reversible process, where \( Q \) is the heat transfer and \( T \) is the absolute temperature. Understanding entropy helps in determining the feasibility and direction of a thermodynamic process.

internally reversible process

An internally reversible process is one where the system changes state in such a way that it can be reversed by an infinitesimal change without leaving any net effect on the system or its surroundings. These processes are idealized, meaning no real-world processes are perfectly reversible due to friction, unrestrained expansion, and other internal dissipative effects. When considering entropy change:

• For \( Q > 0 \): The system absorbs heat, resulting in an increase in entropy. \( \text{Δ}S > 0 \).
• For \( Q = 0 \): No heat transfer means the entropy change is zero. \( \text{Δ}S = 0 \).
• For \( Q < 0 \): The system releases heat, resulting in a decrease in entropy. \( \text{Δ}S < 0 \).

These relationships help predict the direction of entropy change for various processes.

internal irreversibilities

Internal irreversibilities refer to factors within the system that prevent it from returning to its original state without changes in its surroundings. These include friction, non-quasi-static processes, and heat transfer through a finite temperature difference. Presence of internal irreversibilities often leads to generation of entropy, denoted by \( S_{\text{gen}} \). Key points related to internal irreversibilities are:

• When \( Q > 0 \): The entropy change is indeterminate since it also includes entropy generated from irreversibilities (\( \text{Δ}S = \frac{Q}{T} + S_{\text{gen}} \)).
• When \( Q = 0 \): Entropy change is positive due to entropy generation from irreversibilities \( \text{Δ}S = S_{\text{gen}} > 0 \).
• When \( Q < 0 \): The process could lead to a decrease in entropy due to heat loss, but internal irreversibilities might result in an indeterminate total entropy change (\( \text{Δ}S = \frac{Q}{T} + S_{\text{gen}} \)).

This explains why internal irreversibilities complicate the prediction of entropy change.

heat transfer

Heat transfer is the movement of thermal energy from one object or system to another. It's a central concept in thermodynamics and can significantly influence the entropy change in a system. There are three main modes of heat transfer: conduction, convection, and radiation. Key points concerning entropy changes with heat transfer:

• When heat is added to a system (\( Q > 0 \)), the increase in energy causes an increase in entropy.
• When no heat is transferred (\( Q = 0 \)), there is no change in entropy due to thermal energy.
• When heat is removed from a system (\( Q < 0 \)), it results in a decrease in entropy.

Understanding heat transfer helps in analyzing how energy moves through systems and how it affects entropy.