Thermodynamics: Entropy and Clausius Theorem

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Questions and Answers

What does the change of entropy (dS) primarily depend on?

The volume of the system

The amount of work done on the system

The temperature of the system (T) (correct)

The pressure of the system

Which statement correctly describes a reversible process?

It allows for interaction with surroundings

It generates irreversible heat

It can retrace its intermediate states (correct)

It occurs at a rapid rate

Which of the following best describes an extensive parameter?

It depends on the size or extent of the system (correct)

It is constant under varying conditions

It remains unchanged when the system is divided

It measures the disorder in a system

What happens to entropy when heat is added to a system?

Entropy increases Signup and view all the answers

In thermal equilibrium, what is the change in entropy of the system?

Zero Signup and view all the answers

What must be true for a system to reach maximum probability?

The system must be isolated from its surroundings Signup and view all the answers

What does S = k log W indicate about entropy (S) relative to the number of arrangements (W)?

Entropy increases as W increases Signup and view all the answers

Which of the following events would cause an increase in a system’s entropy?

Heating the system Signup and view all the answers

What does entropy describe in a thermodynamic system?

The ability of the system to do work Signup and view all the answers

According to Clausius theorem, what is the net change in entropy for a reversible cycle?

Equal to zero Signup and view all the answers

During an isothermal expansion in a Carnot cycle, how is the working substance's entropy affected?

It increases by (Q1/T1) Signup and view all the answers

What occurs during adiabatic expansion in terms of entropy?

No change in entropy Signup and view all the answers

How does the efficiency of an irreversible cycle compare to that of a reversible cycle?

It is less than a reversible cycle Signup and view all the answers

What is the net change in entropy of the system in an irreversible cycle?

It is greater than zero Signup and view all the answers

In an irreversible process, what happens to the entropy of the source when heat is absorbed?

It decreases Signup and view all the answers

What does the second law of thermodynamics state in terms of entropy?

The total entropy of an isolated system can never decrease Signup and view all the answers

What does the formula $S2 - S1 = ∫ (dQ/T)$ represent?

The change in entropy of a system along a reversible path Signup and view all the answers

Which statement is consistent with the second law of thermodynamics?

Entropy of a closed system always increases or remains constant Signup and view all the answers

According to Clausius's statement, what cannot exist?

A perfect refrigerator without external work Signup and view all the answers

What does the inequality $dQ/T ext{ } >= 0$ signify?

The change in entropy of a reversible process is always non-negative Signup and view all the answers

What does $dQ = dU + dW$ represent in thermodynamics?

The first law of thermodynamics linking heat, internal energy, and work Signup and view all the answers

How is the change in entropy of a gas calculated when heat $dQ$ is added reversibly?

$dS = dQ/T$ Signup and view all the answers

What condition is required for the equation $dS = dQ/T$ to hold true?

The process must be reversible Signup and view all the answers

What does the term 'entropy' measure in a thermodynamic system?

The disorder or randomness of the system Signup and view all the answers

What happens to entropy during irreversible processes?

Entropy increases Signup and view all the answers

In which part of Carnot's cycle does entropy remain constant?

Adiabatic Expansion Signup and view all the answers

What does Nernst's theorem state regarding absolute zero?

Entropy must be constant at absolute zero Signup and view all the answers

What is the efficiency formula for Carnot's heat engine?

η = (Q1 - Q2) / Q1 Signup and view all the answers

Which of the following statements is true regarding entropy and temperature?

Entropy increases with an increase in temperature Signup and view all the answers

What is the net change in entropy during one complete Carnot's cycle?

Zero Signup and view all the answers

What does the Kelvin-Planck statement imply?

Heat cannot flow from a cooler body to a hotter body without additional work Signup and view all the answers

Which of the following is NOT a consequence of the Second Law of Thermodynamics?

Heat can flow spontaneously from cold to hot Signup and view all the answers

Study Notes

Concept of Entropy

Entropy is a thermodynamic variable that changes between the initial and final states of a thermodynamic process.
It is a measurable property of the system that describes its ability to do work.
Entropy is fundamental to understanding the operation of heat engines.

Clausius Theorem

The Clausius theorem states that in any reversible cycle, the net change in entropy is zero.
The theorem is demonstrated using the Carnot cycle, which consists of two isothermal and two adiabatic processes.
During isothermal processes, heat is transferred, resulting in a change in entropy, but these changes cancel out over the complete cycle.
Adiabatic processes, where no heat is exchanged, contribute zero change in entropy.

Change of Entropy in an Irreversible Process

In an irreversible cycle, the efficiency is less than a reversible cycle operating between the same temperatures.
This means that the heat rejected to the sink (Q2/T2) is greater than the heat absorbed from the source (Q1/T1).
Even though the working substance returns to its initial state, the net change in entropy of the system (source, sink, and working substance) is positive, indicating an overall increase in entropy.

Second Law of Thermodynamics in terms of Entropy

The change in entropy of a system during a reversible process is given by the integral of dQ/T, where dQ is the amount of heat absorbed or rejected at temperature T.
The second law of thermodynamics mathematically states dS = dQ/T.
The principle of entropy states that dQ/T >= 0, meaning entropy always increases or remains constant in a process.
This principle encompasses both the Kelvin-Planck and Clausius statements of the second law.
The Kelvin-Planck statement states that a perfect heat engine cannot convert all heat into work, as this would violate the entropy principle.
The Clausius statement states that a perfect refrigerator cannot exist without external work, as it would also violate the entropy principle.

Entropy of a Perfect Gas

The change in entropy of 1 gram of a perfect gas undergoing a reversible process can be calculated using the formula dS= dQ/T, where dQ is the amount of heat supplied at temperature T.
dQ can be expressed in terms of internal energy (dU) and work done against external pressure (dW).

Entropy as a Measure of Disorder

Entropy (S) represents the extent of disorder or randomness in a system.
It is an extensive parameter, meaning its value changes when the system is partitioned.

Third Law of Thermodynamics

The third law of thermodynamics states that the entropy of a perfect crystal at absolute zero is zero.
The change in entropy is given by dS = dQ/T, where dQ is the amount of heat supplied to the system at absolute temperature T.
Adding heat increases entropy, while removing heat decreases entropy.

Reversible Process

A reversible process occurs when the system passes through the same intermediate states, allowing for a retraceable path back to the initial state.
Key conditions for reversibility include:
- Infinitely slow rate
- No heat loss through conduction, convection, or radiation
- No work done against friction
- No diffusion of fluids
- No conduction or convection
- No irreversible heat generation

Law of Increase of Entropy

In an isolated system, entropy can increase, but it never decreases.
The change in entropy is given by S = k log W, where W is the thermodynamic probability of the system.
When a system reaches thermal equilibrium, entropy reaches a maximum value and remains constant.
Irreversible processes, such as those moving from an inequilibrium to an equilibrium state, always result in an increase in entropy.

Examples of Increase of Entropy

Transfer of heat from a hotter to a colder body increases entropy due to the irreversible nature of this process.
Expansion of a gas into a vacuum also increases entropy, as the gas's disorder increases.

Temperature-Entropy (T-S) Diagram

The T-S diagram visually represents the changes in entropy and temperature during a thermodynamic process.
The Carnot cycle can be represented on a T-S diagram, showing the four stages of the cycle: isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression.
The area enclosed by the Carnot cycle represents the net work done during the cycle.
The efficiency of a Carnot engine can be determined from the T-S diagram.

Unattainability of Absolute Zero

Absolute zero (0 Kelvin) is unattainable due to the third law of thermodynamics.
The adiabatic demagnetization process, which involves lowering the temperature of a substance by removing its magnetic field, cannot reach absolute zero.
This is because the entropy of a perfect crystal must be constant at absolute zero, as dictated by Nernst's theorem.

Notes:

Entropy remains constant in reversible processes but increases in irreversible processes.
The units of entropy are calories per Kelvin (°K) or Joules per Kelvin (J/K).
The Second Law of Thermodynamics includes the Kelvin-Planck and Clausius statements, both of which demonstrate the impossibility of perpetual motion machines.
The Third Law of Thermodynamics states that the entropy of a perfect crystal at absolute zero is zero.

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Description

Explore the fundamental concepts of entropy in thermodynamics, including its definition, significance, and the Clausius theorem. Understand how entropy changes in both reversible and irreversible processes, and learn about its role in the efficiency of heat engines.