ENTROPY

1. Introduction to Entropy

Entropy is one of the most important and fundamental properties in thermodynamics. It is a measure of disorder, randomness, or molecular chaos in a system. Entropy also provides a quantitative way to determine the direction of a process and the extent of irreversibility.

The concept of entropy was introduced by Rudolf Clausius to express the limitations of the Second Law of Thermodynamics.

In simple terms:

Energy tells us how much work is possible.
Entropy tells us how much of that energy is unavailable for work.

2. Need for Entropy

The First Law of Thermodynamics deals with energy conservation, but it does not indicate:

Direction of heat flow
Feasibility of a process
Degree of irreversibility

Entropy overcomes these limitations by:

Predicting whether a process is possible or impossible
Quantifying irreversibility
Defining the quality of energy

3. Definition of Entropy

Entropy (S) is defined as a thermodynamic property whose change is given by the ratio of heat transfer to absolute temperature for a reversible process.

Mathematical Definition:

$dS = \frac{\delta Q_{rev}}{T}$ dS=TδQrev

Where:

$dS$ dS = Change in entropy (kJ/K)
$\delta Q_{rev}$ δQrev = Reversible heat transfer (kJ)
$T$ T = Absolute temperature (K)

4. Unit of Entropy

SI Unit: kJ/K
Specific entropy: kJ/kg·K

5. Entropy as a Property

Entropy is a point function or state property, meaning:

Its value depends only on the state of the system
It is independent of the path followed by the process

6. Entropy Change of a System

(a) Reversible Process

$\Delta S = \int \frac{\delta Q_{rev}}{T}$ ΔS=∫TδQrev

(b) Irreversible Process

$\Delta S > \int \frac{\delta Q}{T}$ ΔS>∫TδQ

This shows that entropy generation always occurs in irreversible processes.

7. Clausius Inequality

The Clausius inequality is a mathematical statement of the Second Law. $\oint \frac{\delta Q}{T} \leq 0$ ∮TδQ≤0

Equality holds for reversible cycles
Inequality holds for irreversible cycles

This inequality forms the basis for defining entropy.

8. Entropy Change for Various Processes

(a) Constant Temperature (Isothermal Process)

$\Delta S = \frac{Q}{T}$ ΔS=TQ

(b) Constant Pressure Process

$\Delta S = C_p \ln\left(\frac{T_2}{T_1}\right)$ ΔS=Cpln(T1T2)

(c) Constant Volume Process

$\Delta S = C_v \ln\left(\frac{T_2}{T_1}\right)$ ΔS=Cvln(T1T2)

(d) Ideal Gas Process

$\Delta S = C_p \ln\left(\frac{T_2}{T_1}\right) – R \ln\left(\frac{P_2}{P_1}\right)$ ΔS=Cpln(T1T2)−Rln(P1P2)

9. Entropy Change of Universe

The universe consists of:

System
Surroundings

$\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings}$ ΔSuniverse=ΔSsystem+ΔSsurroundings

According to the Second Law:

Reversible process: $\Delta S_{universe} = 0$ ΔSuniverse=0
Irreversible process: $\Delta S_{universe} > 0$ ΔSuniverse>0
Impossible process: $\Delta S_{universe} < 0$ ΔSuniverse<0

10. Principle of Increase of Entropy

Entropy of an isolated system always increases for irreversible processes and remains constant for reversible processes.

This principle explains:

Natural direction of processes
Why heat flows from hot to cold
Why perpetual motion machines of second kind are impossible

11. Entropy and Irreversibility

Irreversibility is caused by:

Friction
Unrestrained expansion
Mixing of fluids
Heat transfer across finite temperature difference

All irreversible processes lead to entropy generation.

12. Entropy Generation

$S_{gen} = \Delta S_{system} – \int \frac{\delta Q}{T}$ Sgen=ΔSsystem−∫TδQ

$S_{gen} = 0$ Sgen=0 → Reversible process
$S_{gen} > 0$ Sgen>0 → Irreversible process

Entropy generation is a measure of lost work.

14. Isentropic Process

An isentropic process is:

Reversible and adiabatic
Entropy remains constant

$\Delta S = 0$ ΔS=0

Used to model:

Turbines
Compressors
Nozzles

15. Applications of Entropy

Performance analysis of heat engines
Efficiency improvement of power plants
Design of turbines, compressors, refrigerators
Determination of irreversibility and losses