In this article we will discuss about:- 1. Introduction to Entropy 2. Clausius Theorem for a Process and for a Cycle 3. Entropy—A Property 4. Clausius Inequality 5. Change of Entropy in a System 6. Property Relations from Energy Equations 7. General Equations for Change in Entropy 8. Change in Entropy during Various Processes 9. Representation of Various Processes on T-S Chart and Few Other Details.

Contents:

1. Introduction to Entropy
2. Clausius Theorem for a Process and for a Cycle
3. Entropy—A Property
4. Clausius Inequality
5. Change of Entropy in a System
6. Property Relations from Energy Equations
7. General Equations for Change in Entropy
8. Change in Entropy during Various Processes
9. Representation of Various Processes on T-S Chart
10. Pure Substance and Change of Entropy
11. Third Law of Thermodynamics (Entropy)

#### 1. Introduction to Entropy:

Analysis of I-law leads to the definition of a derived property known as Internal energy. Analysis of second law will lead to the definition of another derived property known as Entropy.

Clausius discovered that, when a small amount of heat δQ is supplied to a system, which is at an absolute temperature T, then it will undergo a process, and the ratio δQ/T is same for all reversible processes. He assigned the value δQ/T = dS and called S as entropy.

The term entropy is taken, from Greek word ‘tropee’ meaning transformation. Thus, when a small amount of heat δQ is transformed to a system, the entropy changes by an amount dS. This change in entropy is regarded as the transformation content of the system.

Note that entropy is a thermodynamics property, it increases with the addition of heat and it decreases with the removal of heat.

#### 2. Clausius Theorem for a Process and for a Cycle:

Let a system be taken from an equilibrium state (1) to another equilibrium state (2) following the reversible path 1-2. Let (a) and (b) be two reversible adiabatic, which pass through the points (1) and (2) respectively. A reversible isotherm (c) is drawn, such that the area under 1-3-4-2 is equal to the area under the curve 1-2. From the first law,

Thus any reversible path can be replaced by a zig-zag path between the same end states, consisting of a revers­ible adiabatic process followed by a reversible isothermal process, which is again followed by a reversible adia­batic process, such that the heat transferred during the isothermal process is the same as that transferred during the original process.

Consider a reversible cycle as shown in Fig. 5.2. It is divided into a large number of strips by means of reversible adiabatic. Each strip may be closed at the top and bottom by means of reversible isotherms. The original closed cycle is thus replaced by a zig-zag closed path, consisting of alternate adiabatic and isothermal processes, such that the heat transferred during all the isothermal processes is equal to the heat transferred in the original cycle.

Now, for cycle 1 – 2 – 3 – 4, δQ1 is heat absorbed reversibly at T1 and δQ2 is heat rejected reversibly at T2.

If similar equation are written for all the elemental Carnot cycles, then for the whole (complete) original cycle.

Thus the cyclic integral of δQ/T for a reversible cycle is equal to zero. This is known as Clausious Theorem.

#### 3. Entropy—A Property:

To prove this, we have to prove that the change of entropy does not depend upon path but it depends upon end states. Then, we will able to say that, entropy is a property of the system.

Consider a system which changes its state from state point (1) to state point (2) by following the reversible path a and returns from state point (2) to state point (1) by following the reversible path b.

Then the two paths 1- a – 2 and 2 – b -1 together will form a cycle.

Now from Clausius theorem,

Note:

May be read as integral for the reversible path 1 – a 2 – b – 1.

The above integral may be replaced as the sum of two integrals one for the path a and other for the path b.

The magnitude of δQ/T (i.e., dS) is same for the paths a and b and it does not depend upon the end states, hence it is point function and we know that properties are point-functions, hence it is a property of the system.

#### 4. Clausius Inequality:

According to Clausius Theorem,

Thus expression is known as Clausius Inequality. It implies whether any cyclic process is reversible or irreversible or impossible.

#### 5. Change of Entropy in a System:

We know that, change in entropy for a reversible process is given by,

Now to find the value of change in entropy in an Irreversible process.

Consider a system, which change its state from state point (1) to state point (2) by following the reversible path a and returns from state point (2) to state point (1) by following the irreversible path b as shown in Fig. 5.6.

Since cyclic integral of any property is zero and entropy is a property we can write,

Combining Eqs (1) and (6), we can write,

dS ≥ δQ/T

Where equality sign is for reversible process and inequality sign is for an Irreversible process (from Eq. 6).

Note:

The effect of irreversibility is always to increase the entropy of the system.

We know that, in an isolated system, matter, work or heat cannot cross the boundary of the system. Hence according to the first law of thermodynamics, the IE of the system will remain constant.

Since for an isolated system δQ = 0, from equation,

Equation (2) states, that entropy of an isolated system either increases or remains constant and never decreases. This is known as the Principle of increase of entropy.

We know that all natural processes are irreversible processes and during irreversible processes entropy increases and hence entropy of the universe always increases.

Consider an isolated system comprising of two gases O2 and H2 in a separated box as shown in Fig. 5.7. When the partition or barrier is removed, the molecules of both gages get more space to move randomly and therefore collision take place between the mole molecules of the same gas as well as of both the gases, and after some time, equilibrium will be established.

If we assume that initially both gases have the same volume and temperature then after removing the partition each gas occupies double the previous volume while the temperature and the I.E. of the system remains the same, but we find that the entropy increase.

The reason is obvious, as the mixing process is irreversible and irreversible process is always associated with the increase in entropy.

Thus an irreversible process always tends to take the system (isolated) to state of greater disorder. It is a ten­dency on the part of nature to proceed to a state of greatest disorder. And an isolate system always tends to a state of greater entropy. So there is a close link between entropy and disorder.

It may be stated roughly that “The entropy of a system is a measure of the degree of molecular disorder existing in the system”. When heat is supplied to the system, the disorderly motion of molecules increases and so the entropy of the system increases. The reverse occurs when heat is removed from the system.

#### 6. Property Relations from Energy Equations:

For deriving the equation for change of entropy it is essential to know equation based on I and II law of thermody­namics.

We know that, change in entropy for a reversible process is given by,

#### 7. General Equations for Change in Entropy:

We know that, the first law applied to a closed system undergoing a process gives,

Q – W = ΔU

Now consider a unit mass of a gas which changes its states from pressure P1, specific volume v1 and tempera­ture T1 to a new state P2, V2, T2.

For unit mass q = Δu + w

In the differential form

#### 8. Change in Entropy during Various Processes:

Now to derive the equations for change of entropy during,

#### 9. Representation of Various Processes on T-S Chart:

Various processes are shown on T-S diagram. Note that constant volume lines on T-S diagram have steep slope compared to constant pressure lines.

#### 10. Pure Substance and Change of Entropy:

We define a pure substance as a system which is homogeneous in composition and chemical aggregation that remains invariable with time.

In many cases we have to deal with system of fixed chemical composition with changing phases, as for example, in case of vapour power generation and refrigeration cycles. In the former system water is commonly used whereas in the latter system ammonia, freon etc. are used. In these applications we find the working substance as liquid in one part of the cycle and as vapour or mixture of vapour and liquid in another part of the cycle and as vapour or mixture of vapour and liquid in another part of the cycle. The change in phase of the substance in the cycle is of physical nature only as its chemical composition remains unchanged. These fall in the category of pure substance.

We define a pure substance as a system which is homogeneous in composition and chemical aggregation than remain invariable with time.

A pure substance can exist in three different states of aggregation viz. solid, liquid and gas. It may exist in a single phase like solid, liquid, gas or more than one phase which are in equilibrium with each other. Thus a mixture of water and ice, water and steam, water and ice are all examples of pure substance.

Third law of thermodynamics states “The entropy of a pure substance in complete thermodynamic equilibrium becomes zero at the absolute zero of temperature”.

This law enables the absolute entropies of pure substances to be calculated from the fundamental definition of entropy, with SO set equal to zero.

Therefore, the change of entropy of a pure substance when it undergoes a reversible process, can be calculated as in case of ideal gases.

Thus, ΔSw = Change in entropy of 1 kg water when heated at constant pressure from 0° C to saturation temperature.

#### 11. Third Law of Thermodynamics (Entropy):

Statistical analysis suggests that the entropy of a substance tends to zero as absolute zero temperature is reached.

This point has been so intensely investigated, that it is probably safe to say that at absolute zero temperature, the entropy of a pure substance in some perfect crystalline form becomes zero, a generalisation known as the Third law.

It is found that the specific heats approach zero as temperature tends to zero and the difference Cp – Cv also approaches zero, because the co-efficient of thermal expansion approaches zero.