What is the Entropy | Entropy in Statistical Mechanics, Meaning, Definition, Thermodynamic Relation

What is Entropy?

Entropy of a system is a function of the thermodynamical coordinates defining the state of the systems viz, the pressure, temperature, volume or internal energy and its change between two states is equal to the integral of the quantity the dQ/T between the states along any  reversible path joining them.

Carnot's reversible cycle

Physically the entropy of a system is a measure of disorder of its molecular motion. Greater is disorder of molecular motion of the system greater is the entropy. The change in entropy passing from one state A to another state B is given by

`S_B-S_A=\int_A^B\frac{dQ}T`

Where dQ is the quantity of heat absorbed or ejected at a temperature T in going from state A to state B. We also note that :

1. The entropy of a system remains constant during an adiabatic change.
2. The entropy of a system remains constant in all reversible processes.
3. The entropy of a system increase in all irreversible processes.

Since irreversible processes are continuously occurring in nature, the entropy of the universe is increasing. The unit of entropy depends on the unit of heat employed and the absolute temperature. It is measured in kilo-cal/K or joule/K.

Entropy in Statistical Mechanics

The entropy (σ) of a system in classical statistical mechanics in statistical equilibrium is defined as

σ = log ΔΓ

Where, (ΔΓ) in the volume of the phase space accessible to the system i.e. it is volume corresponding to energies between E and E+dE.

Entropy of a perfect Gas

Let us consider 1 g-mole of a perfect gas occupying a volume V at a pressure P and temperature T. Let a quantity of heat dQ be given to the gas, then by the first law of thermodynamics, we have

dQ = dU + dW

Now if C_v is the specific heat of the gas at constant volume, dT is the rise in temperature, dV is the change in volume then

dU = C_vdT       and    dW = PdV

Hence,

dQ = C_vdT + PdV

If S is the entropy per g-mole of the gas then change in entropy, when its thermodynamic coordinates change from (P₁ V₁ T₁) to (P₂ V₂ T₂) is give by

`\triangle S=\int\frac{dQ}T`

`=\int_{P_1\V_1\T_1}^{P_2\V_2\T_2}\frac{(C_vdT+PdV)}T`

`\triangle S=\int C_v\frac{dT}T+\int P\frac{dV}T`

Properties of Entropy

• It is a thermodynamic function.
• It is a state function.
• It depends on the state of the system and not the path that is followed.
• It is represented by S but in the standard state, it is represented by S°.
• It’s SI unit is J/Kmol. It’s CGS unit is cal/Kmol.
• Entropy is an extensive property which means that it scales with the size or extent of a system.

FAQs

Que:- Does freezing increase entropy?

Ans:- Water has a greater entropy than ice and so entropy favours melting. Freezing is an exothermic process; energy is lost from the water and dissipated to the surroundings. Therefore, as the surroundings get hotter, they are gaining more energy and thus the entropy of the surroundings is increasing.

Que:- Can entropy ever decrease?

Ans:- It just says that the total entropy of the universe can never decrease. Entropy can decrease somewhere, provided it increases somewhere else by at least as much. The entropy of a system decreases only when it interacts with some other system whose entropy increases in the process. That is the law.

Que:- Can entropy be infinite?

Ans:- Since no finite system can have an infinite number of microstates, it’s impossible for the entropy of the system to be infinite. In fact, entropy tends toward finite maximum values as a system approaches equilibrium.

Que:- Can entropy be negative?

Ans:- So if entropy is the amount of disorder, negative entropy means something has less disorder or more order. The shirt is now less disordered and in a state of negative entropy, but you are more disordered and thus the system as a whole is in a state of either zero entropy or positive entropy.

Read also

• Ensembles in Statistical Mechanics
• Derivation of Mayer's Formula
• Introduction to Statistical Mechanics