Entropy changes during Reversible adiabatic expansion

Entropy Change During Reversible Adiabatic Expansion

Entropy is the measure of the randomness or disorderliness of a system. It is a state quantity, which is a physical quantity that defines the state of a system. The system is a part of the universe that is under observation. Systems can be of three types: open, closed, and isolated systems.

Entropy 

Entropy is a quantity that measures the thermodynamic disorderliness of a physical quantity. It is the measure of change in heat energy with the temperature change. 

Entropy S= dQ/dT 

Reversible adiabatic expansion

An adiabatic expansion is a thermodynamic system in which the change in the state of matter is brought about by the pressure, volume, and temperature without transferring the heat energy between the system or its surroundings.

Heat transfer Q is zero in the adiabatic process, so the work done is by the change in internal energy U.

An adiabatic process is an ideal process in which there is no heat transfer between the system and surroundings, and the change in internal energy is ΔU = U2 – U1

And W = -ΔU = U1-U2

In this process, since there is no heat transfer(Q=0), entropy is said to be zero(ΔS=0)

Examples: Hot water flask, turbines.

Adiabatic expansion process

Expansion by the gas is the increase in volume with a decrease in pressure, and since the process is adiabatic or carried out in an isolated system, there is no heat transfer. Work is done by the system where the expansion of gas takes place using the system’s internal energy, decreasing the system’s temperature. This is called the cooling effect. 

Work done by the gas W=P.dV and dQ=0

Adiabatic process:

P Vˠ   = constant

Work done in Adiabatic process 

Considering a mole of an ideal gas in a cylinder with a frictionless and insulating piston, P is the pressure, and dx is the small distance moved by the piston,

dw = PAdx =PdV

Where A is the cross-section and dV = Adx

The total work done is

      W = ∫v₁v₂ PdV……(1)

 In  an adiabatic change,

 PV r = G or P = G* V – r Putting it in eq..1

w=∫v₁v₂ G∗V−rdv 

w=∫v₁v₂ G∗V−rdv

=G∫v₁v₂ V−rdv

= G/(1-r) * [ V(1-r)]v₁v₂

 = G/ (1-r) * [ V₂ (1 – r)  –  v₁ (1 – r)]

= G/ (r -1) * [v₁ (1 – r) – v₂ ( 1 – r)]

= 1/ (r -1) * [G* v₁ (1 – r) – G * v₂ (1 – r)] …….(2)

G = P₁ V₁ r = P₂ V₂ r… putting in eq (2)

W (adia) = 1/ (r – 1) * [( P₁Vr₁ * V₁ (1-r ) – P₂Vr₂ * V₂ (1-r)]

W(adia) = 1 / (r – 1) *(P1V1−P2V2)……(3) 

Derivation of an Adiabatic Equation

From 1st  law of thermodynamics,

dQ = dU + dW….(1)

For 1 mole of gas,  we get

dW = PdV

dU = nCVdT, and CV = dU/dT => dU =  nCVdT = CVdT (as n=1)

In case of an adiabatic process, dQ = 0;

From the ideal gas equation:

PV= RT…..(2) (n =1)

Now applying the values of dU and dW in eq (1):

0 = CVdT + PdV…. (3)

Differentiating both the sides in equation (2), we get:

PdV + VdP = RdT

dT= (PdV + VdP)/ R

Putting the value of dT in q.. (3)

CV (PdV + VdP)/ R + PdV = 0

CV (VdP) + CV + R (PdV)… (4)

we know that CP = CV + R put it in eq. (4)

CV (VdP) + CP (PdV) = 0

Divide both the sides by CVPV, we get:

dP/P + (Cp/Cv)* dV/V = 0

As we know Cp / Cv = r

dP/P + r* dV/V = 0…….(5)

Integrate both sides in eq…(5)

= ∫ dP/P + r  ∫ dV/V = C

= Loge P + r Logₑ V = C

= Loge PV r = C 

= PV r = ec

=  PV r = G 

Relation Between P and T

Ideal gas equation is,

PV = RT ( for 1 mol gas)

V = RT/P…(1)

Put the value of equation (1)….in the equation  PVr= K

P(RT/P) r = G

= P (1 – r) Tr = G (Constant)

Relation Between V and T

For 1 mole of gas,  PV= RT

P =RT/V

Put PV r =G, we get

RT/V * Vr = G or T*V(r – 1) = G/ R

= TV(r – 1)  = G  (Constant)

It describes the adiabatic relation between V and t of an Ideal gas.

Adiabatic reversible process

A reversible adiabatic process is said to be an isentropic process. In an adiabatic process, the change in heat energy is zero(Q=0), so the entropy change is also said to be zero. Such a process where entropy is fixed is said to be an isentropic process.

Entropy is dQ/dT 

dQ will be zero as there is no heat transfer dQ/dT = 0

dS= 0

Examples include the expansion of steam in steam turbines and gas in gas turbines.

Conclusion

Entropy measures the state of randomness or a measure of disorder. It is an important scientific concept that helps calculate the state of disorder or uncertainty. Expansion by the gas is the increase in volume with a decrease in pressure but in adiabatic processes or carried out in an isolated system, there is no heat transfer. Therefore,  Entropy change during  reversible adiabatic process is zero. But remember that the entropy of a non isolated system can change during a reversible process and an irreversible change to an isolated system will increase the entropy.