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ChemistryEdu Logo Thermodynamics | Entropy#

Entropy#

  • The measure of degree of randomness of a system is known as its entropy.
  • Entropy is an extensive property. Thus, more is the number of molecules, more is the randomness or entropy.
  • Entropy of gas is greater than that of liquid or solid. Similarly, entropy of liquid is greater than that of the solid.
  • Entropy is a state function, ie, its value depends only on initial and final states.
  • Mathematically, entropy (S) can be expressed as:
dS=dqrevT
dS=dqrevT
ΔS=dqrevT
ΔS=nCpdTT

Here, qrev = Heat exchanged by the system reversibly and T = temperature

  • For a spontaneous process, change in entropy will be positive (ΔS > 0).
  • Entropy of a system is maximum at equilibrium.
  • Entropy gives the quantitative idea about the unavailable energy of a system, ie, the energy which cannot be used for performing useful work.

Calculation of change in entropy for an ideal gas#

dS=dqrevT

By first law of thermodynamics, dqrev = dU - W

dS=dUdWT
dS=dU(PdV)T
dS=dU+PdVT

Since, dU = nCvdT, we can write:

dS=nCvdT+PdVT

Integrating both sides:

S1S2dS=T1T2nCvTdT+V1V2PTdV
S2S1=T1T2nCvTdT+V1V2nRTVTdV
ΔS=nCvT1T21TdT+nRV1V21VdV
ΔS=nCvlnT2T1+nRlnV2V1

Let's derive the formula for ΔS in terms of Cp:

ΔS=n(CpR)lnT2T1+nRlnV2V1
ΔS=nCplnT2T1nRlnT2T1+nRlnV2V1
ΔS=nCplnT2T1+nRlnV2T1T2V1

Since, P ∝ (T/V):

ΔS=nCplnT2T1+nRlnP1P2

Case 1. For an isochoric process

In an isochoric process, volume remains constant. So, V1 = V2.

ΔS=nCvlnT2T1+nRlnV2V1
ΔS=nCvlnT2T1

Case 2. For an isobaric process

In an isobaric process, pressure remains constant. So, P1 = P2.

ΔS=nCplnT2T1+nRlnP1P2
ΔS=nCplnT2T1

Case 3. For an isothermal Process

In an isothermal process, temperature remains constant. So, T1 = T2.

ΔS=nCvlnT2T1+nRlnV2V1
ΔS=nRlnV2V1
Or, ΔS=nRlnP1P2

Case 4. When two gases are mixed at same temperature in a container

  • Let us consider two gases A and B with moles nA and nB be mixed together in a container of volume V.

  • For gas A, let initial pressure, P1 = (nART / V) and final pressure P2 = ((nA + nB)RT / V)

ΔSA=nARlnP1P2
ΔSA=nARlnnARTV(nA+nB)RTV
ΔSA=nARlnnAnA+nB

We know that nA / (nA + nB) = mole fraction of A (ΧA).

ΔSA=nARlnΧA
  • Similarly for gas B, entropy change is given by:
ΔSB=nBRlnΧB
  • Total entropy change is given by:
ΔStotal=ΔSA+ΔSB
ΔStotal=nARlnΧA+nBRlnΧB
  • Total molar entropy (entropy per mole) is given by:
ΔSm=nAnA+nBRlnΧA+nBnA+nBRlnΧB
ΔSm=ΧARlnΧA+ΧBRlnΧB

Change in entropy of system, surrounding and universe#

  • Let the change in entropy of universe be denoted by ΔStotal.
ΔStotal=ΔSsystem+ΔSsurrounding

Case 1. For a reversible process

For a reversible process, total entropy change, ΔStotal is zero.

ΔStotal=ΔSsystem+ΔSsurrounding=0
ΔSsystem=ΔSsurrounding

Case 2. For an irreversible process

  • For an irreversible process, entropy change of surrounding (ΔSsurrounding) is given by:
ΔSsurrounding=qirrT

where, qirr = heat absorbed by the surrounding during irreversible process

  • Total entropy change or entropy change of universe is given by:
ΔStotal=ΔSsystem+ΔSsurrounding
ΔStotal=1TdqrevqirrT
ΔStotal=qrevTqirrT
  • It is worth noting that all irreversible processes (or, natural processes) are spontaneous processes. So, ΔStotal will always be positive.
ΔStotal>0
qrevTqirrT>0
qrevT>qirrT
qrev>qirr
  • For an adiabatic irreversible process, qirr = 0 because surrounding does not exchange heat with the system. So, ΔSsurrounding = 0.

So, for an adiabatic irreversible process:

ΔStotal=ΔSsystem

Case 3. For free expansion

  • In free expansion, W = 0, q = 0 and ΔU = 0.

  • Surrounding exchange no heat with the system. So, qirr = 0. Therefore, ΔSsurrounding = 0.

  • Total entropy change is given by:

ΔStotal=ΔSsystem+ΔSsurrounding
Stotal=ΔSsystem

Thermal Death of Universe#

  • All natural processes are irreversible and spontaneous. So, change in entropy of natural processes will always be greater than zero.
  • In other words, the entropy of the universe is constantly increasing for irreversible processes.
  • Entropy is constantly increasing, ie, unavailable energy is constantly increasing. This leads to the conclusion that a time will come when no energy will be available for doing work. This is known as thermal death of universe.