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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:
\[dS = {dq_{rev} \over T}\]
\[ \int dS = \int {dq_{rev} \over T}\]
\[ ΔS = \int {dq_{rev} \over T}\]
\[ ΔS = \int {nC_pdT \over T}\]

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#

\[dS = {dq_{rev} \over T}\]

By first law of thermodynamics, dqrev = dU - W

\[dS = {dU - dW \over T}\]
\[dS = {dU - (-PdV) \over T}\]
\[dS = {dU + PdV \over T}\]

Since, dU = nCvdT, we can write:

\[dS = {nC_vdT + PdV \over T}\]

Integrating both sides:

\[ \int\limits_{S_1}^{S_2}dS = \int\limits_{T_1}^{T_2}{nC_v \over T}dT + \int\limits_{V_1}^{V_2}{P \over T}dV\]
\[ S_2 - S_1 = \int\limits_{T_1}^{T_2}{nC_v \over T}dT + \int\limits_{V_1}^{V_2}{nRT \over VT}dV\]
\[ΔS = nC_v \int\limits_{T_1}^{T_2}{1 \over T}dT + nR \int\limits_{V_1}^{V_2}{1 \over V}dV\]
\[ {ΔS = nC_vln{T_2 \over T_1} + nRln{V_2 \over V_1}}\]

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

\[ΔS = n(C_p-R)ln{T_2 \over T_1} + nRln{V_2 \over V_1}\]
\[ΔS = nC_pln{T_2 \over T_1} - nRln{T_2 \over T_1} + nRln{V_2 \over V_1}\]
\[ΔS = nC_pln{T_2 \over T_1} + nRln{V_2T_1 \over T_2V_1}\]

Since, P ∝ (T/V):

\[ {ΔS = nC_pln{T_2 \over T_1} + nRln{P_1 \over P_2}}\]

Case 1. For an isochoric process

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

\[ΔS = nC_vln{T_2 \over T_1} + nRln{V_2 \over V_1}\]
\[ΔS = nC_vln{T_2 \over T_1}\]

Case 2. For an isobaric process

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

\[ΔS = nC_pln{T_2 \over T_1} + nRln{P_1 \over P_2}\]
\[ΔS = nC_pln{T_2 \over T_1}\]

Case 3. For an isothermal Process

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

\[ΔS = nC_vln{T_2 \over T_1} + nRln{V_2 \over V_1}\]
\[ΔS = nRln{V_2 \over V_1}\]
\[Or,\ ΔS = nRln{P_1 \over P_2}\]

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)

\[ΔS_A = n_ARln{P_1 \over P_2}\]
\[ΔS_A = n_ARln{{n_ART \over V} \over {(n_A + n_B)RT \over V}}\]
\[ΔS_A = n_ARln{n_A \over n_A + n_B}\]

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

\[ΔS_A = n_ARlnΧ_A\]
  • Similarly for gas B, entropy change is given by:
\[ΔS_B = n_BRlnΧ_B\]
  • Total entropy change is given by:
\[ΔS_{total} = ΔS_A + ΔS_B\]
\[ {ΔS_{total} = n_ARlnΧ_A + n_BRlnΧ_B} \]
  • Total molar entropy (entropy per mole) is given by:
\[ΔS_m = {n_A \over n_A + n_B}RlnΧ_A + {n_B \over n_A + n_B}RlnΧ_B\]
\[ {ΔS_m = Χ_ARlnΧ_A + Χ_BRlnΧ_B} \]

Change in entropy of system, surrounding and universe#

  • Let the change in entropy of universe be denoted by ΔStotal.
\[ ΔS_{total} = ΔS_{system} + ΔS_{surrounding} \]

Case 1. For a reversible process

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

\[ ΔS_{total} = ΔS_{system} + ΔS_{surrounding} = 0 \]
\[ ΔS_{system} = - ΔS_{surrounding} \]

Case 2. For an irreversible process

  • For an irreversible process, entropy change of surrounding (ΔSsurrounding) is given by:
\[ΔS_{surrounding} = {-q_{irr} \over T}\]

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

  • Total entropy change or entropy change of universe is given by:
\[ ΔS_{total} = ΔS_{system} + ΔS_{surrounding} \]
\[ ΔS_{total} = {{1 \over T} \int dq_{rev}} - {q_{irr} \over T} \]
\[ ΔS_{total} = {q_{rev} \over T} - {q_{irr} \over T} \]
  • It is worth noting that all irreversible processes (or, natural processes) are spontaneous processes. So, ΔStotal will always be positive.
\[ ΔS_{total} > 0\]
\[ {q_{rev} \over T} - {q_{irr} \over T} > 0\]
\[ {q_{rev} \over T} > {q_{irr} \over T} \]
\[ q_{rev} > q_{irr} \]
  • 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:

\[ΔS_{total} = ΔS_{system}\]

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:

\[ ΔS_{total} = ΔS_{system} + ΔS_{surrounding} \]
\[ S_{total} = ΔS_{system} \]

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.