Entropy

Entropy is a state function that quantifies the dispersal of energy and the number of accessible microstates for a system. For a closed system undergoing a reversible process, the differential entropy change is dS = δQ_rev/T. Entropy increases for spontaneous (irreversible) processes in isolated systems; the total entropy generation S_gen ≥ 0 (S_gen = 0 for reversible processes). The third law states that the entropy of a perfect crystalline substance approaches zero as temperature approaches absolute zero. Entropy has units J·K⁻1 (absolute).

Governing FormulaKey relations (SI units): - Differential (reversible): dS = δQ_rev / T - Entropy change (ideal gas, cp constant): ΔS = cp ln(T2/T1) - R ln(P2/P1) - Alternate ideal-gas form (cv constant): ΔS = cv ln(T2/T1) + R ln(V2/V1) - Isothermal ideal-gas: ΔS = n R ln(V2/V1) = n R ln(P1/P2) - Constant-pressure reversible heating: ΔS = ∫ cp/T dT = cp ln(T2/T1) - Entropy of mixing (ideal ideal gases, isothermal, no reaction): ΔS_mix = -n_total R Σ xi ln(xi) - Entropy generation (universe): ΔS_univ = ΔS_system + ΔS_surroundings = S_gen ≥ 0 Constants/units used: R = 8.314 J·mol⁻1·K⁻1.

Knowledge Check

10 Questions

1.Which expression correctly gives the infinitesimal change in entropy for a closed system undergoing a reversible process?

2.For an ideal gas with constant molar heat capacity cp, the entropy change between states (T1,P1) and (T2,P2) is:

3.One mole of a monatomic ideal gas (cp = 5/2·R) is heated reversibly at constant pressure from 300 K to 600 K. What is the molar entropy change? (Use R = 8.314 J·mol⁻1·K⁻1.)

4.Two different ideal gases, 1.00 mol A and 1.00 mol B, initially separated by a partition in the same container at constant temperature, are allowed to mix irreversibly after the partition is removed (isothermal, no reactions). What is the entropy change of the mixing (ΔS_mix)? (Use R = 8.314 J·mol⁻1·K⁻1.)

5.Which statement about entropy generation S_gen is correct?

6.A reversible transfer of 1000 J of heat occurs from a hot reservoir at 500 K to a cold reservoir at 300 K (treat reservoirs as constant-temperature bodies). What is the change in entropy of the universe (ΔS_univ)?

7.One mole of an ideal gas undergoes an isothermal, reversible expansion where its volume doubles (V2 = 2·V1). What is the entropy change of the gas? (Use R = 8.314 J·mol⁻1·K⁻1.)

8.Which formulation of the Third Law of Thermodynamics is correct?

9.One mole of an ideal gas initially at 10.0 L expands freely (into vacuum, irreversible adiabatic free expansion) to 20.0 L. What is the entropy change of the gas? (Use R = 8.314 J·mol⁻1·K⁻1.)

10.A 2.00 kg copper block (assume cp = 385 J·kg⁻1·K⁻1 constant) is heated reversibly from 300 K to 400 K. What is the entropy change of the block?