Two moles of an ideal gas undergo free expansion from 10 L to 100 L at 300 K. The values of Δ S_system and Δ S_surroundings are (R = universal gas constant):

Solution & Explanation

Concept: free (Joule) expansion into vacuum. Entropy is a state function, so ΔS_system is computed via a reversible isothermal path between the same end states. For isothermal expansion of an ideal gas: ΔS_system = nR ln(V₂/V₁). Here n = 2, V₂/V₁ = 100/10 = 10: ΔS_system = 2R ln(10) = 2R × 2.303 = 4.606R (positive — gas expands, disorder rises). For the surroundings: in free expansion the gas pushes against vacuum, so work w = 0; for an ideal gas at constant T, ΔU = 0, hence q = ΔU - w = 0. No heat is exchanged with the surroundings, so: ΔS_surroundings = q_surr/T = 0. Therefore ΔS_system = 4.606R and ΔS_surr = 0 — answer (D). Note: ΔS_total = +4.606R > 0, consistent with the spontaneity of an irreversible expansion. Trap: do not write ΔS_surr = -ΔS_sys (that would only hold for a reversible exchange of heat, which does not occur here).