Q. 25 One mole of helium gas in an isolated system undergoes a reversible isothermal expansion at 25°C from an initial volume of 2.0 liters to a final volume of 10.0 liters. The change in entropy (ΔS) of the surroundings is ____ J K⁻¹ (up to two decimal places). (Given: Gas constant R = 8.314 J K⁻¹ mol⁻¹)

Entropy Change of Surroundings in Reversible Isothermal Expansion of Helium Gas

One mole of helium gas undergoes reversible isothermal expansion at 25°C from 2.0 L to 10.0 L in an isolated system, resulting in zero entropy change for the surroundings. This key thermodynamics concept from GATE XL 2018 is essential for understanding entropy in isolated systems.

Problem Breakdown

The system contains 1 mole of ideal helium gas (monatomic) expanding reversibly and isothermally (T = 25°C = 298 K) from V_i = 2.0 L to V_f = 10.0 L. Being isolated means no heat or matter exchange with surroundings occurs. For reversible isothermal expansion of an ideal gas, system entropy change is ΔS_sys = nR ln(V_f/V_i) > 0 since volume increases.

Correct Answer and Calculation

Calculate ΔS_surroundings directly: In an isolated system, Q_surroundings = 0 because no heat flows to/from surroundings. Thus, ΔS_surr = -Q_sys / T = 0, where Q_sys = nRT ln(V_f/V_i) is absorbed by the system internally. The answer is 0.00 J K⁻¹.

For reference, ΔS_sys = (1 mol)(8.314 J K⁻¹ mol⁻¹) ln(10.0/2.0) = 8.314 × ln(5) = 8.314 × 1.6094 ≈ 13.38 J K⁻¹, but surroundings remain unaffected.

Why Surroundings Entropy is Zero

Isolated systems conserve total entropy internally: ΔS_total = ΔS_sys + ΔS_surr = ΔS_sys + 0 > 0, satisfying the second law. No options are provided, but common distractors include:

• 13.38 J K⁻¹: ΔS_sys (system entropy, not surroundings).
• -13.38 J K⁻¹: -ΔS_sys (if mistakenly assuming heat leaves surroundings).
• 38.30 J K⁻¹: For 2 moles and 10x volume ratio, as in similar problems.

Key Thermodynamics Insights

• Reversible isothermal: ΔU = 0, W = -Q_sys = -nRT ln(V_f/V_i).
• Isolated constraint forces ΔS_surr = 0, emphasizing process boundaries.
• Practice for exams: Always identify system boundaries first.

This GATE question tests nuanced understanding of entropy partitioning in isolated setups.