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GATE 2024 Chemical Engineering, Q57 · Thermodynamics

The correct answer is -101 to -99.

Question

Ethylene obeys the truncated virial equation-of-state PV/RT = 1 + BP/RT, where P is the pressure, V is the molar volume, T is the absolute temperature and B is the second virial coefficient. The universal gas constant R =83.14bar⋅cm3/(mol⋅K).= 83.14 bar\cdot cm³/(mol\cdot K). At 340 K, the slope of the compressibility factor vs. pressure curve is −3.538×10−3bar−1.−3.538 \times 10⁻³ bar⁻¹. Let GRG^{R} denote the molar residual Gibbs free energy. At these conditions, the value of dGR/dPdG^{R}/dP at constant temperature, in cm3/mol,cm³/mol, rounded off to 1 decimal place, is ___________

Official answer: -101 to -99

Why

GATE 2024 official key: −101.0 to −99.0. Since Z=1+(B/RT)P, dZ/dP=B/RT=−3.538×10−3⇒B=−3.538×10−3×83.14×340≈−100.0cm3/mol.dZ/dP=B/RT=−3.538 \times 10⁻³ \Rightarrow B=−3.538 \times 10⁻³ \times 83.14 \times 340\approx −100.0 cm³/mol. For this EOS, GR=BPG^{R}=BP (constant B), so dGR/dP=B≈−100.0cm3/mol.dG^{R}/dP=B\approx −100.0 cm³/mol.

Explanation cross-checked for consistency. Final answer matches the official IIT answer key.

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