Van der Waals constant from P-V diagram · CAP 2019 #17

Physics · thermodynamics

Problem

The equation of state for some gases can be approximately expressed in the following way: (P+an2V2)V=nRT(P + a\frac{n^2}{V^2})V = nRT where P is pressure, V is volume, n is the number of moles of a substance, V/n is molar volume (the volume of 1 mole of gas), T is temperature, R ≈ 8.3144621 J/mol·K is the gas constant, and a is substance-specific van der Waals constant, in units of kPa·ℓ²/mol², where ℓ means litres. The following diagram shows the pressure of carbon disulphide gas in terms of volume per mole at temperature of 300 K. What is the van der Waals constant for this gas? [Figure: P-V diagram showing pressure vs. volume per mole] a) 1300 b) 2000 c) 850 d) 1150
Intuitive explanation

This is the van der Waals equation, which corrects ideal gas behavior by accounting for intermolecular attractions (the aa term). Rearranging for pressure gives P=nRTV−an2V2P = \frac{nRT}{V} - a\frac{n^2}{V^2}, or in terms of molar volume Vm=V/nV_m = V/n: P=RTVm−aVm2P = \frac{RT}{V_m} - \frac{a}{V_m^2}. The key insight is that the aa constant controls the curvature of the PP vs. VmV_m plot—you can extract it by reading two points from the graph and using the equation to solve for aa.

Intuitive explanation

This is the van der Waals equation, which corrects ideal gas behavior by accounting for intermolecular attractions (the aa term). Rearranging for pressure gives P=nRTV−an2V2P = \frac{nRT}{V} - a\frac{n^2}{V^2}, or in terms of molar volume Vm=V/nV_m = V/n: P=RTVm−aVm2P = \frac{RT}{V_m} - \frac{a}{V_m^2}. The key insight is that the aa constant controls the curvature of the PP vs. VmV_m plot—you can extract it by reading two points from the graph and using the equation to solve for aa.

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