Van der Waals Real-Gas Equation & Compressibility Factor (Z)

Keep the units and model explicit

Use P=RT/(Vm−b)−a/Vm² with molar volume Vm. Here R is in L·bar/(mol·K), temperature in kelvins, Vm and b in L/mol, and a in L²·bar/mol². Total vessel volume cannot be substituted for molar volume unless the amount of gas is one mole.

The initial R=0.0831446, a=3.658 and b=0.04286 are supplied teaching parameters. They are not a validated property package for a gas cylinder or natural-gas mixture. Pressure values below are predictions of this model, not measurements.

The excluded-volume term raises pressure relative to RT/Vm, while the attraction term lowers it. The net departure can have either sign; there is no universal 20–50% ideal-gas error.

Evaluate pressure without rounding intermediate results

E2 returns the raw model pressure. With T=310 K and Vm=0.5 L/mol it is approximately 41.751 bar, compared with E3 ideal pressure of 51.549652 bar.

The guard requires five numeric inputs, positive R, T and Vm, nonnegative a and b, and Vm>b. The zero-parameter limit a=b=0 is allowed and reproduces the ideal model. Celsius temperature must first be converted to kelvins.

=IF(COUNT(B2:B6)<>5,"Enter five numeric inputs",IF(OR(B2<=0,B3<=0,B4<=0,B5<0,B6<0,B4<=B6),"Require R,T,Vm>0; a,b>=0; Vm>b",B2*B3/(B4-B6)-B5/B4^2))

Compare model compressibility

E4 evaluates Z=Pmodel×Vm/(RT), about 0.8099 for the initial state. It uses unrounded pressure and propagates invalid-input messages. Nonpositive model pressure is flagged instead of presented as a usable gas compressibility factor.

Z is dimensionless. Z<1 means this model pressure is below the ideal value at the same temperature and molar volume; it is not an experimental error estimate. Increasing Vm to 25 L/mol gives about 0.9960. At large Vm the corrections diminish.

=IF(ISNUMBER(E2),IF(E2<=0,"Nonpositive model pressure; review state",E2*B4/(B2*B3)),E2)

Try independent limiting cases

Set a=0 to isolate excluded volume: Z becomes Vm/(Vm−b)>1 for b>0. Set b=0 to isolate attraction: Z becomes 1−a/(RTVm). Set both to zero to obtain Z=1.

E3 is deliberately independent of a and b. It can still display the ideal comparison when those model parameters are missing; E2 and E4 must remain unavailable.

=IF(COUNT(B2:B4)<>3,"Enter numeric R,T,Vm",IF(OR(B2<=0,B3<=0,B4<=0),"Require R,T,Vm>0",B2*B3/B4))

Do not infer phase stability from the input guard

Vm>b and positive pressure do not establish a stable single gas phase. Near condensation or a critical region, phase-equilibrium analysis and a suitable validated equation of state are needed. The spreadsheet does not select a stable phase or model a mixture.

E2 retains negative algebraic outputs for diagnosis, while E4 flags them. Do not use this educational calculation alone for vessel sizing, operating limits or pressure protection.

Further reading

• OpenStax Chemistry 2e — Non-Ideal Gas Behavior
• NIST REFPROP documentation — Reference fluid property models