Real Gases & the van der Waals Equation — Theory & Guide
Real gases deviate from the ideal–gas law, especially at high pressure (molecules occupy
non-negligible volume) and low temperature (attractive forces matter). The van der Waals (vdW)
equation introduces two empirical corrections that capture these effects with reasonable accuracy for
many gases.
\[
\left(P+\frac{a\,n^2}{V^2}\right)\!\left(V-nb\right)=nRT
\quad\Longleftrightarrow\quad
P=\frac{nRT}{\,V-nb\,}-\frac{a\,n^2}{V^2}
\]
Meaning of the correction terms
- Finite molecular size (\(b\)): molecules exclude volume, so the free volume is \(V-nb\). This
increases the pressure relative to the ideal prediction. Units: \(\mathrm{L\,mol^{-1}}\)
(table) or \(\mathrm{m^3\,mol^{-1}}\) (SI).
- Attractions (\(a\)): molecules attract their neighbors and reduce the force on the wall; the pressure
is corrected by \(-\,a(n/V)^2\). Larger \(a\) means stronger attractions. Units:
\(\mathrm{bar\,L^2\,mol^{-2}}\) (table) or \(\mathrm{Pa\,m^6\,mol^{-2}}\) (SI).
When is a gas “ideal” vs “nonideal”?
- Nearly ideal: low \(P\) (large \(V\)), high \(T\), weakly interacting gases (e.g., He, H₂).
- Strongly nonideal: high \(P\) (crowding, finite size) and/or low \(T\) (attractions).
Compressibility factor
A convenient deviation measure is the compressibility factor
\[
Z=\frac{PV}{nRT}
\]
For an ideal gas \(Z=1\). With dominant attractions \(Z<1\); with dominant repulsions/finite size \(Z>1\).
Unit conventions used by the calculator
- Table constants are entered as \(a\) in \(\mathrm{bar\,L^2\,mol^{-2}}\), \(b\) in \(\mathrm{L\,mol^{-1}}\).
Internally we convert to SI:
\[
a_{\text{SI}} = 0.1\,a_{\text{(bar\,L}^2\text{)}}\ \mathrm{Pa\,m^6\,mol^{-2}},\qquad
b_{\text{SI}} = 10^{-3} b\ \mathrm{m^3\,mol^{-1}}.
\]
- The gas constant is \(R=8.314\,462\,618\ \mathrm{J\,mol^{-1}\,K^{-1}}=\mathrm{Pa\,m^3\,mol^{-1}\,K^{-1}}\).
- You may enter \(P\) in Pa, kPa, bar, or atm; \(V\) in L or m³; \(T\) in K or °C (converted to K).
How to think about the two terms
- Sign check for \(P\):
\[
P_{\text{vdW}}= \underbrace{\frac{nRT}{V-nb}}_{\text{> ideal (smaller volume)}}
\;-\; \underbrace{a\left(\frac{n}{V}\right)^{\!2}}_{\text{pressure lowered by attractions}}.
\]
At high \(P\) (small \(V\)), the \(b\)-term dominates and \(Z>1\). At low \(T\), the \(a\)-term dominates and \(Z<1\).
- Physical constraint: you must have \(V-nb > 0\); otherwise the input is unphysical.
Typical constants (25 °C, gases)
| Gas | \(a\) (bar·L²·mol⁻²) | \(b\) (L·mol⁻¹) | Notes |
|---|
| H₂ | 0.2452 | 0.0265 | Small, weak attractions |
| He | 0.0346 | 0.0238 | Very close to ideal |
| N₂ | 1.370 | 0.0387 | Moderate nonideality |
| O₂ | 1.382 | 0.0319 | Moderate nonideality |
| CO₂ | 3.658 | 0.0429 | Stronger attractions |
| Cl₂ | 6.34 | 0.0542 | Used in the example problem |
Worked idea (pressure from vdW)
For \(n=1.00\ \mathrm{mol}\) of \(\mathrm{Cl_2}\) in \(V=2.00\ \mathrm{L}\) at \(T=273\ \mathrm{K}\) with
\(a=6.34\ \mathrm{bar\,L^2\,mol^{-2}}\), \(b=0.0542\ \mathrm{L\,mol^{-1}}\):
\[
P = \frac{nRT}{V-nb} - \frac{a n^2}{V^2} \approx 10.1\ \mathrm{bar},
\]
a bit lower than the ideal-gas value (\(\sim 11.3\ \mathrm{bar}\)) because the attractive \(a\)-term reduces the pressure.
Advanced (critical constants from vdW)
For 1 mol, the vdW model predicts
\[
T_c=\frac{8a}{27Rb},\qquad P_c=\frac{a}{27b^2},\qquad V_c=3b,
\]
which provide a useful rough link between \(a,b\) and a gas’s critical point.
(The model is qualitative near criticality; real fluids show more complex behavior.)
