The Van der Waals equation

The ideal gas model makes two strong assumptions: the molecules are point-like (zero volume) and do not interact with each other (zero force at a distance). It is an excellent approximation for rarefied gases far from condensation, but fails close to the liquefaction point. In 1873 Johannes van der Waals proposed a simple correction, able to qualitatively describe the liquid-gas transition; the equation earned him the Nobel Prize in 1910.

Law — Van der Waals equation

For $n$ moles of real gas at temperature $T$ in a volume $V$: $\ev{\left(P + \frac{a\,n^2}{V^2}\right)(V - n\,b) = n\,R\,T}$

The two constants $a$ and $b$ are characteristic of the gas and correct the two ideal assumptions:

• $a$ (in J·m$^3$/mol$^2$) measures the mutual attraction between molecules. Near the walls, molecules are held back by their companions in the interior “bulk”, which reduces the pressure actually measured. The term $a n^2/V^2$ is added to the measured $P$ to reconstruct the ideal pressure.
• $b$ (in m$^3$/mol) is the excluded volume per mole: molecules are not point-like, they take up space. The volume actually available for motion is $V - n b$, not $V$.

Key formula — Van der Waals

$\left(P+\frac{an^2}{V^2}\right)(V-nb) = nRT$ $a$: attraction $\Rightarrow$ lower effective pressure. $b$: bulk $\Rightarrow$ lower effective volume.

In the limit of a rarefied gas ($V \gg nb$ and $V^2 \gg an^2/P$) both corrections become negligible and the equation reduces to $PV = nRT$: the ideal model re-emerges as a special case.

Example — Estimate of $b$ for nitrogen

For N$_2$ the experimental value is $b \approx 3{,}9\cdot 10^{-5}$ m$^3$/mol. Dividing by Avogadro’s number gives the “effective” volume of a molecule: $b/N_A \approx 6{,}5\cdot 10^{-29}\;\text{m}^3 \;\Rightarrow\; d \approx \sqrt[3]{6\,b/(\pi N_A)} \approx 3{,}5\cdot 10^{-10}\;\text{m}$ The diameter estimated from $b$ ($\sim 0{,}35$ nm) coincides with that found from viscosity and the mean free path. When independent methods give the same answer, atoms stop being a conjecture.

In summary

Real gas = ideal gas + two corrections. $a$ (molecular attraction) and $b$ (excluded volume) are enough to qualitatively capture the liquid-gas transition. Van der Waals is the “bridge” between the ideal gas and condensed matter.

Collegamenti

Argomenti: Teoria cinetica dei gas Concetti: Legge dei gas perfetti · Energia interna

Esercizi collegati: Heating under the piston · Worked exercise — Helium in a braking truck · Problem — Moles and molecules in a cylinder