Eötvös rule

The first assumption of the Eötvös rule is:

1. The surface tension is a linear function of the temperature.

This assumption is approximately fulfilled for most known liquids. When plotting the surface tension versus the temperature a fairly straight line can be seen which has a surface tension of zero at the critical temperature.

The Eötvös rule also gives a relation of the surface tension behaviour of different liquids in respect to each other:

2. The temperature dependence of the surface tension can be plotted for all liquids in a way that the data collapses to a single master curve. To do so either the molar mass, the density, or the molar volume of the corresponding liquid has to be known.

More accurate versions are found on the main page for surface tension.

The Eötvös rule

If V is the molar volume and T_c the critical temperature of a liquid the surface tension γ is given by

${\displaystyle \gamma V^{2/3}=k(T_c-T)}$

where k is a constant valid for all liquids, with a value of 2.1×10⁻⁷ J/(K·mol^2/3).

More precise values can be gained when considering that the line normally passes the temperature axis 6 K before the critical point:

${\displaystyle \gamma V^{2/3}=k(T_c-6\mathrm{K}-T)}$

The molar volume V is given by the molar mass M and the density ρ

${\displaystyle V=M/\rho }$

The term ${\displaystyle \gamma V^{2/3}}$ is also referred to as the "molar surface tension" γ_mol :

${\displaystyle {\gamma }_{mol}=\gamma V^{2/3}}$

A useful representation that prevents the use of the unit mol^−2/3 is given by the Avogadro constant N_A :

${\displaystyle \gamma =k^{\prime }{\left(\frac{M}{\rho N_A}\right)}^{-2/3}(T_c-6\mathrm{K}-T)=k^{\prime }{\left(\frac{N_A}{V}\right)}^{2/3}(T_c-6\mathrm{K}-T)}$

As John Lennard-Jones and Corner showed in 1940 by means of the statistical mechanics the constant k′ is nearly equal to the Boltzmann constant.

Water

For water, the following equation is valid between 0 and 100 °C.

${\displaystyle \gamma =0.07275\mathrm{N}/\mathrm{m}\cdot (1-0.002\cdot (T-291\mathrm{K}))}$