The Eötvös rule, named after the Hungarian physicist Loránd (Roland) Eötvös (1848–1919) enables the prediction of the

surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension (physics), tension is what allows objects with a higher density than water such as razor blades and insects (e.g. Ge ...

of an arbitrary

liquid Liquid is a state of matter with a definite volume but no fixed shape. Liquids adapt to the shape of their container and are nearly incompressible, maintaining their volume even under pressure. The density of a liquid is usually close to th ...

pure substance at all

temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...

s. The

density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...

molar mass In chemistry, the molar mass () (sometimes called molecular weight or formula weight, but see related quantities for usage) of a chemical substance ( element or compound) is defined as the ratio between the mass () and the amount of substance ...

and the

critical temperature Critical or Critically may refer to: *Critical, or critical but stable, medical states **Critical, or intensive care medicine *Critical juncture, a discontinuous change studied in the social sciences. *Critical Software, a company specializing in ...

of the liquid have to be known. At the critical point the surface tension is zero. 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

γ is given by :

\gamma V^ = 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: :

\gamma V^ = k(T_c - 6 \ \mathrm - T)\,

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

V = M / \rho\,

The term

\gamma V^

is also referred to as the "molar surface tension" γ_mol : :

\gamma_ = \gamma V^\,

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

Avogadro constant The Avogadro constant, commonly denoted or , is an SI defining constant with an exact value of when expressed in reciprocal moles. It defines the ratio of the number of constituent particles to the amount of substance in a sample, where th ...

N_A : :

\gamma = k' \left( \frac \right)^(T_c - 6 \ \mathrm - T) = k' \left( \frac \right)^(T_c - 6 \ \mathrm - T)

As John Lennard-Jones and Corner showed in 1940 by means of

statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

the constant ''k''′ is nearly equal to the

Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...

Water

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

\gamma = 0.07275 \ \mathrm \cdot (1-0.002 \cdot (T - 291 \ \mathrm))

History

As a student, Eötvös started to research surface tension and developed a new method for its determination. The Eötvös rule was first found phenomenologically and published in 1886. In 1893 William Ramsay and Shields showed an improved version considering that the line normally passes the temperature axis 6 K before the critical point. John Lennard-Jones and Corner published (1940) a derivation of the equation by means of

. In 1945 E. A. Guggenheim gave a further improved variant of the equation.

References

{{DEFAULTSORT:Eotvos rule Physical chemistry Thermodynamic equations