Eötvös rule

Temperature dependence of the surface tension using the example of benzene

The Eötvös rule , named after the Hungarian physicist Loránd (Roland) Eötvös (1848–1919), allows the surface tension of any liquid substance to be predicted at any temperature . All that is required is to know the density , molar mass and critical temperature of the liquid. At the critical point the surface tension is zero.

The first statement in the rule is:

1. The surface tension depends linearly on the temperature.

This rule is at least approximately fulfilled for most known cases. When the surface tension is plotted against the temperature, at least approximately a straight line results, which results in a surface tension of zero at the critical temperature.

The Eötvös equation not only describes the dependence of the surface tension of a liquid on the temperature, but also makes another essential and more comprehensive statement:

2. The temperature dependence of the surface tension can be plotted for all liquids in such a way that approximately the same straight line always results. For this, either the molar mass and density of the liquid or its molar volume must be known.

Eötvös' rule thus follows the theorem of the corresponding states , according to which with a suitable choice of reduced quantities - here the so-called molar interfacial tension - all substances obey the same equations.

With the help of these two rules, one can predict the surface tension of any liquid at any temperature.

Eötvös equation

If the molar volume and the critical temperature of the liquid, then its surface tension is γ according to the simple Eötvös equation ${\displaystyle V_{\text{m}}}$ ${\displaystyle T_{\text{c}}}$

${\displaystyle \gamma \cdot {V_{\text{m}}}^{2/3}=k(T_{\text{c}}-T).}$

According to Eötvös, the Eötvös constant valid for all liquids has a value of

${\displaystyle {\begin{aligned}k&=2{,}1\cdot 10^{-7}\,\mathrm {J/(K\cdot mol^{2/3})} \\&=2{,}1\,\mathrm {erg/(K\cdot mol^{2/3})} \end{aligned}}}$

with the units

• J for joules
• K for Kelvin
• mol
• erg (in the sometimes still used cgs system ).

Somewhat more precise values ​​are obtained if one takes into account that the straight line usually already intersects the temperature axis 6 K before the critical point:

${\displaystyle \gamma \cdot {V_{\text{m}}}^{2/3}=k(T_{\text{c}}-6\ \mathrm {K} -T).}$

The molar volume is given by the molar mass  M and the density ρ: ${\displaystyle V_{\text{m}}}$

${\displaystyle V_{\text{m}}=M/\rho .}$

The term is also known as molar interfacial tension : ${\displaystyle \gamma {V_{\text{m}}}^{2/3}}$

${\displaystyle \gamma _{\text{m}}=\gamma \cdot {V_{\text{m}}}^{2/3}.}$

So the Eötvös equation can be written as:

${\displaystyle \gamma _{\text{m}}=k(T_{\text{c}}-6\ \mathrm {K} -T).}$

A meaningful representation that avoids the unfavorable occurrence of the unit mol ^{−2/3 can} be obtained with the help of Avogadro's constant N _A :

${\displaystyle {\begin{aligned}\gamma &=k'\left({\frac {M}{\rho N_{\text{A}}}}\right)^{-2/3}(T_{\text{c}}-6\ \mathrm {K} -T)\\&=k'\left({\frac {N_{\text{A}}}{V_{\text{m}}}}\right)^{2/3}(T_{\text{c}}-6\ \mathrm {K} -T)\end{aligned}}}$

As John Lennard-Jones and Corner showed in 1940 with statistical mechanics , the constant is  roughly equal to the Boltzmann constant : ${\displaystyle k'}$

${\displaystyle k'=k\cdot {N_{\text{A}}}^{-2/3}\approx k_{\text{B}}.}$

For the water example, the following numerical equation results after inserting all quantities:

${\displaystyle {\gamma }=0{,}07275\cdot \left(1-0{,}002\cdot \left({T}-291\right)\right)}$

with the units

T in Kelvin

${\displaystyle {\gamma }}$ in ${\displaystyle N/m}$

This corresponds to a good approximation with the experimentally measured surface tensions.

Historical

Eötvös began to study surface tension as a student. He developed a new way of determining surface tension, the reflection method. The Eötvös equation was initially found purely phenomenologically and published in 1886. In 1893 William Ramsay and John Shields (1850–1909) showed the improved version, which takes into account that the straight line usually intersects the temperature axis before the critical point. Even Albert Einstein was concerned with the temperature dependence of the surface tension. In 1940 John Lennard-Jones and Corner published a derivation of the equation using statistical mechanics . Masao Katayama (1877–1961) showed in 1916 an empirically found variant of the Eötvös equation for the case that the density of the vapor is not negligible compared to the density of the liquid. Based on this, EA Guggenheim announced another variant of the equation in 1945, which is now called the Katayama-Guggenheim equation:

${\displaystyle \gamma =\gamma _{0}(1-T/T_{\text{c}})^{11/9}.}$

Individual evidence

1. ^ ^{A b} Edward A. Guggenheim: The Principle of Corresponding States . In: The Journal of Chemical Physics . tape 13 , no. 7 , 1945, ISSN  0021-9606 , p. 253-261 , doi : 10.1063 / 1.1724033 .
2. ^ ^{A b c} John Edward Lennard-Jones and James Corner: The calculation of surface tension from intermolecular forces . In: Transactions of the Faraday Society (1905-1971) . tape 36 , 1940, p. 1156-1162 , doi : 10.1039 / TF9403601156 .
3. ↑ Roland Eötvös: About the connection between the surface tension of liquids and their molecular volume . In: G. Wiedemann (Ed.): Annalen der Physik . tape 263 , no. 3 . Johann Ambrosius Barth, 1886, p. 448–459 , doi : 10.1002 / andp.18862630309 .
4. ↑ Albert Einstein: Comment on the law of Eötvös . In: Annals of Physics . tape 339 , no. 1 . Johann Ambrosius Barth, 1911, p. 165-169 , doi : 10.1002 / andp.19113390109 .