User:WilfriedC/Playground/Szyszkowski equation

The Szyskowski Equation has been used by Meissner and Michaels to describe the decrease of the surface tension in aqueos solutions of carboxylic acids, alcohols, esters, and few other components from pure water to increasing concentrations of the solved components. It describes the exponential decrease of the surface tension at smaller concentrations quite well but should be used only at concentrations below 1 mole%.

Equation

$\sigma _{m}=\sigma _{w}*\left(1.-0.411*log\left(1+{\frac {x}{a}}\right)\right)$

with

• $\sigma _{m}$: Surface tension of the mixture
• $\sigma _{w}$: Surface tension of pure Water (approx. 72 mN/m, see also example below)
• $a$: Component specific constant (see table below)
• $x$: Mole fraction of the solved component

The equation can be rearranged to be explicit in a:

$a={\frac {x}{10^{\left({\frac {\sigma _{m}/\sigma _{w}-1}{-0.411}}-1\right)}}}$

This allows the direct calculation of that component specific parameter a from experimental data.

Parameters

Meissner and Michaels published the following a constants:

Szyszkowski Equation Constants

Compound	a * 104	Compound	a * 104
Propionic acid	26	n-Propyl alcohol	26
Isopropyl alcohol	26	Methyl acetate	26
n-Propyl amine	19	Methyl ethyl ketone	19
n-Butyric acid	7	Isobutyric acid	7
n-Butyl alcohol	7	Isobutyl alcohol	7
Propyl formate	8.5	Ethyl acetate	8.5
Methyl propionate	8.5	Diethyl ketone	8.5
Ethyl propionate	3.1	Propyl acetate	3.1
n-Valeric acid	1.7	Isovaleric acid	1.7
n-Amyl alcohol	1.7	Isoamyl alcohol	1.7
Propyl propionate	1.0	n-Caproic acid	0.75
n-Heptanoic acid	0.17	n-Octanoic acid	0.034
n-Decanoic acid	0.0025

Example

The following table and diagram show experimentally determined surface tensions in the mixture of water and propionic acid.

Surface Tension of Propionic Acid solved in Water xAcid [mol/mol] xWater [mol/mol] σ [mN/m] T [K] a 1(Pure Acid) 0 26.68 293.15 298.E-04 0.18229 0.81771 33.08 293.15 90.7E-04 0.17102 0.82898 33.15 293.15 85.6E-04 0.1494 0.8506 33.54 293.15 77.2E-04 0.12803 0.87197 34.18 293.15 69.7E-04 0.10947 0.89053 34.56 293.15 61.4E-04 0.0869 0.9131 35.3 293.15 51.8E-04 0.0747 0.9253 35.77 293.15 46.3E-04 0.06602 0.93398 36.43 293.15 43.2E-04 0.05436 0.94564 38.51 293.15 42.2E-04 0.0455 0.9545 40.07 293.15 40.3E-04 0.03268 0.96732 42.34 293.15 35.1E-04 0.02075 0.97925 46.78 293.15 32.8E-04 0.01457 0.98543 50.17 293.15 31.4E-04 0.00861 0.99139 53.61 293.15 25.9E-04 0 1(Pure Water) 72.58 293.15

This example shows a good agreement between the published value a=26E-4 and the calculated value a=25.9E-4 at the smallest given concentration of x_Acid=0.00861 ^mol/_mol but at higher concentrations of propionic acid the value of a increases considerably and shows that the equation with a constant a should be used only for small solute concentrations.