Szyszkowski equation

The Szyskowski Equation^[1] has been used by Meissner and Michaels^[2] to describe the decrease in surface tension of aqueous solutions of carboxylic acids, alcohols and esters at varying mole fractions. It describes the exponential decrease of the surface tension at low concentrations reasonably but should be used only at concentrations below 1 mole%.^[3]

Equation

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

with:

σ_m is surface tension of the mixture
σ_w is surface tension of pure water
a is component specific constant (see table below)
x is mole fraction of the solvated component

The equation can be rearranged to be explicit in a:

{\begin{aligned}a&={\frac {x}{10^{{\left(-1+{\frac {{\frac {\sigma _{m}}{\sigma _{w}}}-1}{-0.411}}\right)}}}}\\&=x10^{{\left(1+{\frac {{\frac {\sigma _{m}}{\sigma _{w}}}-1}{0.411}}\right)}}\end{aligned}}

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

The equation can also be written as:

\gamma =\gamma _{0}-{\frac {RT}{\omega }}\ln(1+K_{{ad}}c)

with:

γ is surface tension of the mixture
γ₀ is surface tension of pure water
R is ideal gas constant 8.31 J/(mol*K)
T is temperature in K
ω is cross-sectional area of the surfactant molecules at the surface

The surface tension of pure water is dependent on temperature. At room temperature (298 K), it is equal to 71.97 mN/m ^[4]

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^[5]
x_Acid [^mol/_mol]	x_Water [^mol/_mol]	σ [^mN/_m]	T [K]	a.10⁴
1^{(Pure Acid)}	0	26.68	293.15	298.
0.18229	0.81771	33.08	293.15	90.7
0.17102	0.82898	33.15	293.15	85.6
0.1494	0.8506	33.54	293.15	77.2
0.12803	0.87197	34.18	293.15	69.7
0.10947	0.89053	34.56	293.15	61.4
0.0869	0.9131	35.3	293.15	51.8
0.0747	0.9253	35.77	293.15	46.3
0.06602	0.93398	36.43	293.15	43.2
0.05436	0.94564	38.51	293.15	42.2
0.0455	0.9545	40.07	293.15	40.3
0.03268	0.96732	42.34	293.15	35.1
0.02075	0.97925	46.78	293.15	32.8
0.01457	0.98543	50.17	293.15	31.4
0.00861	0.99139	53.61	293.15	25.9
0	1^{(Pure Water)}	72.58	293.15

This example shows a good agreement between the published value a=2.6*10⁻³ and the calculated value a=2.59*10⁻³ at the smallest given mole fraction of 0.00861 but at higher concentrations of propionic acid the value of an increases considerably, showing deviations from the predicted value.

References

↑ B. von Szyszkowski, Z. physik. Chemie, 64, 385 (1908)
↑ H.P. Meissner, A.S. Michaels, Surface Tension s of Pure Liquids and Liquid Mixtures", Ind. Eng. Chem., 41(12), 2782, (1949)
↑ Bruce E. Poling, John M. Prausnitz, John P. O’Connell, The Properties of Gases and Liquids, 5th Edition
↑ NIST Chemistry WebBook http://webbook.nist.gov/chemistry/
↑ Suarez F., Romero C.M., J.Chem.Eng.Data, 56(5), 1778-1786, 2011

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Szyszkowski equation

Equation

Parameters

Example

See also

References