Antoine equation

The Antoine equation is a class of semi-empirical correlations describing the relation between vapor pressure and temperature for pure substances. The equation was presented in 1888 by the French engineer Louis Charles Antoine [fr] (1825–1897).

The Antoine equation is $${\displaystyle \log _{10}p=A-{\frac {B}{C+T}},}$$ where p is the vapor pressure, T is temperature (in °C or in K according to the value of C), and A, B and C are component-specific constants.

The simplified form with C set to zero, $${\displaystyle \log _{10}p=A-{\frac {B}{T}},}$$ is the August equation, after the German physicist Ernst Ferdinand August (1795–1870). The August equation describes a linear relation between the logarithm of the pressure and the reciprocal temperature. This assumes a temperature-independent heat of vaporization. The Antoine equation allows an improved, but still inexact description of the change of the heat of vaporization with the temperature.

The Antoine equation can also be transformed in a temperature-explicit form with simple algebraic manipulations: $${\displaystyle T={\frac {B}{A-\log _{10}\,p}}-C.}$$

Usually, the Antoine equation cannot be used to describe the entire saturated vapour pressure curve from the triple point to the critical point, because it is not flexible enough. Therefore, multiple parameter sets for a single component are commonly used. A low-pressure parameter set is used to describe the vapour pressure curve up to the normal boiling point and the second set of parameters is used for the range from the normal boiling point to the critical point.

The normal boiling point of ethanol is T_B = 78.32 °C. At this temperature, the two sets of parameters above produce the following vapor pressures: $${\displaystyle {\begin{aligned}P&=10^{\left(8.20417-{\frac {1642.89}{78.32+230.300}}\right)}=760.0\ {\text{mmHg}},\\P&=10^{\left(7.68117-{\frac {1332.04}{78.32+199.200}}\right)}=761.0\ {\text{mmHg}}\end{aligned}}}$$ (760 mmHg = 101.325 kPa = 1.000 atm = normal pressure).

This example shows a severe problem caused by using two different sets of coefficients. The described vapor pressure is not continuous—at the normal boiling point the two sets give different results. This causes severe problems for computational techniques which rely on a continuous vapor pressure curve.

Two solutions are possible: The first approach uses a single Antoine parameter set over a larger temperature range and accepts the increased deviation between calculated and real vapor pressures. A variant of this single set approach is using a special parameter set fitted for the examined temperature range. The second solution is switching to another vapor pressure equation with more than three parameters. Commonly used are simple extensions of the Antoine equation (see below) and the equations of DIPPR or Wagner.