Antoine equation

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.

• Typical deviations of a parameter fit over the entire range (experimental data for Benzene)
• 
  Deviations of an August equation fit (2 parameters)
• 
  Deviations of an Antoine equation fit (3 parameters)
• 
  Deviations of a DIPPR 105 equation fit (4 parameters)

The coefficients of Antoine's equation are normally given in mmHg—even today where the SI is recommended and pascals are preferred. The usage of the pre-SI units has only historic reasons and originates directly from Antoine's original publication.

It is however easy to convert the parameters to different pressure and temperature units. For switching from degrees Celsius to kelvin it is sufficient to subtract 273.15 from the C parameter. For switching from millimeters of mercury to pascals it is sufficient to add the common logarithm of the factor between both units to the A parameter:

${\displaystyle A_{\mathrm {Pa} }=A_{\mathrm {mmHg} }+\log _{10}{\frac {101325}{760}}=A_{\mathrm {mmHg} }+2.124903.}$

The parameters for °C and mmHg for ethanol

• A, 8.20417
• B, 1642.89
• C, 230.300

are converted for K and Pa to

• A, 10.32907
• B, 1642.89
• C, −42.85

The first example calculation with T_B = 351.47 K becomes

${\displaystyle \log _{10}(P)=10{.}3291-{\frac {1642{.}89}{351{.}47-42{.}85}}=5{.}005727378=\log _{10}(101328\ \mathrm {Pa} ).}$

A similarly simple transformation can be used if the common logarithm should be exchanged by the natural logarithm. It is sufficient to multiply the A and B parameters by ln(10) = 2.302585.

The example calculation with the converted parameters (for K and Pa):

• A, 23.7836
• B, 3782.89
• C, −42.85

becomes

${\displaystyle \ln P=23{.}7836-{\frac {3782{.}89}{351{.}47-42{.}85}}=11{.}52616367=\ln(101332\,\mathrm {Pa} ).}$

(The small differences in the results are only caused by the used limited precision of the coefficients).

To overcome the limits of the Antoine equation some simple extension by additional terms are used:

${\displaystyle {\begin{aligned}P&=\exp {\left(A+{\frac {B}{C+T}}+D\cdot T+E\cdot T^{2}+F\cdot \ln \left(T\right)\right)}\\P&=\exp \left(A+{\frac {B}{C+T}}+D\cdot \ln \left(T\right)+E\cdot T^{F}\right).\end{aligned}}}$

The additional parameters increase the flexibility of the equation and allow the description of the entire vapor pressure curve. The extended equation forms can be reduced to the original form by setting the additional parameters D, E and F to 0.

A further difference is that the extended equations use the e as base for the exponential function and the natural logarithm. This doesn't affect the equation form.

• NIST Chemistry WebBook
• Dortmund Data Bank
• Directory of reference books and data banks containing Antoine constants
• Several reference books and publications, e. g.
  • Lange's Handbook of Chemistry, McGraw-Hill Professional
  • Wichterle I., Linek J., "Antoine Vapor Pressure Constants of Pure Compounds"
  • Yaws C. L., Yang H.-C., "To Estimate Vapor Pressure Easily. Antoine Coefficients Relate Vapor Pressure to Temperature for Almost 700 Major Organic Compounds", Hydrocarbon Processing, 68(10), Pages 65–68, 1989

• Vapour pressure of water
• Arden Buck equation
• Lee–Kesler method
• Goff–Gratch equation
• Raoult's law
• Thermodynamic activity

• Gallica, scanned original paper
• NIST Chemistry Web Book
• Calculation of vapor pressures with the Antoine equation