User:Milton Beychok/Sandbox

The Antoine equation is a mathematical expression (derived from the Clausius-Clapeyron equation) of the relation between the vapor pressure and the temperature for pure substances. The basic form of the equation is:


 * $$\log P = A-\frac{B}{C+T}$$

and it can be transformed into this temperature-explicit form:


 * $$T = \frac{B}{A-\log P} - C$$

where: $$P$$ is the absolute vapor pressure of a substance
 * $$T$$ is the temperature of the substance
 * $$A$$, $$B$$ and $$C$$ are substance-specific coefficients (i.e., constants or parameters)
 * $$\log$$ is typically either $$\log_{10}$$ or $$\log_e$$

A simpler form of the equation with only two coefficents is sometimes used:


 * $$\log P = A-\frac{B}{T}$$

which can be transformed to:


 * $$T = \frac{B}{A-\log P}$$

Validity ranges
The Antoine equation can't be used to describe the entire saturated vapor pressure curve from the triple point to the critical point because it is not flexible enough. Therefore two parameter sets for a single component are commonly used. A low-pressure parameter set is used to describe the vapor pressure curve up to the normal boiling point and the second set of parameters is used for the range from the nomal boiling point to the critical point.

Example coefficients
The constants are given in °C and mmHg.

Example Calculation
The normal boiling point of Ethanol is TB=78.32 °C.


 * $$P = 10^{8{.}20417 - \frac{1642{.}89}{78{.}32 + 230{.}300}} = 760{.}0\,\mathrm{mmHg}$$


 * $$P = 10^{7{.}68117 - \frac{1332{.}04}{78{.}32 + 199{.}200}} = 761{.}0\,\mathrm{mmHg}$$

(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.

Units
The coefficients of Antoine's equation are normally given in mmHg and °C - even today where the SI is recommended and Pascal and Kelvin 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 degree Celsius to Kelvin it is sufficient to subtract 273.15 from the C parameter. For switching from millimeter of mercury to Pascal it is sufficient to add the common logarithm of the factor between both units to the A parameter:

$$A_{Pa} = A_{mmHg} + log_{10}\frac{101325}{760} = A_{mmHg} + 2.124903$$

The parameters for °C and mmHg for Ethanol
 * {| cellpadding="2" rules="all" style="border: 1px solid #aaa;"

!A !B !C
 * 8.20417||1642.89||230.300
 * }
 * }

are converted for K and Pa to


 * {| cellpadding="2" rules="all" style="border: 1px solid #aaa;"

!A !B !C
 * 10.32907||1642.89||-42.85
 * }
 * }

The first example calculation with TB = 351.47 K becomes
 * $$log_{10}P = 10{.}3291 - \frac{1642{.}89}{351{.}47 - 42{.}85} = 101328\,\mathrm{Pa}$$

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 )


 * {| cellpadding="2" rules="all" style="border: 1px solid #aaa;"

!A !B !C
 * 23.7836||3782.89||-42.85
 * }
 * }

becomes


 * $$ln P = 23{.}7836 - \frac{3782{.}89}{351{.}47 - 42{.}85} = 101332\,\mathrm{Pa}$$

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

Sources of Antoine equation coefficients

 * The NIST online chemistry web book.


 * The online Physical Properties Sources Index (PPSI) of the Swiss Federal Institute of Technology.


 * The online Dortmund Data Bank.


 * Several books and publications.