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 cannot be used for the entire vapor pressure range from the triple point to the critical point because it is not flexible enough. Therefore two sets of coefficients are commonly used: one set for vapor pressures at temperatures below the normal boiling point (NBP) and one set for vapor presures at temperatures above the normal boiling point.

Example sets of coefficients

The table below lists Antoine equation coefficients for water and for ethanol with each having two sets of coefficients: one for the temperature range below the normal boiling point (NBP) and one for the temperature range above the NBP. The temperature ranges are denoted by the indicated minimum and and maximum temperatures.

	A	B	C	T minimum	T maximum
Water below the NBP	8.07131	1730.63	233.426	1	100
Water above the NBP	8.14019	1810.94	244.485	99	374
Ethanol below the NBP	8.20417	1642.89	230.300	-57	80
Ethanol below the NBP	7.68117	1332.04	199.200	77	243

The above coefficients are for temperatures in °C and absolute pressures in mmHg when using $\log _{10}$ as the logarithmic function.

Example calculations

The NBP of ethanol is 78.32 °C. Using the Antoine coefficient range for below the NBP from the above table, the vapor pressure at the NBP temperature is:

$\log P=8.20417-{\frac {1642.89}{230.000+78.32}}=2.8808$

and

$P=10^{2.8808}=760.02\;\mathrm {mmHg}$

Using the Antoine coefficient range for above the NBP from the above table, the vapor pressure at the NBP temperature is:

$\log P=7.68117-{\frac {1332.04}{199.200+78.32}}=2.8814$

and

$P=10^{2.8814}=760.98\;\mathrm {mmHg}$

(760 mmHg = 1.000 atm = typical atmospheric pressure at sea level)

This example shows the problem caused by using two different sets of coefficients. The two sets of coefficients give different results at the NBP temperature. This causes 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 parameter set specially fitted for the desired temperature range. The second approach is to use the equations of Wagner or of the AIChE's Design Institute for Physical Properties (DIPPR).

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

A	B	C
8.20417	1642.89	230.300

are converted for K and Pa to

A	B	C
10.32907	1642.89	-42.85

The first example calculation with T_B = 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 )

A	B	C
23.7836	3782.89	-42.85

becomes

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

(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.

External links

• Calculation of vapor pressures with the Antoine equation