The Antoine equation is a class of
semi-empirical
Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences an ...
correlations describing the relation between
vapor pressure
Vapor pressure (or vapour pressure in English-speaking countries other than the US; see spelling differences) or equilibrium vapor pressure is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases ...
and temperature for pure substances. The Antoine equation is derived from the
Clausius–Clapeyron relation
The Clausius–Clapeyron relation, named after Rudolf Clausius and Benoît Paul Émile Clapeyron, specifies the temperature dependence of pressure, most importantly vapor pressure, at a discontinuous phase transition between two phases of matter ...
. The equation was presented in 1888 by the French engineer (1825–1897).
Equation
The Antoine equation is
:
where ''p'' is the vapor pressure, is
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
(in °C or in K according to the value of C) and , and are component-specific constants.
The simplified form with set to zero:
:
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 enthalpy of vaporization (symbol ), also known as the (latent) heat of vaporization or heat of evaporation, is the amount of energy (enthalpy) that must be added to a liquid substance to transform a quantity of that substance into a gas. T ...
. 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:
:
Validity range
Usually, the Antoine equation cannot be used to describe the entire saturated vapour pressure curve from the
triple point
In thermodynamics, the triple point of a substance is the temperature and pressure at which the three phases (gas, liquid, and solid) of that substance coexist in thermodynamic equilibrium.. It is that temperature and pressure at which the subli ...
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
The boiling point of a substance is the temperature at which the vapor pressure of a liquid equals the pressure surrounding the liquid and the liquid changes into a vapor.
The boiling point of a liquid varies depending upon the surrounding envir ...
and the second set of parameters is used for the range from the normal boiling point to the critical point.
Image:VaporPressureFitAugust.png , Deviations of an ''August'' equation fit (2 parameters)
Image:VaporPressureFitAntoine.png , Deviations of an ''Antoine'' equation fit (3 parameters)
Image:VaporPressureFitDIPPR101.png , Deviations of a ''DIPPR 105'' equation fit (4 parameters)
Example parameters
Example calculation
The normal boiling point of ethanol is ''T''
B = 78.32 °C.
:
(760mmHg = 101.325kPa = 1.000atm = normal pressure)
This example shows a severe problem caused by using two different sets of coefficients. The described vapor pressure is not
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** 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—even today where the
SI is recommended and
pascal
Pascal, Pascal's or PASCAL may refer to:
People and fictional characters
* Pascal (given name), including a list of people with the name
* Pascal (surname), including a list of people and fictional characters with the name
** Blaise Pascal, Fren ...
s 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
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
of the factor between both units to the A parameter:
The parameters for °C and mmHg for
ethanol
Ethanol (abbr. EtOH; also called ethyl alcohol, grain alcohol, drinking alcohol, or simply alcohol) is an organic compound. It is an Alcohol (chemistry), alcohol with the chemical formula . Its formula can be also written as or (an ethyl ...
* 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
:
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
:
(The small differences in the results are only caused by the used limited precision of the coefficients).
Extension of the Antoine equations
To overcome the limits of the Antoine equation some simple extension by additional terms are used:
:
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.
Sources for Antoine equation parameters
NIST Chemistry WebBook*
Dortmund Data Bank
The Dortmund Data Bank (short DDB) is a factual data bank for thermodynamic and thermophysical data. Its main usage is the data supply for process simulation where experimental data are the basis for the design, analysis, synthesis, and optimizati ...
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
See also
*
Vapour pressure of water
The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). The saturation vapour pressure is the pressure at which water vapour is in thermodynamic ...
*
Arden Buck equation The Arden Buck equations are a group of empirical correlations that relate the saturation vapor pressure to temperature for moist air. The curve fits have been optimized for more accuracy than the Goff–Gratch equation in the range .Buck 1981
A s ...
*
Lee–Kesler method
The Lee–Kesler method
allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure ''P''c, the critical temperature ''T''c, and the acentric factor ''ω'' are known.
Equation ...
*
Goff–Gratch equation The Goff–Gratch equation is one (arguably the first reliable in history) amongst many experimental correlation proposed to estimate the saturation water vapor pressure at a given temperature.
Another similar equation based on more recent data is ...
*
Raoult's law
*
Thermodynamic activity
In chemical thermodynamics, activity (symbol ) is a measure of the "effective concentration" of a species in a mixture, in the sense that the species' chemical potential depends on the activity of a real solution in the same way that it would depen ...
References
External links
Gallica, scanned original paperNIST Chemistry Web BookCalculation of vapor pressures with the Antoine equation
{{DEFAULTSORT:Antoine Equation
Equations
Thermodynamic equations