Antoine equation

Clausius-Clapeyron ( p vs. T )

The Antoine equation describes the functional relationship between temperature and saturation vapor pressure for pure substances and is derived from the Clausius-Clapeyron equation .

the equation

Water vapor pressure
left: p vs. T , right: log ( p ) vs. 1 / T

The simplified form with ${\ displaystyle C = 0 \ colon}$

{\ displaystyle {\ begin {aligned} p & = 10 ^ {A - {\ frac {B} {T}}} \\\ Leftrightarrow \ log _ {10} \ p & = A - {\ frac {B} {T }} \ end {aligned}}}

is called the August equation after the German physicist Ernst Ferdinand August (1795–1870). It is derived directly from the Clausius-Clapeyron relationship and describes a purely linear relationship between the logarithm of pressure and the reciprocal of temperature. The factor B (slope of the straight line) corresponds to the enthalpy of evaporation .

The curvature of the straight line is specified by the Antoine equation by Louis Charles Antoine (1825–1897).

Original equation

Antoine used the shape

{\ displaystyle \ log _ {10} p \ = \ A _ {\ text {orig}} \ left (D - {\ frac {1000} {\ theta}} \ right)}

It turns out that with the conversions

{\ displaystyle {\ begin {aligned} A \ & = A _ {\ text {orig}} \ cdot D \\ B \ & = 1000 \ cdot A _ {\ text {orig}} \\ C \ & = \ theta \ - \ T \ end {aligned}}}

the form used today corresponds to the original equation.

With the values A _orig = 1.1650, D = 5.8524 and C = 216 given by Antoine for benzene , a vapor pressure of benzene results for 80 ° C (corresponds to the boiling point of benzene)

{\ displaystyle {\ begin {aligned} \ log _ {10} p \ & = \ 1 {,} 1650 \ left (5 {,} 8524 - {\ frac {1000} {80 + 216}} \ right) \ \\ Leftrightarrow p \ & = \ 762 {,} 5 \, \ mathrm {mmHg} \ end {aligned}}}

and thus approximately atmospheric pressure .

Form in use today

{\ displaystyle {\ begin {aligned} p & = 10 ^ {A - {\ frac {B} {C + T}}} \\\ Leftrightarrow \ log _ {10} \ p & = A - {\ frac {B} {C + T}} \ end {aligned}}}

With

p : pressure, usually in mmHg given
T : temperature, given in ° C
A , B , C empirical , substance-specific parameter, wherein the easily from B molar vaporization enthalpy derived can be.

Temperature explicit form (inverse function)

The Antoine equation can be rearranged so that temperature can be calculated as a function of pressure:

{\ displaystyle \ Leftrightarrow T = {\ frac {B} {A- \ log _ {10} \, p}} - C}

scope

The Antoine equation can not be the entire saturated vapor pressure curve between the three parameters with its triple point and the critical point describe correctly. Therefore, 2 parameter sets are mostly used for one component. As a rule, one set of parameters is valid below the normal boiling point and a second for the range from the normal boiling point to the critical point. Since inconsistencies then arise at the transition point, the application of the Antoine equation is no longer appropriate today.

Typical deviations of a parameter adjustment over the entire range (experimental data for benzene)
Deviation of an adjustment of the August equation (2 parameters)
Deviation of an adaptation of the Antoine equation (3 parameters)
Deviation of an adaptation of the DIPPR 101 equation (4 parameters)

Examples

	A.	B.	C.	T _min (° C)	T _max (° C)
water	8.07131	1730.63	233,426	1	100
Water (under pressure, p> 760 mmHg)	8.14019	1810.94	244,485	99	374
Ethanol	8.20417	1642.89	230,300	−57	80
Ethanol (under pressure, p> 760 mmHg)	7.68117	1332.04	199,200	77	243

[ p ] = mmHg

Sample calculation

The normal boiling point T _B for ethanol is 78.32 ° C. This results in the following calculation:

{\ displaystyle p = 10 ^ {8 {,} 20417 - {\ frac {1642 {,} 89} {78 {,} 32 + 230 {,} 300}}} = 760 {,} 0 \, \ mathrm { mmHg}}

{\ displaystyle 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 )

units

The coefficients of the Antoine equation are usually given in mmHg and ° C - even today, where the SI system of units is preferred and thus the pressure unit Pascal and the temperature unit Kelvin . The use of the pre-SI units is purely historical and comes straight from Antoine's original publication.

However, it is easy to convert the Antoine parameters to other pressure and temperature units. To change from degrees Celsius to Kelvin, subtract 273.15 from the C parameter. To switch from millimeter mercury to Pascal, it is enough to add the decadic logarithm of the factor between the units to the A parameter:

{\ displaystyle A _ {\ text {Pa}} = A _ {\ text {mmHg}} + \ log _ {10} {\ frac {101325} {760}} = A _ {\ text {mmHg}} + 2 {, } 124903}

(101325 Pa is 760 mmHg)

The parameters in ° C and mmHg for ethanol

A.	B.	C.
8.20417	1642.89	230,300

are converted to for K and Pa

A.	B.	C.
10.32907	1642.89	−42.85

The first example calculation at T _B = 351.47 K becomes

{\ displaystyle \ log _ {10} p = 10 {,} 3291 - {\ frac {1642 {,} 89} {351 {,} 47-42 {,} 85}} = 101328 \, \ mathrm {Pa} }

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

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

A.	B.	C.
23.7836	3782.89	−42.85

becomes

{\ displaystyle \ ln p = 23 {,} 7836 - {\ frac {3782 {,} 89} {351 {,} 47-42 {,} 85}} = 101332 \, \ mathrm {Pa}}

(The small differences in the results arise exclusively from the limited accuracy of the coefficients used.)

Extensions of the equation

In order to get around the limitations of the Antoine equation, there are simple extensions to further terms.

${\ displaystyle p = \ exp {\ left (A + {\ frac {B} {C + T}} + D \ cdot T + E \ cdot T ^ {2} + F \ cdot \ ln \ left (T \ right ) \ right)}}$

${\ displaystyle p = \ exp \ left (A + {\ frac {B} {C + T}} + D \ cdot \ ln \ left (T \ right) + E \ cdot T ^ {F} \ right)}$

The other parameters increase the flexibility of the equations and thus allow the description of the entire vapor pressure curve from the triple point to the critical point. In addition, the extended equations can be reduced to the original Antoine equation by setting the additional parameters D , E and F to 0.

Another difference to the original form is that the extended equations use the exponential function and the natural logarithm. However, this does not affect the form of the equation.

swell

NIST Chemistry WebBook
Dortmund database
Various reference works and publications, e.g.
- Lange's Handbook of Chemistry. McGraw-Hill Professional
- Ivan Wichterle, Jan Linek: Antoine Vapor Pressure Constants of pure compounds. Academia, Prague 1971
- Carl L. Yaws, Haur-Chung Yang: To Estimate Vapor Pressure Easily. Antoine Coefficients Relate Vapor Pressure to Temperature for Almost 700 Major Organic Compounds. In: Hydrocarbon Processing. Vol. 68, No. 10, 1989, ISSN 0018-8190 , pp. 65-68.

literature

Ch. Antoine: Tensions des vapeurs: nouvelle relation entre les tensions et les températures. In: Comptes Rendus des Séances de l'Académie des Sciences. Vol. 107, 1888, pp. 681-684 , 778-780 , 836-837 .