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== Vapor pressure-temperature chart ==
== Vapor pressure-temperature chart ==
A plot of vapor pressures for various temperatures is shown in '''Fig. 1''' for ''n''-butane. The temperature at which the vapor pressure is equal to 1 atm (14.696 psia or 101.32 kPa) is known as the normal boiling point.  
 
A plot of vapor pressures for various temperatures is shown in '''Fig. 1''' for ''n''-butane. The temperature at which the vapor pressure is equal to 1 atm (14.696 psia or 101.32 kPa) is known as the normal boiling point.


<gallery widths="300px" heights="200px">
<gallery widths="300px" heights="200px">
Line 9: Line 10:


== Clapeyron equation ==
== Clapeyron equation ==
The Clapeyron equation gives a rigorous quantitative relationship between vapor pressure and temperature:  
The Clapeyron equation gives a rigorous quantitative relationship between vapor pressure and temperature:  


[[File:Vol1 page 0241 eq 003.png]]....................(1)
[[File:Vol1 page 0241 eq 003.png|RTENOTITLE]]....................(1)


where:
where:
*''p''<sub>''v''</sub> = vapor pressure
*''p''<sub>''v''</sub> = vapor pressure
*''T'' = absolute temperature
*''T'' = absolute temperature
*Δ''V'' = increase in volume caused by vaporizing 1 mole
*Δ''V'' = increase in volume caused by vaporizing 1 mole
*''L''<sub>''v''</sub> = molal latent heat of vaporization.  
*''L''<sub>''v''</sub> = molal latent heat of vaporization.


Assuming [[Ideal gases|ideal gas behavior]] of the vapor and neglecting the liquid volume, the Clapeyron equation can be simplified over a small temperature range to give the approximation  
Assuming [[Ideal_gases|ideal gas behavior]] of the vapor and neglecting the liquid volume, the Clapeyron equation can be simplified over a small temperature range to give the approximation  


[[File:Vol1 page 0241 eq 004.png]]....................(2)
[[File:Vol1 page 0241 eq 004.png|RTENOTITLE]]....................(2)


which is known as the Clausius-Clapeyron equation. Integrating this equation gives
which is known as the Clausius-Clapeyron equation. Integrating this equation gives


[[File:Vol1 page 0242 eq 001.png]]....................(3)
[[File:Vol1 page 0242 eq 001.png|RTENOTITLE]]....................(3)


where ''b'' is a constant of integration that depends on the particular fluid and the data range. This equation suggests that a plot of logarithm of vapor pressure against the reciprocal of the absolute temperature would approximate a straight line. Such a plot is useful in interpolating and extrapolating data over short ranges. However, the shape of this relationship for a real substance over a significant temperature range is more S-shaped than straight. Therefore, the use of the Clausius-Clapeyron equation is not recommended when other methods are available, except over short temperature ranges in regions where the ideal gas law is valid.  
where ''b'' is a constant of integration that depends on the particular fluid and the data range. This equation suggests that a plot of logarithm of vapor pressure against the reciprocal of the absolute temperature would approximate a straight line. Such a plot is useful in interpolating and extrapolating data over short ranges. However, the shape of this relationship for a real substance over a significant temperature range is more S-shaped than straight. Therefore, the use of the Clausius-Clapeyron equation is not recommended when other methods are available, except over short temperature ranges in regions where the ideal gas law is valid.


== Cox chart ==
== Cox chart ==
Cox<ref name="r1" /> further improved the method of estimating vapor pressure by plotting the logarithm of vapor pressure against an arbitrary temperature scale. The vapor-pressure/temperature plot forms a straight line, at least for the reference compound (and usually for most of the materials related to the reference compound). This is especially true for petroleum hydrocarbons. A Cox chart, using water as a reference material, is shown in '''Fig. 2'''. In addition to forming nearly straight lines, compounds of the same family appear to converge on a single point. Thus, it is necessary to know only vapor pressure at one temperature to estimate the position of the vapor-pressure line. This approach is very useful and can be much better than the Clapeyron method. Its accuracy is dependent to a large degree on the readability of the chart.  
 
Cox<ref name="r1">Cox, E.R. 1923. Pressure-Temperature Chart for Hydrocarbon Vapors. Ind. Eng. Chem. 15 (6): 592-593. http://dx.doi.org/10.1021/ie50162a013</ref> further improved the method of estimating vapor pressure by plotting the logarithm of vapor pressure against an arbitrary temperature scale. The vapor-pressure/temperature plot forms a straight line, at least for the reference compound (and usually for most of the materials related to the reference compound). This is especially true for petroleum hydrocarbons. A Cox chart, using water as a reference material, is shown in '''Fig. 2'''. In addition to forming nearly straight lines, compounds of the same family appear to converge on a single point. Thus, it is necessary to know only vapor pressure at one temperature to estimate the position of the vapor-pressure line. This approach is very useful and can be much better than the Clapeyron method. Its accuracy is dependent to a large degree on the readability of the chart.


<gallery widths="300px" heights="200px">
<gallery widths="300px" heights="200px">
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== Calingheart and Davis equation ==
== Calingheart and Davis equation ==
The Cox chart was fit with a three-parameter function by Calingeart and Davis.<ref name="r2" /> Their equation is


[[File:Vol1 page 0242 eq 002.png]]....................(4)
The Cox chart was fit with a three-parameter function by Calingeart and Davis.<ref name="r2">Calingaert, G. and Davis, D.S. 1925. Pressure-Temperature Charts—Extended Ranges. Ind. Eng. Chem. 17 (12): 1287-1288. http://dx.doi.org/10.1021/ie50192a037</ref> Their equation is
 
[[File:Vol1 page 0242 eq 002.png|RTENOTITLE]]....................(4)


where ''A'' and ''B'' are empirical constants and, for compounds boiling between 32 and 212°F, ''C'' is a constant with a value of 43 when ''T'' is in K and a value of 77.4 when ''T'' is in °R. This equation generally is known as the Antoine<ref name="r3" /> equation because Antoine proposed one of a very similar nature that used 13 K for the constant ''C''. Knowledge of the vapor pressure at two temperatures will fix ''A'' and ''B'' and permit approximations of vapor pressures at other temperatures. Generally, the Antoine approach can be expected to have less than 2% error and is the preferred approach if the vapor pressure is expected to be less than 1,500 mm Hg (200 kPa) and if the constants are available.  
where ''A'' and ''B'' are empirical constants and, for compounds boiling between 32 and 212°F, ''C'' is a constant with a value of 43 when ''T'' is in K and a value of 77.4 when ''T'' is in °R. This equation generally is known as the Antoine<ref name="r3">Antoine, C. 1888. Tensions des vapeurs; nouvelle relation entre les tensions et les températures. Comptes Rendus des Séances de l'Académie des Sciences 107: 836–850.</ref> equation because Antoine proposed one of a very similar nature that used 13 K for the constant ''C''. Knowledge of the vapor pressure at two temperatures will fix ''A'' and ''B'' and permit approximations of vapor pressures at other temperatures. Generally, the Antoine approach can be expected to have less than 2% error and is the preferred approach if the vapor pressure is expected to be less than 1,500 mm Hg (200 kPa) and if the constants are available.


== Lee-Kesler equation ==
== Lee-Kesler equation ==
Vapor pressures also can be calculated by corresponding-states principles. The most common expansions of the Clapeyron equation lead to a two-parameter expression. Pitzer ''et al.''<ref name="r4" /> extended the expansion to contain three parameters:


[[File:Vol1 page 0243 eq 001.png]]....................(5)
Vapor pressures also can be calculated by corresponding-states principles. The most common expansions of the Clapeyron equation lead to a two-parameter expression. Pitzer ''et al.''<ref name="r4">Pitzer, K.S., Lippman, D.Z., Curl, R.F. Jr. et al. 1955. The Volumetric and Thermodynamic Properties of Fluids. II. Compressibility Factor, Vapor Pressure, and Entropy of Vaporization. J. Am. Chem. Soc. 77: 3433.</ref> extended the expansion to contain three parameters:
 
[[File:Vol1 page 0243 eq 001.png|RTENOTITLE]]....................(5)


where ''p''<sub>''vr''</sub> is the reduced vapor pressure (vapor pressure/critical pressure), ''f''<sup>0</sup> and ''f''<sup>1</sup> are functions of reduced temperature, and ''ω'' is the acentric factor.  
where ''p''<sub>''vr''</sub> is the reduced vapor pressure (vapor pressure/critical pressure), ''f''<sup>0</sup> and ''f''<sup>1</sup> are functions of reduced temperature, and ''ω'' is the acentric factor.


Lee and Kesler<ref name="r5" /> have expressed ''f''<sup>0</sup> and ''f''<sup>1</sup> in analytical forms:
Lee and Kesler<ref name="r5">Lee, B.I. and Kesler, M.G. 1975. A generalized thermodynamic correlation based on three-parameter corresponding states. AIChE J. 21 (3): 510-527. http://dx.doi.org/10.1002/aic.690210313</ref> have expressed ''f''<sup>0</sup> and ''f''<sup>1</sup> in analytical forms:


[[File:Vol1 page 0243 eq 002.png]]....................(6)
[[File:Vol1 page 0243 eq 002.png|RTENOTITLE]]....................(6)


[[File:Vol1 page 0243 eq 003.png|RTENOTITLE]]....................(7)


[[File:Vol1 page 0243 eq 003.png]]....................(7)
which can be solved easily by computer or spreadsheet. Lee-Kesler<ref name="r5">Lee, B.I. and Kesler, M.G. 1975. A generalized thermodynamic correlation based on three-parameter corresponding states. AIChE J. 21 (3): 510-527. http://dx.doi.org/10.1002/aic.690210313</ref> is the preferred method of calculation but should be used only for nonpolar liquids.


which can be solved easily by computer or spreadsheet. Lee-Kesler<ref name="r5" /> is the preferred method of calculation but should be used only for nonpolar liquids.  
The advent of computers, calculators, and spreadsheets makes the use of approximations and charts much less advantageous than it was before the 1970s. Values of acentric factors can be found in Poling ''et al.'',<ref name="r6">Poling, B.E., Prausnitz, J.M., and O’Connell, J.P. 2001. The Properties of Gases and Liquids, fifth edition (paperback). New York: McGraw-Hill Professional.</ref> who also presented many other available vapor-pressure correlations and calculation techniques, with comments about their advantages and limitations.


The advent of computers, calculators, and spreadsheets makes the use of approximations and charts much less advantageous than it was before the 1970s. Values of acentric factors can be found in Poling ''et al.'',<ref name="r6" /> who also presented many other available vapor-pressure correlations and calculation techniques, with comments about their advantages and limitations.
== Example: Estimating vapor pressure of a gas ==


==Example: Estimating vapor pressure of a gas==
The vapor pressure of pure hexane as a function of temperature is 54.04 kPa at 50°C and 188.76 kPa at 90°C. Estimate the vapor pressure of hexane at 100°C, using all the methods outlined previously.
The vapor pressure of pure hexane as a function of temperature is 54.04 kPa at 50°C and 188.76 kPa at 90°C. Estimate the vapor pressure of hexane at 100°C, using all the methods outlined previously.  


''Solution: Clausius-Clapeyron''.
''Solution: Clausius-Clapeyron''.


The Clausius-Clapeyron equation can be solved graphically by plotting a log of vapor pressure vs. reciprocal absolute temperature and extrapolating. It also can be solved by slopes fitting an equation of the form log(''p''<sub>''v''</sub>) = ''c''/''T''+''b'' to the two data points. Because the other three methods must be done in American customary units, the Clausius-Clapeyron method also will be converted to those units.  
The Clausius-Clapeyron equation can be solved graphically by plotting a log of vapor pressure vs. reciprocal absolute temperature and extrapolating. It also can be solved by slopes fitting an equation of the form log(''p''<sub>''v''</sub>) = ''c''/''T''+''b'' to the two data points. Because the other three methods must be done in American customary units, the Clausius-Clapeyron method also will be converted to those units.


*''T''<sub>1</sub> = 50°C = 122°F = 581.67°R
*''T''<sub>1</sub> = 50°C = 122°F = 581.67°R
*l/1 = 0.0017192°R<sup>–1</sup>
*l/1 = 0.0017192°R<sup>–1</sup>
*''T''<sub>2</sub> = 90°C = 653.67°R,  
*''T''<sub>2</sub> = 90°C = 653.67°R,
*1/''T''<sub>2</sub> = 0.0015298°R<sup>–1</sup>,  
*1/''T''<sub>2</sub> = 0.0015298°R<sup>–1</sup>,
*''p''<sub>''v''</sub> at ''T''<sub>1</sub> = 54.04 kPa = 7.8374 psia,  
*''p''<sub>''v''</sub> at ''T''<sub>1</sub> = 54.04 kPa = 7.8374 psia,
*log ''p''<sub>''v''</sub> = 0.89417,  
*log ''p''<sub>''v''</sub> = 0.89417,
*''p''<sub>''v''</sub> at ''T''<sub>2</sub> = 188.76 kPa = 27.3773 psia,  
*''p''<sub>''v''</sub> at ''T''<sub>2</sub> = 188.76 kPa = 27.3773 psia,
*log ''p''<sub>''v''</sub> = 1.43739,  
*log ''p''<sub>''v''</sub> = 1.43739,
*Δlog ''p''<sub>''v''</sub> = –0.543195,  
*Δlog ''p''<sub>''v''</sub> = –0.543195,
*1/''T''<sub>1</sub> –1/''T''<sub>2</sub> = 0.00018936,  
*1/''T''<sub>1</sub> –1/''T''<sub>2</sub> = 0.00018936,
*''c'' = slope = –0.543195/0.00018936 = –2868.52°R.  
*''c'' = slope = –0.543195/0.00018936 = –2868.52°R.
 


Solving for ''b'', log ''p''<sub>''v''</sub> = –2868.52 /''T''+''b'' yields  
Solving for ''b'', log ''p''<sub>''v''</sub> = –2868.52 /''T''+''b'' yields
*''b'' = 5.8257,
*''T''<sub>3</sub> = 100°C = 212°F = 671.67°R,
*1/''T''<sub>3</sub> = 0.0014888.


*''b'' = 5.8257,
*''T''<sub>3</sub> = 100°C = 212°F = 671.67°R,
*1/''T''<sub>3</sub> = 0.0014888.


Solving for ''p''<sub>''v''</sub> at 100°C yields
Solving for ''p''<sub>''v''</sub> at 100°C yields


[[File:Vol1 page 0252 eq 001.png]]
[[File:Vol1 page 0252 eq 001.png|RTENOTITLE]]


hence, ''p''<sub>''v''</sub> = 35.89 psia = 247.46 kPa.  
hence, ''p''<sub>''v''</sub> = 35.89 psia = 247.46 kPa.


Alternatively, if the vapor pressure at 70°C is 105.37 kPa and is known, you can use the 70 to 90°C temperature differential to calculate the slope and intercept and ultimately calculate ''p''<sub>''v''</sub> = 35.79 psia = 246.79 kPa.  
Alternatively, if the vapor pressure at 70°C is 105.37 kPa and is known, you can use the 70 to 90°C temperature differential to calculate the slope and intercept and ultimately calculate ''p''<sub>''v''</sub> = 35.79 psia = 246.79 kPa.


''Solution: Cox Chart''.
''Solution: Cox Chart''.


From Fig. 2<ref name="r1" /> above, the vapor pressure at 100°C can be approximated between 35 and 36 psia. A larger chart is required for more-precise readings.
From Fig. 2<ref name="r1">Cox, E.R. 1923. Pressure-Temperature Chart for Hydrocarbon Vapors. Ind. Eng. Chem. 15 (6): 592-593. http://dx.doi.org/10.1021/ie50162a013</ref> above, the vapor pressure at 100°C can be approximated between 35 and 36 psia. A larger chart is required for more-precise readings.


''Solution: Calingeart and Davis or Antoine equation''.
''Solution: Calingeart and Davis or Antoine equation''.


This can be used by obtaining the Antoine constants from Poling ''et al.''<ref name="r6" /> For ''n''-hexane, with temperature in K, these constants are ''A'' = 15.8366, ''B'' = 2697.55, and ''C'' = –48.78. Then,
This can be used by obtaining the Antoine constants from Poling ''et al.''<ref name="r6">Poling, B.E., Prausnitz, J.M., and O’Connell, J.P. 2001. The Properties of Gases and Liquids, fifth edition (paperback). New York: McGraw-Hill Professional.
[edit]</ref> For ''n''-hexane, with temperature in K, these constants are ''A'' = 15.8366, ''B'' = 2697.55, and ''C'' = –48.78. Then,


[[File:Vol1 page 0252 eq 002.png]]
[[File:Vol1 page 0252 eq 002.png|RTENOTITLE]]


and ''p''<sub>''v''</sub> = 36.68 psia = 252.73 kPa.  
and ''p''<sub>''v''</sub> = 36.68 psia = 252.73 kPa.


''Solution: Lee-Kesler''.
''Solution: Lee-Kesler''.


The use of the Lee-Kesler<ref name="r5" /> equation requires ''p''<sub>''c''</sub>, ''T''<sub>''c''</sub>, and ''ω'' for ''n''-hexane. These can be obtained from Table 1 in [[Gas properties]].
The use of the Lee-Kesler<ref name="r5">Lee, B.I. and Kesler, M.G. 1975. A generalized thermodynamic correlation based on three-parameter corresponding states. AIChE J. 21 (3): 510-527. http://dx.doi.org/10.1002/aic.690210313</ref> equation requires ''p''<sub>''c''</sub>, ''T''<sub>''c''</sub>, and ''ω'' for ''n''-hexane. These can be obtained from Table 1 in [[Gas_properties|Gas properties]].
*''p''<sub>''c''</sub> = 436.9 psia (29.7 atm)
*''T''<sub>''c''</sub> = 453.7°F or 913.3°R or 507.4 K
*''ω'' = 0.3007.  


For 100°C,  
*''p''<sub>''c''</sub> = 436.9 psia (29.7 atm)
:''T''<sub>''r''</sub> = 0.7351  
*''T''<sub>''c''</sub> = 453.7°F or 913.3°R or 507.4 K
:(''T''<sub>''r''</sub>) 6 = 0.15782  
*''ω'' = 0.3007.
 
For 100°C,
 
:''T''<sub>''r''</sub> = 0.7351
:(''T''<sub>''r''</sub>) 6 = 0.15782
:ln ''T''<sub>''r''</sub> = –0.30775
:ln ''T''<sub>''r''</sub> = –0.30775


[[File:Vol1 page 0252 eq 003.png]]
[[File:Vol1 page 0252 eq 003.png|RTENOTITLE]]


[[File:Vol1 page 0253 eq 001.png]]
[[File:Vol1 page 0253 eq 001.png|RTENOTITLE]]


: ''p''<sub>''v''</sub> = (0.0816)(29.7) = 2.4235 atm = 35.62 psia = 245.6 kPa.  
:''p''<sub>''v''</sub> = (0.0816)(29.7) = 2.4235 atm = 35.62 psia = 245.6 kPa.


''Experimental value'': 35.69 psia = 246.1 kPa.
''Experimental value'': 35.69 psia = 246.1 kPa.


''Conclusions''. Lee-Kesler gives the best answer, but the Clausius-Clapeyron method is also very accurate to within 0.17 psi, which is typical if the extrapolation is close to the appropriate range.  
''Conclusions''. Lee-Kesler gives the best answer, but the Clausius-Clapeyron method is also very accurate to within 0.17 psi, which is typical if the extrapolation is close to the appropriate range.
 
== Nomenclature ==


==Nomenclature==
{|
{|
|''p''<sub>''v''</sub>
|=
|vapor pressure
|-
|-
|''T''  
| ''p''<sub>''v''</sub>
|=  
| =
|absolute temperature
| vapor pressure
|-
| ''T''
| =
| absolute temperature
|-
|-
|''V''  
| ''V''
|=  
| =
|increase in volume caused by vaporizing 1 mole
| increase in volume caused by vaporizing 1 mole
|-
|-
|L''<sub>''v''</sub>  
| L<sub>v</sub>
|=  
| =
|molal latent heat of vaporization.  
| molal latent heat of vaporization.
|-
|-
|''A''
| ''A''
|=
| =
|empirical constant
| empirical constant
|-
|-
|''B''
| ''B''
|=
| =
|empirical constant
| empirical constant
|-
|-
|''p''<sub>''vr''</sub>
| ''p''<sub>''vr''</sub>
|=
| =
|reduced vapor pressure (vapor pressure/critical pressure)
| reduced vapor pressure (vapor pressure/critical pressure)
|-
|-
|''f''<sup>0</sup>  
| ''f''<sup>0</sup>
|=
| =
|function of reduced temperature
| function of reduced temperature
|-
|-
|''f''<sup>1</sup>
| ''f''<sup>1</sup>
|=
| =
|function of reduced temperature
| function of reduced temperature
|-
|-
|''ω''
| ''ω''
|=
| =
|acentric factor
| acentric factor
|-
|-
|''T''<sub>''r''</sub>  
| ''T''<sub>''r''</sub>
|=  
| =
|reduced temperature  
| reduced temperature
|}
|}


==References==
== References ==
<references>
<ref name="r1">Cox, E.R. 1923. Pressure-Temperature Chart for Hydrocarbon Vapors. ''Ind. Eng. Chem.'' '''15''' (6): 592-593. http://dx.doi.org/10.1021/ie50162a013</ref>


<ref name="r2">Calingaert, G. and Davis, D.S. 1925. Pressure-Temperature Charts—Extended Ranges. ''Ind. Eng. Chem.'' '''17''' (12): 1287-1288. http://dx.doi.org/10.1021/ie50192a037 </ref>
<references />


<ref name="r3">Antoine, C. 1888. Tensions des vapeurs; nouvelle relation entre les tensions et les températures. ''Comptes Rendus des Séances de l'Académie des Sciences'' '''107''': 836–850.</ref>
== Noteworthy papers in OnePetro ==


<ref name="r4">Pitzer, K.S., Lippman, D.Z., Curl, R.F. Jr. et al. 1955. The Volumetric and Thermodynamic Properties of Fluids. II. Compressibility Factor, Vapor Pressure, and Entropy of Vaporization. ''J. Am. Chem. Soc.'' '''77''': 3433.</ref>
Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read


<ref name="r5">Lee, B.I. and Kesler, M.G. 1975. A generalized thermodynamic correlation based on three-parameter corresponding states. ''AIChE J.'' '''21''' (3): 510-527. http://dx.doi.org/10.1002/aic.690210313</ref>
== External links ==


<ref name="r6">Poling, B.E., Prausnitz, J.M., and O’Connell, J.P. 2001. The Properties of Gases and Liquids, fifth edition (paperback). New York: McGraw-Hill Professional.</ref>
Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro
</references>


==Noteworthy papers in OnePetro==
== See also ==
Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read


==External links==
[[Gas_properties|Gas properties]]
Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro


==See also==
[[Real_gases|Real gases]]
[[Gas properties]]


[[Real gases]]
[[Calculating_gas_properties|Calculating gas properties]]


[[Calculating gas properties]]
[[PEH:Gas_Properties]]


[[PEH:Gas Properties]]
[[Category:5.2.1 Phase behavior and PVT measurements]]

Latest revision as of 16:37, 4 June 2015

At a given temperature, the vapor pressure of a pure compound is the pressure at which vapor and liquid coexist at equilibrium. The term "vapor pressure" should be used only with pure compounds and is usually considered as a liquid (rather than a gas) property. For a pure compound, there is only one vapor pressure at any temperature. This page discusses calculation of vapor pressure from other properties.

Vapor pressure-temperature chart

A plot of vapor pressures for various temperatures is shown in Fig. 1 for n-butane. The temperature at which the vapor pressure is equal to 1 atm (14.696 psia or 101.32 kPa) is known as the normal boiling point.

Clapeyron equation

The Clapeyron equation gives a rigorous quantitative relationship between vapor pressure and temperature:  

RTENOTITLE....................(1)

where:

  • pv = vapor pressure
  • T = absolute temperature
  • ΔV = increase in volume caused by vaporizing 1 mole
  • Lv = molal latent heat of vaporization.

Assuming ideal gas behavior of the vapor and neglecting the liquid volume, the Clapeyron equation can be simplified over a small temperature range to give the approximation  

RTENOTITLE....................(2)

which is known as the Clausius-Clapeyron equation. Integrating this equation gives

RTENOTITLE....................(3)

where b is a constant of integration that depends on the particular fluid and the data range. This equation suggests that a plot of logarithm of vapor pressure against the reciprocal of the absolute temperature would approximate a straight line. Such a plot is useful in interpolating and extrapolating data over short ranges. However, the shape of this relationship for a real substance over a significant temperature range is more S-shaped than straight. Therefore, the use of the Clausius-Clapeyron equation is not recommended when other methods are available, except over short temperature ranges in regions where the ideal gas law is valid.

Cox chart

Cox[1] further improved the method of estimating vapor pressure by plotting the logarithm of vapor pressure against an arbitrary temperature scale. The vapor-pressure/temperature plot forms a straight line, at least for the reference compound (and usually for most of the materials related to the reference compound). This is especially true for petroleum hydrocarbons. A Cox chart, using water as a reference material, is shown in Fig. 2. In addition to forming nearly straight lines, compounds of the same family appear to converge on a single point. Thus, it is necessary to know only vapor pressure at one temperature to estimate the position of the vapor-pressure line. This approach is very useful and can be much better than the Clapeyron method. Its accuracy is dependent to a large degree on the readability of the chart.

Calingheart and Davis equation

The Cox chart was fit with a three-parameter function by Calingeart and Davis.[2] Their equation is

RTENOTITLE....................(4)

where A and B are empirical constants and, for compounds boiling between 32 and 212°F, C is a constant with a value of 43 when T is in K and a value of 77.4 when T is in °R. This equation generally is known as the Antoine[3] equation because Antoine proposed one of a very similar nature that used 13 K for the constant C. Knowledge of the vapor pressure at two temperatures will fix A and B and permit approximations of vapor pressures at other temperatures. Generally, the Antoine approach can be expected to have less than 2% error and is the preferred approach if the vapor pressure is expected to be less than 1,500 mm Hg (200 kPa) and if the constants are available.

Lee-Kesler equation

Vapor pressures also can be calculated by corresponding-states principles. The most common expansions of the Clapeyron equation lead to a two-parameter expression. Pitzer et al.[4] extended the expansion to contain three parameters:

RTENOTITLE....................(5)

where pvr is the reduced vapor pressure (vapor pressure/critical pressure), f0 and f1 are functions of reduced temperature, and ω is the acentric factor.

Lee and Kesler[5] have expressed f0 and f1 in analytical forms:

RTENOTITLE....................(6)

RTENOTITLE....................(7)

which can be solved easily by computer or spreadsheet. Lee-Kesler[5] is the preferred method of calculation but should be used only for nonpolar liquids.

The advent of computers, calculators, and spreadsheets makes the use of approximations and charts much less advantageous than it was before the 1970s. Values of acentric factors can be found in Poling et al.,[6] who also presented many other available vapor-pressure correlations and calculation techniques, with comments about their advantages and limitations.

Example: Estimating vapor pressure of a gas

The vapor pressure of pure hexane as a function of temperature is 54.04 kPa at 50°C and 188.76 kPa at 90°C. Estimate the vapor pressure of hexane at 100°C, using all the methods outlined previously.

Solution: Clausius-Clapeyron.

The Clausius-Clapeyron equation can be solved graphically by plotting a log of vapor pressure vs. reciprocal absolute temperature and extrapolating. It also can be solved by slopes fitting an equation of the form log(pv) = c/T+b to the two data points. Because the other three methods must be done in American customary units, the Clausius-Clapeyron method also will be converted to those units.

  • T1 = 50°C = 122°F = 581.67°R
  • l/1 = 0.0017192°R–1
  • T2 = 90°C = 653.67°R,
  • 1/T2 = 0.0015298°R–1,
  • pv at T1 = 54.04 kPa = 7.8374 psia,
  • log pv = 0.89417,
  • pv at T2 = 188.76 kPa = 27.3773 psia,
  • log pv = 1.43739,
  • Δlog pv = –0.543195,
  • 1/T1 –1/T2 = 0.00018936,
  • c = slope = –0.543195/0.00018936 = –2868.52°R.

Solving for b, log pv = –2868.52 /T+b yields

  • b = 5.8257,
  • T3 = 100°C = 212°F = 671.67°R,
  • 1/T3 = 0.0014888.

Solving for pv at 100°C yields

RTENOTITLE

hence, pv = 35.89 psia = 247.46 kPa.

Alternatively, if the vapor pressure at 70°C is 105.37 kPa and is known, you can use the 70 to 90°C temperature differential to calculate the slope and intercept and ultimately calculate pv = 35.79 psia = 246.79 kPa.

Solution: Cox Chart.

From Fig. 2[1] above, the vapor pressure at 100°C can be approximated between 35 and 36 psia. A larger chart is required for more-precise readings.

Solution: Calingeart and Davis or Antoine equation.

This can be used by obtaining the Antoine constants from Poling et al.[6] For n-hexane, with temperature in K, these constants are A = 15.8366, B = 2697.55, and C = –48.78. Then,

RTENOTITLE

and pv = 36.68 psia = 252.73 kPa.

Solution: Lee-Kesler.

The use of the Lee-Kesler[5] equation requires pc, Tc, and ω for n-hexane. These can be obtained from Table 1 in Gas properties.

  • pc = 436.9 psia (29.7 atm)
  • Tc = 453.7°F or 913.3°R or 507.4 K
  • ω = 0.3007.

For 100°C,

Tr = 0.7351
(Tr) 6 = 0.15782
ln Tr = –0.30775

RTENOTITLE

RTENOTITLE

pv = (0.0816)(29.7) = 2.4235 atm = 35.62 psia = 245.6 kPa.

Experimental value: 35.69 psia = 246.1 kPa.

Conclusions. Lee-Kesler gives the best answer, but the Clausius-Clapeyron method is also very accurate to within 0.17 psi, which is typical if the extrapolation is close to the appropriate range.

Nomenclature

pv = vapor pressure
T = absolute temperature
V = increase in volume caused by vaporizing 1 mole
Lv = molal latent heat of vaporization.
A = empirical constant
B = empirical constant
pvr = reduced vapor pressure (vapor pressure/critical pressure)
f0 = function of reduced temperature
f1 = function of reduced temperature
ω = acentric factor
Tr = reduced temperature

References

  1. 1.0 1.1 1.2 Cox, E.R. 1923. Pressure-Temperature Chart for Hydrocarbon Vapors. Ind. Eng. Chem. 15 (6): 592-593. http://dx.doi.org/10.1021/ie50162a013
  2. Calingaert, G. and Davis, D.S. 1925. Pressure-Temperature Charts—Extended Ranges. Ind. Eng. Chem. 17 (12): 1287-1288. http://dx.doi.org/10.1021/ie50192a037
  3. Antoine, C. 1888. Tensions des vapeurs; nouvelle relation entre les tensions et les températures. Comptes Rendus des Séances de l'Académie des Sciences 107: 836–850.
  4. Pitzer, K.S., Lippman, D.Z., Curl, R.F. Jr. et al. 1955. The Volumetric and Thermodynamic Properties of Fluids. II. Compressibility Factor, Vapor Pressure, and Entropy of Vaporization. J. Am. Chem. Soc. 77: 3433.
  5. 5.0 5.1 5.2 Lee, B.I. and Kesler, M.G. 1975. A generalized thermodynamic correlation based on three-parameter corresponding states. AIChE J. 21 (3): 510-527. http://dx.doi.org/10.1002/aic.690210313
  6. 6.0 6.1 Poling, B.E., Prausnitz, J.M., and O’Connell, J.P. 2001. The Properties of Gases and Liquids, fifth edition (paperback). New York: McGraw-Hill Professional. Cite error: Invalid <ref> tag; name "r6" defined multiple times with different content

Noteworthy papers in OnePetro

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External links

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See also

Gas properties

Real gases

Calculating gas properties

PEH:Gas_Properties