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Vapor pressure
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.
Contents
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:
where:
- p_{v} = vapor pressure
- T = absolute temperature
- ΔV = increase in volume caused by vaporizing 1 mole
- L_{v} = 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
which is known as the Clausius-Clapeyron equation. Integrating this equation gives
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
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:
where p_{vr} is the reduced vapor pressure (vapor pressure/critical pressure), f^{0} and f^{1} are functions of reduced temperature, and ω is the acentric factor.
Lee and Kesler^{[5]} have expressed f^{0} and f^{1} in analytical forms:
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(p_{v}) = 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_{1} = 50°C = 122°F = 581.67°R
- l/1 = 0.0017192°R^{–1}
- T_{2} = 90°C = 653.67°R,
- 1/T_{2} = 0.0015298°R^{–1},
- p_{v} at T_{1} = 54.04 kPa = 7.8374 psia,
- log p_{v} = 0.89417,
- p_{v} at T_{2} = 188.76 kPa = 27.3773 psia,
- log p_{v} = 1.43739,
- Δlog p_{v} = –0.543195,
- 1/T_{1} –1/T_{2} = 0.00018936,
- c = slope = –0.543195/0.00018936 = –2868.52°R.
Solving for b, log p_{v} = –2868.52 /T+b yields
- b = 5.8257,
- T_{3} = 100°C = 212°F = 671.67°R,
- 1/T_{3} = 0.0014888.
Solving for p_{v} at 100°C yields
hence, p_{v} = 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_{v} = 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,
and p_{v} = 36.68 psia = 252.73 kPa.
Solution: Lee-Kesler.
The use of the Lee-Kesler^{[5]} equation requires p_{c}, T_{c}, and ω for n-hexane. These can be obtained from Table 1 in Gas properties.
- p_{c} = 436.9 psia (29.7 atm)
- T_{c} = 453.7°F or 913.3°R or 507.4 K
- ω = 0.3007.
For 100°C,
- T_{r} = 0.7351
- (T_{r}) 6 = 0.15782
- ln T_{r} = –0.30775
- p_{v} = (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
p_{v} | = | vapor pressure |
T | = | absolute temperature |
V | = | increase in volume caused by vaporizing 1 mole |
L_{v} | = | molal latent heat of vaporization. |
A | = | empirical constant |
B | = | empirical constant |
p_{vr} | = | reduced vapor pressure (vapor pressure/critical pressure) |
f^{0} | = | function of reduced temperature |
f^{1} | = | function of reduced temperature |
ω | = | acentric factor |
T_{r} | = | reduced temperature |
References
- ↑ ^{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
- ↑ 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
- ↑ 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.
- ↑ 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.
- ↑ ^{5.0} ^{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
- ↑ ^{6.0} ^{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
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