Benzene has a vapor pressure of 0.086 bar and 0.9162 bar at
290 Kelvin and 350 Kelvin, respectively.
Estimate the vapor pressure of benzene at 320 Kelvin.
Solution:
Background: When the
vapor pressure of a pure fluid is in equilibrium with the liquid phase, the
equality of chemical potential, temperature, and pressure in both phases leads
to the Clausius-Clapeyron equation: dPvp over dT
is equal to delta Hv over (T delta Vv). This could also be written as d ln Pvp over d(1/T) is equal to minus delta Hv
over (R delta Zv). In this equation, delta Hv
and delta Zv refer to differences in the
enthalpies and compressibility factors of saturated vapor and saturated
liquid. Most vapor pressure estimation
and correlation equations stem from an integration of this equation. When this is done, an assumption must be made
regarding the dependence of the group delta Hv
and delta Zv on temperature. Also, a constant of integration which must be
evaluated by using one vapor pressure-temperature point is obtained.
A common
practice is to use both the normal boiling point and the critical point to
obtain generalized constants. This gives
an equation lnPvpr is equal to h time (one
minus one over Tr). Pvpr is
the reduced vapor pressure. It can be
calculated using the boiling point and critical pressure. This is a two=-parameter corresponding states
equation for vapor pressure.
To achieve
more accuracy, several investigators have proposed three-parameter forms. The Pitzer
expansion is one of the more successful:
lnPvpr is equal to f(0)(Tr) plus omega f(1)(Tr). The functions f(0) and f(1) have been expressed both in tabular form and
analytical form by Lee and Kesler. The values of the acentric
factor are also tabulated in the literature.
Antoine
proposed a simple modification of the equation as lnPvp
is equal to A minus B over (T plus C).
A vapor
pressure estimation is also available that is based on information from two and
even four other reference fluids.
Antoine
proposed a simple modification of this equation which has been widely used over
limited temperature ranges. ln Pvp
is equal to A minus B over (T plus C).
Discussion
and Recommendations: The least accurate
is the Clapeyron equation, especially at lower
temperatures. The Antoine equation
should not be used above 2.0 to 2.7 bar when the constants are obtained from
experimental data below that pressure.
The Wagner equation is the most accurate, although all the predictive
methods, i.e., Lee-Kesler, Gomez-Thodos,
and two-reference fluid methods, perform satisfactorily.
The simplest approach is to assume that the group delta Hv
and delta Zv is constant and
independent of temperature. Then, with
the constant of integration denoted as A, we obtain ln Pvp is equal to A minus B over T,
where B is equal to delta Hv and delta Zv. This equation is sometimes called Clapeyron equation.
Using the given data, B is found to be equal to 4004.36 and
A as 11.34 when pressure is used in atmospheres.