# 5.33: Untitled Page 95

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## Chapter 5

Here *K*

is referred to as the Henry’s law *constant* even though it is not a *eq,A*

constant since it depends on the temperature and composition of the liquid, i.e.

*K*

*F T , x , x , x , .... x*

(5‐32)

*eq,A*

*A B C*

*N*1

This treatment of gas‐liquid systems is extremely brief and devoid of the rigor that will be encountered in a comprehensive discussion of phase equilibrium.

However, we now have sufficient information to solve a few simple mass balance problems that involve two‐phase systems.

**5.4 Saturation, Dew Point and Bubble Point of Liquid Mixtures** When a *pure liqui* d phase (species *A*) is in equilibrium with the *pure gas* phase, the partial pressure of the component in the gas phase is equal to the vapor pressure of the pure component.

*p*

*p*

(5‐33)

*A*

*A,vap*

This result is consistent with setting *x * 1 in Eq. 5‐23 so that the liquid is pure *A*

component *A*. In general, when air is the gas phase we will assume that the gas phase is saturated with the liquid component and that the concentration of the air in the liquid is negligible. When the liquid phase is a mixture, Raoult’s law (Eq. 5‐26) must be used to compute the composition of the gas phase.

If a liquid mixture is in equilibrium with its own vapors, the total pressure is equal to the sum of the partial pressures of the individual components.

*A * *N*

*A * *N*

*p *

*p*

*p*

*x *

(5‐34)

*A*

*A,vap*

*A*

*A * 1

*A * 1

When a liquid mixture is heated, the vapor pressure of the components in the mixture increases and the sum of the partial pressures, given by Eq. 5‐34, increases accordingly. When the sum of the partial pressures of the components of the mixture is equal to the atmospheric pressure, the liquid mixture boils. The difference between a liquid mixture and a pure liquid is that the boiling temperature of a mixture is not constant. For a mixture in equilibrium with its own vapors, the *bubble point* of a mixture is the pressure at which the liquid starts to vaporize. Similarly, for a vapor mixture the *dew point* of the mixture is the pressure at which the vapors start to condense. These terms are also used when the liquid is in contact with air, i.e. it is customary to refer to the *bubble point* as

*Two‐Phase Systems* & *Equilibrium Stages* 171

the temperature at which the liquid mixture starts to boil and *dew point* as the temperature at which the first condensed liquid appears.

EXAMPLE 5.5. Bubble point of a water‐alcohol mixture

A mixture of ethanol ( C H OH ) and water ( H O ) with the mole 2

5

2

fractions given by *x*

*x*

0 *. * 5 is slowly heated under well‐stirred

Et

H2O

conditions in an open beaker. Using Antoine’s equation to determine the vapor pressure of the components and Raoult’s law to estimate the partial pressures, we can estimate the bubble point of this mixture, i.e. the temperature at which the first bubbles will start forming at the bottom of the beaker as well as the composition of the first bubbles.

The vapor pressure of pure ethanol and water can be computed using Antoine’s equation. The partial pressure of the components in the gas phase in equilibrium with the liquid mixture are computed using Raoult’s law given by Eq. 5‐26. The bubble point will be determined as the temperature at which the sum of the partial pressures of ethanol and water is equal to atmospheric pressure, i.e.,

*p*

*p*

*p*

H2O

Et

*atm*

or

(1)

*x*

*p*

*x p*

760 mmHg

H2O H2O *,vap*

Et

Et *,vap*

This problem can be solved by substitution of Antoine’s equation for the vapor pressures of the pure components in Eq. 1, and then solving for the temperature. A much simpler route consists of guessing values of the temperature until we satisfy Eq. 1. This procedure is easily done using a spreadsheet as illustrated in Table 5.5. The values of Antoine’s coefficients for water and ethanol are available in Table A3 of Appendix A and are given by

Water:

*A * 7 *. * 94915 *, B * 1657 *. * 46 *, * 227 *. * 03

(2)

Ethanol:

*A * 8 *. * 1629 *, B * 1623 *. * 22 *, * 227 03

*. *

(3)

The computed values of the vapor pressure are listed in Table 5.5. We could continue the computation by inserting additional rows between the temperatures *T * 86 *. * 8 C and *T * 86 *. * 9 C . However, for the purpose of this