10.2. EQUILIBRIUM RELATIONS BETWEEN PHASES

10.2A. Phase Rule and Equilibrium

In order to predict the concentration of a solute in each of two phases in equilibrium, experimental equilibrium data must be available. If the two phases are not at equilibrium, the rate of mass transfer is proportional to the driving force, which is the departure from equilibrium. In all cases involving equilibria, two phases are involved, such as gas–liquid or liquid–liquid. The important variables affecting the equilibrium of a solute are temperature, pressure, and concentration.

The equilibrium between two phases in a given situation is restricted by the phase rule:

Equation 10.2-1

where P is the number of phases at equilibrium, C the number of total components in the two phases when no chemical reactions are occurring, and F the number of variants or degrees of freedom of the system. For example, for the gas–liquid system of CO₂–air–water, there are two phases and three components (considering air as one inert component). Then, by Eq. (10.2-1),

This means that there are 3 degrees of freedom. If the total pressure and temperature are set, only one variable is left that can be set arbitrarily. If the mole fraction composition x_A of CO₂ (A) in the liquid phase is set, the mole fraction composition y_A or pressure p_A in the gas phase is automatically determined.

The phase rule does not tell us the partial pressure p_A in equilibrium with the selected x_A. The value of p_A must be determined experimentally. The two phases can, of course, be gas–liquid, liquid–solid, and so on. For example, the equilibrium distribution of acetic acid between a water phase and an isopropyl ether phase has been determined experimentally for various conditions.

10.2B. Gas–Liquid Equilibrium

1. Gas–liquid equilibrium data

To illustrate the obtaining of experimental gas–liquid equilibrium data, the system SO₂–air–water will be considered. An amount of gaseous SO₂, air, and water are put in a closed container and shaken repeatedly at a given temperature until equilibrium is reached. Samples of the gas and liquid are analyzed to determine the partial pressure p_A in atm of SO₂ (A) in the gas and mole fraction x_A in the liquid. Figure 10.2-1 shows a plot of data from Appendix A.3 of the partial pressure p_A of SO₂ in the vapor in equilibrium with the mole fraction x_A of SO₂ in the liquid at 293 K (20°C).

2. Henry's law

Often the equilibrium relation between p_A in the gas phase and x_A can be expressed by a straight-line Henry's law equation at low concentrations:

Equation 10.2-2

where H is the Henry's law constant in atm/mole fraction for the given system. If both sides of Eq. (10.2-2) are divided by total pressure P in atm,

Equation 10.2-3

where H' is the Henry's law constant in mole frac gas/mole frac liquid and is equal to H/P. Note that H' depends on total pressure, whereas H does not.

In Fig. 10.2-1 the data follow Henry's law up to a concentration x_A of about 0.005, where H = 29.6 atm/mol frac. In general, up to a total pressure of about 5 × 10⁵ Pa (5 atm) the value of H is independent of P. Data for some common gases with water are given in Appendix A.3.

EXAMPLE 10.2-1. Dissolved Oxygen Concentration in Water What will be the concentration of oxygen dissolved in water at 298 K when the solution is in equilibrium with air at 1 atm total pressure? The Henry's law constant is 4.38 × 104 atm/mol fraction. Solution: The partial pressure pA of oxygen (A) in air is 0.21 atm. Using Eq. (10.2-2), Solving, xA = 4.80 × 10−6 mol fraction. This means that 4.80 × 10−6 mol O2 is dissolved in 1.0 mol water plus oxygen, or 0.000853 part O2/100 parts water.