Physical Chemistry , 1st ed.

168 CHAPTER 7 Equilibria in Multiple-Component Systems

What are the degrees of freedom that can be specified? We already know that

pressure and temperature are common degrees of freedom. But for multiple-

component systems, we also need to specify the relative amounts of each com-

ponent, usually in terms of moles. Figure 7.1 illustrates this for a simple system.

If a chemical equilibrium is present, then not all of the components are

truly independent. Their relative amounts are dictated by the stoichiometry of

the balanced chemical reaction. Before applying the Gibbs phase rule, we need

to identify the number of independent components. This is done by removing

the dependentcomponent from consideration. A dependent component is one

that is made from any other component(s) in the system. In Figure 7.1, the

water and ethanol are not in any chemical equilibrium involving both these

compounds, so they are independent components. However, for the equilibrium

H 2 O () H(aq) OH(aq)

the amounts of hydrogen and hydroxide ions are related by the chemical re-

action. Thus, instead of having three independent components, we have only

two: H 2 O and either Hor OH(the other can be determined by the fact

that the reaction is at equilibrium). Examples 7.1 and 7.2 illustrate degrees of

freedom.

Example 7.1

Consider a mixed drink that has ethanol (C 2 H 5 OH), water, and ice cubes in

it. Assuming that this describes your system, how many degrees of freedom

are necessary to define your system? What might the degrees of freedom be?

Solution

There are two individual components: C 2 H 5 OH and H 2 O. There are also two

phases, solid (the ice cubes) and liquid (the water/ethanol solution). There are

no chemical equilibria to consider, so we don’t have to worry about depen-

dencies among the components. Therefore, from the Gibbs phase rule, we have

FCP 2  2  2  2

F 2

What might be specified? If the temperature is specified, then we know the

pressure of the system, because we also know that liquid and solid H 2 O are

in equilibrium. We can use the phase diagram of H 2 O to determine the nec-

essary pressure if the temperature is given. Another specification might be an

amount of one component. We usually know a total amount of material in a

system. By specifying one component’s amount we can subtract to find the

other component’s amount. By specifying these two degrees of freedom, we

completely define our system.

Example 7.2

Iron(III) sulfate, Fe 2 (SO 4 ) 3 , decomposes upon heating to make iron(III) ox-

ide and sulfur trioxide by the following reaction:

Fe 2 (SO 4 ) 3 (s) Fe 2 O 3 (s) 3SO 3 (g)

Using the phase labels given in the equilibrium reaction, how many degrees

of freedom does this equilibrium have?

JQPJ

JQPJ

System

H 2 O C^2 H^5 OH

C 2 H 5 OH

C 2 H 5 OH

H 2 O

C 2 H 5 OH

H 2 O

Degrees of freedom at equilibrium:

• Temperature


• Pressure


• Amount (mole fraction) of one component
  (mole fraction of other can be determined)
  ∴ 3 degrees of freedom

From Gibbs phase rule:
F  2  1  2  3 degrees of freedom
CP
Figure 7.1 A simple multiple-component
system of water and ethanol. The Gibbs phase
rule applies to this system, too.