Concise Physical Chemistry

c09 JWBS043-Rogers September 13, 2010 11:26 Printer Name: Yet to Come

COEXISTENCE CURVES 125

system ice and water has one component, identifiable as H 2 O, and two phases, solid

and liquid. With his famous phase rule (see below), Gibbs specified the number

ofdegrees of freedomof thermochemical systems. Degrees of freedom are like the

independent variables of a system of algebraic equations or the basis vectors of a

complete set. Once a limited number of them have been specified, the others are no

longer free to take on arbitrary values. They are linear combinations of the set that is

already known. For this reason it is thenumberof independent variables that we get

from the phase rule, not their identity. We are free to choose which of the identifiable

quantities shall be taken as independent variables, but only up to the number specified

by the phase rule.

In the case of liquid water, one might argue that there are three components, H+,

OH−, and H 2 O, or even four, including H 3 O+. We ignore these and similar forms

because they are obviously related to each other and are not independent. In pure

water, we know that [H+]=[OH−] because the only source of the ions is dissociation:

H 2 O→H++OH−

That makes it clear that if we measure either one but only one, we know the other.

When we do this, we find

[

H+

]

= 10 −^7 ; therefore we know that

[

OH−

]

= 10 −^7 and

we can write a dissociation equilibrium constant for pure water^2 :

Kw=

[

H+

][

OH−

]

[H 2 O]

≈

[

H+

] 2

[H 2 O]

=

[

10 −^7

] 2

1

= 10 −^14

where the concentrations are expressed as moles per mole of solvent (water).

Pursuing these ideas just a little further, what we call “concentrations” in solution

chemistry are really activities that, as unitless numbers (ratios relative to a standard

state), give a unitlessKw.

If we reformulate so as to include higher polymers such as H 5 O+ 2 , and so on,

their concentrations are not independent either because dissociation constants exist

by which we could calculate their concentration. These calculations are, in principle,

possible, even though we may not have enough information to carry them out.

However you look at it, specifying one molar concentration specifies them all. There

is one component in a pure phase.

9.2 COEXISTENCE CURVES

The thermodynamic state of water vapor, H 2 O(g), like that of any pure substance, has

two degrees of freedom. Any state function, including the Gibbs free energy, can be

expressed as a mathematical function of two independent variables, for example,μ=

f(p,T). We can also writeE=f(p,T)orCp=f(p,T). In principle, equations

(^2) By our conventional definition, pH≡−log[H+]; this leads us to say that an aqueous solution is neutral
when its pH=7.