Let's try to motivate the Gibbs phase rule. Consider a typical one−component phase diagram.
In the one−phase regions one can vary either the temperature, or the pressure, or both (within limits) without crossing a phase line. We
Let's try to motivate the Gibbs phase rule. Consider a typical one−component phase diagram.
In the one−phase regions one can vary either the temperature, or the pressure, or both (within limits) without crossing a phase line. We say that in these regions there is a variance of 2. We have indicated in the solid, liquid, and gas regions that there is one phase and the variance is two. Along a phase line we have two phases in equilibrium with each other, so on a phase line the number of phases is 2. However, if we want to stay on a phase line, we can't change the temperature and pressure arbitrarily. If we change the temperature − and keep two phases − then the pressure must change also to keep us on the phase line. Otherwise we go off the line and we no longer have two phases in equilibrium. So on a phase line the number of phases is 2, but the variance is 1. At the triple point there are three phases in equilibrium, but there is only one point on the diagram where we can have three phases in equilibrium with each other. Therefore, at the triple point the variance is zero. Notice that the variance seems to be related to the number of phases, such as,
v = 3 − p, or (1)(You will see why we wrote it this way in a minute.)
v = 2 + 1 − p. (2)
Now let's look at another example. Consider the boiling diagram of a two−component ideal liquid−liquid solution.
In the one−phase regions (liquid or vapor) we have a variance of three because we can change the temperature, the composition, X, and the pressure (by moving in and out of the plane of the screen). In the two phase region the variance is only two because if we change the temperature the compositions of both the liquid and vapor phases must track as indicated by the tie−lines connecting the liquid and vapor composition lines. Here it looks like the variance satisfies,
v = 4 − p, or (3)Notice that equations 2 and 4 are the same except for the second term. In Equation 2 there is one component and the second term is 1. In Equation 4 there are two components and the second term in Equation 4 is a 2. It looks like the second term may indicate the number of components in the system. We might guess that the general form for Equations 2 and 4 might be,
v = 2 + 2 − p. (4)
v = 2 + c − p.We shall see below whether or not our guess is correct.
Before we try to derive a general form for 