Question

How are the conditions of pressure and temperature shown on a [phase] diagram at which two phases coexist in equilibrium?

Answer 1

Well, luckily for us, phase diagrams are made to be straightforward to read. The curves are literally the lines of two-phase coexistence.

They indicate one degree of freedom (i.e. parallel to the curve at each instantaneous displacement), in accordance with the Gibbs' phase rule.

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Consider the phase diagram of water:

If you are unsure how many phases there are in a given region on a phase diagram, refer to the Gibbs' phase rule to check:

`bb(f = c - p + 2)`

• `f` is the number of degrees of freedom, i.e. the number of coordinate directions you are allowed to move in a phase diagram without moving out of the bounds of consideration.
• `c` is the number of components in the substance (could be a solution full of electrolytes... could be interacting).
• `p` is the number of phases.

Below are example calculations for cases we should already be able to verify physically.

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TEST CHECK 1: TRIPLE POINT

For example, consider the triple point of water at `0.01^@ "C"` and `"0.0060 atm"`.

• `f = 0`, because you can't move around in a triple point in any way, unless you are no longer at the triple point.
• `c = 1` for pure substances, because there is only one of itself in itself.

This indicates that the number of phases at a triple point is:

`bb(p) = c + 2 - f = 1 + 2 - 0`

`= bb3` phases in equilibrium at the triple point.

And in fact, that's why it's called a triple point, because three phases coexist at a triple point.

TEST CHECK 2: COEXISTENCE CURVE

Or, consider the liquid-vapor coexistence curve, `bar(AE)`. We should expect TWO coexisting phases: liquid and vapor. Thus, we expect `p = 2`. We have:

• `f = 1`, because if you move parallel to the curve for each instantaneous displacement, you won't move off the curve.
• `c = 1`, because our water sample is assumed to be a pure substance.

And we get:

`bb(p) = c + 2 - f = 1 + 2 - 1`

`= bb(2)` phases in equilibrium with each other, as expected.

TEST CHECK 3: SINGLE-PHASE REGION

Or, consider just being in a single-phase region, like at `50^@ "C"` and `"1.00 atm"`. We should just have one phase: the liquid. We have:

• `f = 2`, because you can move in spirals if you wanted, which requires hybridizing two particular coordinate directions, while still staying in the same two-dimensional region.
• `c = 1`, and we know why at this point.

So,

`bb(p) = c + 2 - f = 1 + 2 - 2`

`= bb(1)` phase exists by itself, which we already said... it's the liquid phase.