Question

For a chemical system with 5 phases at the invariant point, the number of possible univariant reactions in pressure-temperature space is ____ (answer in integer).

Hint:

Use F = C - P + 2 with F = 0 to find the components, then count how many ways one phase can be dropped from the full set of five.

Answer 1

Correct Answer: 5

Step 1: Recall the Gibbs phase rule.
For a chemical system, the phase rule is written as
\[ F = C - P + 2 \]
where \(F\) is the number of degrees of freedom, \(C\) is the number of independent components, and \(P\) is the number of phases present.

Step 2: Apply the invariant point condition.
At an invariant point in pressure-temperature space, both pressure and temperature are fixed, so the degrees of freedom are zero, \(F = 0\). Substituting into the phase rule,
\[ 0 = C - P + 2 \quad \Rightarrow \quad P = C + 2 \]
We are told the invariant point involves \(P = 5\) phases, so the number of components is
\[ C = P - 2 = 5 - 2 = 3 \]

Step 3: Recall Schreinemakers' rule for univariant reactions.
Each univariant reaction leaving an invariant point uses one fewer phase than the invariant point itself, that is \(P - 1\) phases, so that its own degrees of freedom become \(F = 1\), tracing a curve rather than a point. Every such reaction is formed by leaving out exactly one of the \(P\) phases present at the invariant point, and no two reactions leave out the same phase.

Step 4: Count the reactions.
The number of ways to choose which single phase is left out of a set of \(P\) phases is
\[ \binom{P}{P-1} = \binom{5}{4} = 5 \]
So there are exactly 5 distinct univariant reactions radiating from the invariant point, each one missing a different phase from the full set of five.

Step 5: Final conclusion.
The number of possible univariant reactions in pressure-temperature space is
\[ \boxed{5} \]