Problem Statement
State Gibbs’ phase rule $F = C – P + 2$ and apply it to: (a) water at the triple point, (b) water–ice equilibrium, (c) a single-phase pure gas.
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: Gibbs phase rule: $F = C – \mathcal{P} + 2$, where $C$ = number of components, $\mathcal{P}$ = number of phases, $F$ = degrees of freedom (intensive variables that can be independently varied).
Step 2 — Apply the relevant physical law or equation: (a) Water at triple point: $C=1$, $\mathcal{P}=3$ (solid+liquid+gas).
Step 3 — Solve algebraically for the unknown: $$F = 1-3+2 = 0$$
Step 4 — Substitute numerical values with units: No freedom: triple point is a unique fixed point in $p$-$T$ space. ✓ ($T_{tp}=273.16\ \text{K}$, $p_{tp}=611\ \text{Pa}$)
Step 5 — Compute and check the result: (b) Ice-water equilibrium: $C=1$, $\mathcal{P}=2$.
Step 6: $$F = 1-2+2 = 1$$
Worked Calculation
$$F = 1-3+2 = 0$$
$$F = 1-2+2 = 1$$
$$F = 1-1+2 = 2$$
Gibbs phase rule: $F = C – \mathcal{P} + 2$, where $C$ = number of components, $\mathcal{P}$ = number of phases, $F$ = degrees of freedom (intensive variables that can be independently varied).
(a) Water at triple point: $C=1$, $\mathcal{P}=3$ (solid+liquid+gas).
$$F = 1-3+2 = 0$$
No freedom: triple point is a unique fixed point in $p$-$T$ space. ✓ ($T_{tp}=273.16\ \text{K}$, $p_{tp}=611\ \text{Pa}$)
(b) Ice-water equilibrium: $C=1$, $\mathcal{P}=2$.
$$F = 1-2+2 = 1$$
One variable free (e.g., choose $T$, then $p$ is fixed by the melting curve). ✓
(c) Single-phase pure gas: $C=1$, $\mathcal{P}=1$.
$$F = 1-1+2 = 2$$
Two variables free: can independently set $T$ and $p$. ✓
Answer
$$\boxed{F = 1-1+2 = 2}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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