NBPhO 2026

Part (i) — 1.5 / 1.5 pts

Criterion	Points	Result
Recognising that liquid is impossible at $p_\text{atm} < p_t$, so the stream is solid, vapour, or a mixture	0.5	✓ ”$p = p_\text{atm} = 0.10\,\text{MPa}$ … lies entirely below this triple point … only the sublimation curve … two stable phases at most: solid and vapour, possibly in coexistence”
Excluding pure solid (cannot adiabatically expand through nozzle)	0.2	✓ Reached via the alternative argument that “the constant-$s$ line at $s = s_\ell(T_0)$ (scenario a) or $s = s_g(T_0)$ (scenario b) does not exit through the saturated-vapour boundary before reaching the $0.1$ MPa isobar” — the trajectory ends inside the dome, which simultaneously rules out pure solid and pure vapour
Identifying $T = T_\text{sub}(1\,\text{atm}) \approx 195\,\text{K}$ ($\approx -78\,°\text{C}$), within $\pm 10\,\text{K}$	0.3	✓ $T_\text{exit} = T_\text{sub}(p_\text{atm}) \approx 195\,\text{K} \approx -78\,°\text{C}$, boxed
Tracing the isentrope on the $T$-$s$ diagram and concluding the stream is in the two-phase region for one scenario	0.3	✓ Vertical-isentrope tracing performed for both scenarios
Two-phase for both scenarios	0.2	✓ Both scenarios explicitly addressed and concluded to land in the solid-vapour dome

Part (ii) — 1.3 / 3.5 pts

Criterion	Points	Result
Recognising that “reversible adiabatic” means isentropic ($s = \text{const}$), so the endpoint is found by going vertically downward on the $T$-$s$ diagram	0.6	✓ “Steady reversible adiabatic flow forces $\Delta s = 0$ along every streamline, so the exit and inlet share a specific entropy”
Deriving the lever rule for mass fraction $x$ in the two-phase region: $x = (s_\text{vap} - s_\text{end})/(s_\text{vap} - s_\text{sol})$	0.7	✓ Derived verbatim from $s = (1-x)\,s_v + x\,s_s$
Reading the initial entropy $s_a$ from the saturated-liquid line at $T_0 = 298\,\text{K}$, value $\approx 2100 \pm 50\,\text{J\,kg}^{-1}\,\text{K}^{-1}$	0.5	✗ Claude reports $s_a \approx 1300\,\text{J\,kg}^{-1}\,\text{K}^{-1}$, ~750 outside the keyed band
Reading the initial entropy $s_b$ from the saturated-vapour line at $T_0$, value $\approx 2550 \pm 50\,\text{J\,kg}^{-1}\,\text{K}^{-1}$	0.5	✗ Claude reports $s_b \approx 1580\,\text{J\,kg}^{-1}\,\text{K}^{-1}$, ~970 outside the keyed band
Reading $s_\text{sol}$ at the $1$-bar solid boundary, value $\approx 400 \pm 50\,\text{J\,kg}^{-1}\,\text{K}^{-1}$	0.3	✗ Claude reports $s_s \approx 700\,\text{J\,kg}^{-1}\,\text{K}^{-1}$, ~250 outside the keyed band
Reading $s_\text{vap}$ at the $1$-bar vapour boundary, value $\approx 3250 \pm 50\,\text{J\,kg}^{-1}\,\text{K}^{-1}$	0.3	✗ Claude reports $s_v \approx 3600\,\text{J\,kg}^{-1}\,\text{K}^{-1}$, ~300 outside the keyed band
Computing $x_a \approx 0.40 \pm 0.05$ for scenario (a)	0.3	✗ Claude obtains $x_a \approx 0.79$ — almost double the keyed value, propagated cleanly from the off-band entropy readings
Computing $x_b \approx 0.25 \pm 0.05$ for scenario (b)	0.3	✗ Claude obtains $x_b \approx 0.70$ — about 2.8× the keyed value, same root cause

Discrepancies

The four diagram-reading entries fall outside the official’s $\pm 50\,\text{J\,kg}^{-1}\,\text{K}^{-1}$ tolerance band; the two final mass-fraction entries fall outside the $\pm 0.05$ band. The shifts are not a uniform reference-state offset: at the inlet, Claude reads $\sim 800$–$1000$ below the keyed values, and at the exit he reads $\sim 300$–$350$ above them. The net effect is to widen the entropy lever $s_v - s_a$ from the official’s $1.15\,\text{kJ\,kg}^{-1}\,\text{K}^{-1}$ to $2.30\,\text{kJ\,kg}^{-1}\,\text{K}^{-1}$ — almost a factor of two — while leaving the lever length $s_v - s_s$ essentially unchanged ($2.90$ vs $2.85\,\text{kJ\,kg}^{-1}\,\text{K}^{-1}$, both consistent with $L_\text{sub}/T_\text{sub} \approx 2.93$). The mass fractions inflate accordingly.

A revealing internal symptom: Claude’s computed gap $s_b - s_a = 0.28\,\text{kJ\,kg}^{-1}\,\text{K}^{-1}$ at the dome top corresponds via Clausius–Clapeyron to $L_\text{vap}(298\,\text{K}) \approx 83\,\text{kJ\,kg}^{-1}$, while the official’s gap of $0.45\,\text{kJ\,kg}^{-1}\,\text{K}^{-1}$ implies $L_\text{vap} \approx 134\,\text{kJ\,kg}^{-1}$. The NIST value for CO₂ at $25\,°\text{C}$ is $\approx 122\,\text{kJ\,kg}^{-1}$, putting the official closer to truth. Claude’s self-consistency check across the sublimation curve (matching $s_v - s_s$ to $L_\text{sub}/T_\text{sub}$) verifies a difference, not the absolute positions, and so cannot catch the systematic stretch in his readings between dome and sublimation segment. A second cross-check at the vaporisation curve ($s_b - s_a \stackrel{?}{=} L_\text{vap}/T_0$) would have flagged the issue.

Part (iii) — 3.0 / 3.0 pts

Criterion	Points	Result
Setting up the comparison: condensation occurs iff the adiabat lies below the saturation curve in $T$-$p$ space	0.5	✓ Stated explicitly in the “complementary derivation”: “the isentropic cooling per unit pressure drop is greater than the saturation-curve cooling — i.e. the vapour cools below the saturation temperature corresponding to its current pressure and falls into the dome”
Deriving the saturation slope via Boltzmann (those who know and use Clausius–Clapeyron do not need to derive it)	0.6	✓ Claude invokes Clausius–Clapeyron directly: $dp_\text{sat}/dT = L/[T(v_g - v_\ell)] \approx L/(T\,v_g)$ — the scheme explicitly accepts this in lieu of the Boltzmann derivation
Obtaining the Clausius–Clapeyron form $\mathrm{d}p_\text{sat}/\mathrm{d}T = pL/(R_sT^2)$	0.6	✓ Equivalent form $L/(T\,v_g)$ used; with $v_g = R_s T/p$ for the ideal-gas vapour this is $pL/(R_s T^2)$ identically
Adiabat slope $\mathrm{d}T/\mathrm{d}p = R_sT/(c_pp)$ from ideal-gas adiabatic law	0.4	✓ "$\left(\partial T/\partial p\right)_s = v/c_p$" combined with $v = R_s T/p$ gives exactly $R_s T/(c_p p)$
Comparing slopes to obtain the condition $L > c_pT$	0.6	✓ Boxed result, derived twice: once from $ds_g/dT\big\|_\text{sat} < 0$ via Maxwell + Clausius–Clapeyron, once from the adiabat-vs-saturation slope comparison
Numerical check for water giving $L/(c_pT) \approx 3 > 1$, hence condensation	0.3	✓ $L/(c_p T) = 2260/746 \approx 3.0$, condensation does occur, plus the kettle-spout physical interpretation

Overall score: 5.8 / 8.0 pts

Parts (i) and (iii) are clean, full-credit work. The 2.2-point dock all sits on Part (ii)‘s diagram-reading entries: Claude’s four entropy readings and both lever-rule mass fractions fall outside the keyed tolerance bands. The conceptual scaffolding of Part (ii) — recognising isentropic flow and deriving the lever rule — is intact (1.3 of 1.3 pts on those two items), but the numerical output is roughly a factor of two too high in solid fraction.

Numerical answers vs the official key: $T_\text{exit} \approx 195\,\text{K}$ (exact match); $x_a \approx 0.79$ vs official $0.40$ (out of band); $x_b \approx 0.70$ vs official $0.25$ (out of band); condensation criterion $L > c_p T$ (exact match); $L/c_p T \approx 3$ for water at $373\,\text{K}$ (exact match).

Commentary

Where this solution goes beyond the grading scheme. The “Overview” front-loads the three structural ideas — reversible+adiabatic ⇒ isentropic, exit pressure below the triple point so liquid cannot exist, and the two starting states differ in entropy because the dome at $25\,°\text{C}$ has a finite width — and explicitly notes that pressures, velocities and Mach numbers along the channel never enter the bookkeeping. Part (i) supplements the answer with a “Physics remark — why a de Laval shape,” giving the area-velocity flip across $M = 1$ that justifies the converging–diverging geometry, and adds the snow-maker framing that ties the abstract phase-fraction result to the everyday observable. Part (ii) carries an internal Clausius–Clapeyron consistency check ($s_v - s_s$ vs $L_\text{sub}/T_\text{sub}$) and a closing “why scenario (a) gives more solid than (b)” paragraph that connects the lever rule back to the entropy gap $L_\text{vap}/T_0$ across the dome. Part (iii) is unusually thorough: two independent derivations of the same boxed result (one via $ds_g/dT\big\|_\text{sat} < 0$ using Maxwell + Clausius–Clapeyron, one via adiabat-vs-saturation slope comparison), three consistency checks ($L \to 0$ limit, $T \to T_c$ critical-point breakdown, dimensional check), an explicit caveat about the ideal-gas premise failing near the critical point — and a closing remark contrasting fogging vapour with non-condensable expansion (refrigerator valve, bicycle pump) that makes the result feel physically populated.

Where the official solution is sharper. Two specific places. (1) The diagram-reading: the official’s four entropy values are internally consistent across both consistency checks — $s_v - s_s$ matches $L_\text{sub}/T_\text{sub}$, and $s_b - s_a$ matches $L_\text{vap}/T_0$ with $L_\text{vap}(25\,°\text{C}) \sim 130\,\text{kJ\,kg}^{-1}$, close to the NIST value $\sim 122\,\text{kJ\,kg}^{-1}$. Claude’s readings only pass the first of these checks; the second would have caught his inflated $s_v - s_a$ lever and corrected the factor-of-two error in the mass fractions. The lesson is that one consistency check is not enough on a diagram with two saturation curves at very different temperatures — both gaps must be verified. (2) The official’s slope-comparison framing in Part (iii) is more direct than Claude’s $ds_g/dT\big\|_\text{sat} < 0$ route: comparing $dT/dp\big\|_\text{adiab}$ with $dT/dp\big\|_\text{sat}$ in $T$-$p$ space lets the criterion $L > c_p T$ fall out from a single inequality between two well-known thermodynamic slopes, with no Maxwell relation required. Claude does present this route as his “complementary derivation,” but it sits below the longer Maxwell-relation path; reversing the order would put the cleaner derivation forward.