Physics · Molecular physics and thermodynamics · Grade 10

Phase transitions

Why does water boil at 120 °C, not 100 °C, in a pressure cooker?

The situation

Peas or meat take a long time to boil in a pot, but in a pressure cooker they cook much faster. On a mountain top the tea water boils before reaching 100 °C, while the ice cubes in the freezer shrink slowly without melting. So the temperature at which water boils and freezes depends on pressure. Under what conditions do ice, water and vapour live together — and when does water stop boiling altogether?

Topic

Phase transitions. Water, ice and vapour: the phase diagram

Key ideas

Phase diagram
Latent heat
Critical point

Formulas

\frac{d p}{d T} = \frac{L}{T (v_{2} - v_{1})}

Clapeyron equation: the slope of the phase boundary. $L$ is the heat of transition, $v_{1}, v_{2}$ are the specific volumes of the phases.

\ln \frac{p_{2}}{p_{1}} \approx - \frac{L}{R_{v}} (\frac{1}{T_{2}} - \frac{1}{T_{1}})

Clausius–Clapeyron: when the vapour is taken to be an ideal gas and $v_{w} ≪ v_{v a p}$ ; $R_{v} = 461.5$ J/(kg·K).

Q = m c Δ T, Q = m L

Heating and phase transition. During the transition ( $p$ = const) the temperature does not change.

F = K - Φ + 2

Gibbs phase rule: for pure water ( $K = 1$ ) on a line $Φ = 2$ , $F = 1$ ; at the triple point $Φ = 3$ , $F = 0$ .

T_{c} = 647.096 K, p_{c} = 22.064 MPa

Critical point: the heat of vaporisation $L \to 0$ , and the densities of water and vapour become equal.

Theory

Phase diagram

Every point on the diagram is a pair of pressure and temperature. In the regions a single phase is stable: ice, liquid or vapour. On the lines two phases live together in equilibrium: the melting, boiling and sublimation lines. All three lines meet at the triple point (0.01 °C; 611.657 Pa), while the boiling line ends at the critical point (373.95 °C; 22.064 MPa). The lines are drawn with the IAPWS formulae — this is real data for water. The states differ in the motion of molecules: in a solid they vibrate about lattice sites, in a liquid they are densely packed but slide past one another, in a gas they fly freely over long distances; which state is stable is determined by the ratio of the energy of thermal motion to the energy of mutual attraction. Pure water is one component ( $K = 1$ ). A multicomponent system has more degrees of freedom: for example, in salt water ( $K = 2$ ) $F = 2$ even on a two-phase line — the boiling point depends on the salt concentration as well as on the pressure (salt water boils at a higher temperature and freezes at a lower one).

Why does the temperature stop?

If the pressure is constant, during a phase transition all the energy of the heater goes into breaking the bonds between molecules: a horizontal plateau appears on the temperature graph. To turn 1 kg of water into vapour at 1 atm, $L \approx 2257$ kJ is needed — 5 times more than heating it from 0 °C to 100 °C ( $\approx 419$ kJ). Compare the measured $Q / m$ in the results window with $L$ . On the line the liquid and vapour are in dynamic equilibrium: as many molecules evaporate per unit time as condense — the vapour is saturated. If we heat, evaporation prevails and the liquid turns into vapour; if we cool, the reverse happens (interconversion of liquid and gas).

Ice melts under pressure

Ice is lighter than water (917 and 1000 kg/m³): on melting the volume decreases, $v_{2} - v_{1} < 0$ . According to the Clapeyron equation, the slope of the melting line is negative — if the pressure rises, ice melts at a lower temperature. The effect is small: to lower the melting point by 1 °C requires about 13 MPa (130 atm). See this in the magnified window on the board.

The critical point and the supercritical fluid

As the critical point is approached, the densities of water and vapour become equal, $L \to 0$ and the surface between them disappears. In the region $T > T_{c}$ , $p > p_{c}$ the substance is a supercritical fluid: there is no boiling and the density changes continuously (near the critical point the heat capacity is very large — the temperature rises slowly, but there is no plateau). So by going round the critical point, a liquid can be turned into vapour without boiling. Model simplifications: the enthalpy of the liquid is taken as independent of pressure, $c_{p} = 1.95$ kJ/(kg·K) for vapour and fluid, the density of vapour and fluid is calculated approximately from the Peng–Robinson equation, the piston moves slowly and the thermostat holds the temperature.

Condition for phase equilibrium and kinds of transition

A one-phase system is homogeneous; one with several phases (water + vapour, ice + water) is heterogeneous. For phases to be in equilibrium, their temperature, pressure and chemical potential — the Gibbs potential per 1 kg, $g = h - T s$ — must be equal: $g_{1} (T, p) = g_{2} (T, p)$ . At given $T$ and $p$ the phase with the smaller $g$ is stable. Watch $Δ g = g_{v a p} - g_{w} \approx R_{v} T \ln (p / p_{s} (T))$ in the panel: if it is positive the liquid is stable, if negative the vapour is stable, and zero is the boiling line. Along the line, the condition $d g_{1} = d g_{2}$ gives the Clapeyron equation. In a first-order phase transition the first derivatives of $g$ — the entropy $s = - \partial g / \partial T$ and the specific volume $v = \partial g / \partial p$ — jump: there is a latent heat $L = T Δ s$ and a jump in density (melting, boiling, sublimation). In a second-order transition $s$ and $v$ are continuous and the second derivatives (heat capacity, compressibility) jump: the Curie point of a ferromagnet (in the “Phase transition and fluctuations” experiment) and the transition to the superconducting state (in the “Superconductivity” experiment). Real-gas isotherms and the van der Waals equation are in the “Real gases” experiment.

Did you know?

In a pressure cooker the pressure is about 2 atm and water boils at ≈ 120 °C — which is why food cooks faster than in an ordinary pot.
Until 2019 the kelvin was defined through the triple point of water: its temperature was taken to be exactly 273.16 K.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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Crystalline and amorphous solids — Grade 9
Thermal expansion — Grade 10
Refrigerator and heat pump — Grade 10
Real gases — Grade 10
Maxwell–Boltzmann distribution — Grade 10
Transport phenomena — Year 1