Phase Transitions — Water Heating Curve

By Prof. Marco Ruzzi.

Theoretical background

At constant pressure, heating a single phase gives dq=C_pdT. On a graph of temperature against supplied heat:

\[\frac{dT}{dq}=\frac{1}{C_p}.\]

For one mole this becomes 1/C_p,m. During melting or vaporization of a pure substance at a fixed pressure, the two phases coexist at a constant transition temperature.

Exercise 5 — Interpret the heating curve

The graph shows the temperature of a quantity of water as it passes from the solid to the gaseous state. Explain the physical meaning of its five segments. Identify the conditions under which the plateau widths represent standard molar transition enthalpies and the slopes determine molar heat capacities.

View the graph at full size.

Solution

Melting plateau. For one mole of water at the standard pressure of 1 bar, the heat absorbed during melting is its standard molar enthalpy of fusion. The transition temperatures in this schematic graph are shown approximately as 0 °C and 100 °C.

Vaporization plateau. Segment 4 represents the liquid–gas transition: the two phases coexist while the temperature remains constant.

Slope of segment 1. Segment 1 describes the solid. For one mole its slope is the reciprocal of the molar heat capacity of ice at constant pressure, rather than that of liquid water.

Observations on melting and vaporization

The system absorbs heat during melting. For one mole at 1 bar, the horizontal width of segment 2 represents Δ_fusH°. Melting is isothermal at T_fus: the absorbed energy changes the intermolecular structure of the crystalline solid rather than increasing its temperature. In ice this concerns the hydrogen-bond network.

The same reasoning applies to vaporization. For one mole under standard-pressure conditions, the horizontal width of segment 4 represents Δ_vapH°, at the constant vaporization temperature T_vap.

Segments 1, 3 and 5 represent heating of the solid, liquid and vapor, respectively. Their slopes are:

\[m_1=\frac{1}{C_{p,\mathrm{solid}}}.\] \[m_3=\frac{1}{C_{p,\mathrm{liquid}}}.\] \[m_5=\frac{1}{C_{p,\mathrm{gas}}}.\]

These heat capacities are molar heat capacities when the amount is one mole. The stated geometrical interpretation of the standard molar transition enthalpies requires both one mole of pure substance and a pressure of 1 bar. For another amount, the plateau width represents the total heat required by that amount.