Phase Transitions — Heating Curve and Enthalpy Changes

Theoretical Background

  • Heat required for temperature change: \(q = m c \Delta T\)

  • Heat absorbed in a phase transition at constant pressure: \(q = n \Delta H_{\text{trans}}\)

  • During an equilibrium phase transition of a pure substance at fixed pressure, the temperature remains constant while heat changes the phase and reorganizes intermolecular interactions.
  • Under these conditions, a heating curve combines sloped segments within a single phase and plateaus during phase changes.

Exercise

Calculate the total heat required to bring 50.0 g of ice from –10.0 °C to liquid water at 25.0 °C.

Given data:

  • $c_{\text{ice}} = 2.09 \, J\,g^{-1}\,K^{-1}$
  • $c_{\text{water}} = 4.18 \, J\,g^{-1}\,K^{-1}$
  • $\Delta H_{\text{fus}} = 6.01 \, kJ\,mol^{-1}$
  • Molar mass of $H_2O = 18.0 \, g\,mol^{-1}$

Step-by-Step Solution

Step 1. Heating ice from –10 °C to 0 °C
\(q_1 = m c_{\text{ice}} \Delta T = (50.0)(2.09)(10.0) = 1045 \, J\)

Step 2. Melting ice at 0 °C
Moles of water: \(n = \frac{50.0}{18.0} = 2.78 \, mol\)

\[q_2 = n \Delta H_{\text{fus}} = (2.78\,\mathrm{mol})(6.01\,\mathrm{kJ\,mol^{-1}}) = 16.7\,\mathrm{kJ}\]

Step 3. Heating liquid water from 0 °C to 25 °C
\(q_3 = m c_{\text{water}} \Delta T = (50.0)(4.18)(25.0) = 5225 \, J\)

Step 4. Total heat
\(q_{\text{tot}} = q_1 + q_2 + q_3\)

Convert to consistent units (kJ):

  • $q_1 = 1.05 \, kJ$
  • $q_2 = 16.7 \, kJ$
  • $q_3 = 5.23 \, kJ$
\[q_{\text{tot}} = 1.05 + 16.7 + 5.23 = 23.0 \, kJ\]

Answer:
\(q_{\text{tot}} = 23.0 \, kJ\)

Notes

  • Over the temperature interval considered, the heating curve has three regions: heating the solid, the melting plateau, and heating the liquid.
  • The largest energy contribution comes from the phase transition (fusion), which requires far more heat than simply raising the temperature.
  • This illustrates the difference between:
    • specific heat (energy per unit mass per degree, linked to temperature changes),
    • latent heat (energy associated with structural reorganization of matter).
  • At constant pressure, the supplied heat equals the enthalpy change, so a heating curve represents how enthalpy is added within and between phases.