Heating and cooling curves Knowledge-Based Study

Study notes

1. What is a Heating Curve?

   A heating curve is a graphical representation of the temperature of a substance as heat is added continuously at a constant rate. The vertical axis represents temperature (typically in °C or K), and the horizontal axis represents either heat added (in joules or kilojoules) or time elapsed.

   For a pure crystalline substance starting in the solid state and heated until it becomes a gas, the curve displays five distinct segments:

   • Segment 1 (Sloped): Heating of the solid phase. The temperature increases until the melting point is reached.
   • Segment 2 (Flat plateau): Solid-liquid phase transition (melting). The solid and liquid phases coexist in dynamic equilibrium.
   • Segment 3 (Sloped): Heating of the liquid phase. The temperature rises from the melting point to the boiling point.
   • Segment 4 (Flat plateau): Liquid-gas phase transition (vaporization/boiling). Liquid and vapor coexist in equilibrium.
   • Segment 5 (Sloped): Heating of the gas phase. The temperature continues to rise indefinitely as heat is added.

   Understanding the Slopes and Plateaus

   The distinction between the sloped and flat regions of a heating curve corresponds directly to changes in kinetic and potential energy at the particle level.

   • Sloped Segments (Single Phase): In these regions, added thermal energy increases the average kinetic energy of the particles. Because temperature is a direct measure of average kinetic energy, the temperature rises. The heat absorbed is calculated using the specific heat capacity formula: $q = mc\Delta T$.
   • Flat Plateaus (Phase Transitions): During a phase change, the temperature remains strictly constant despite the ongoing addition of heat. The added energy does not increase the kinetic energy of the particles; instead, it increases their intermolecular potential energy by overcoming the attractive forces holding the particles together. The heat absorbed during these regions is calculated using latent heat: $q = n\Delta H_{\text{fus}}$ or $q = n\Delta H_{\text{vap}}$.

   Plateau Length Comparison: The plateau for vaporization is substantially longer than that for fusion. Melting only requires partially disrupting the orderly arrangement of particles, whereas vaporization requires overcoming nearly all intermolecular attractions to separate particles completely into the gas phase. Therefore, $\Delta H_{\text{vap}}$ is much larger than $\Delta H_{\text{fus}}$.

   The Cooling Process

   A cooling curve is the exact reverse of a heating curve, tracking temperature as heat is removed at a constant rate from a gas down to a solid. Cooling is an exothermic process.

   • Gas Cooling: Temperature drops as particles lose kinetic energy.
   • Condensation Plateau: Temperature remains constant at the condensation point (identical to the normal boiling point). Potential energy decreases as intermolecular attractions draw particles into the liquid phase.
   • Liquid Cooling: Temperature drops until the freezing point is reached.
   • Freezing Plateau: Temperature remains constant at the freezing point (identical to the melting point) while latent heat of fusion is released to the surroundings.
   • Solid Cooling: The temperature of the solid continues to drop.

   Supercooling Phenomenon

   Occasionally, a liquid cooled carefully may remain liquid below its normal freezing point without solidifying. This metastable state is called supercooling. It occurs when particles lack a nucleation site (a seed crystal or surface irregularity) to initiate the crystal lattice. Once nucleation begins, crystallization occurs rapidly, releasing heat and quickly raising the temperature back to the normal freezing point plateau.

   Relating Slope to Specific Heat Capacity

   On a heating curve plotted against time, the rate of temperature increase ($\frac{\Delta T}{\Delta t}$) represents the slope of each single-phase segment. This slope is inversely proportional to the specific heat capacity ($c$) of the phase:

   $$\text{Slope} = \frac{\Delta T}{\Delta t} = \frac{P}{m \cdot c}$$

   where $P$ is the constant rate of heat input ($\text{J/s}$), $m$ is the mass of the sample, and $c$ is the specific heat capacity of that phase.

   • Steep Slope: Indicates a lower specific heat capacity. The substance requires less heat energy per gram to raise its temperature by 1 °C, so its temperature rises rapidly.
   • Gentle Slope: Indicates a higher specific heat capacity. The substance requires more heat energy to raise its temperature, leading to a slower temperature increase.

   For water, the specific heat capacity of the liquid phase ($4.184\text{ J/(g}\cdot\text{°C)}$) is roughly double that of ice ($2.09\text{ J/(g}\cdot\text{°C)}$) or steam ($2.01\text{ J/(g}\cdot\text{°C)}$). Consequently, the liquid segment of water's heating curve has a significantly gentler slope than the solid or vapor segments.