Phase Changes, Heating Curves, and the Clausius-Clapeyron Equation

Phase Changes Between Solids, Liquids, and Gases

Introduction to Phase Changes

Phase changes, also known as state changes, involve the transformation of a substance from one physical state to another (solid, liquid, or gas) without altering its chemical identity. These transitions are fundamental to understanding the behavior of matter under varying temperature and pressure conditions.

• Fusion (Melting): Solid to liquid

• Vaporization: Liquid to gas

• Sublimation: Solid to gas

• Freezing: Liquid to solid

• Condensation: Gas to liquid

• Deposition: Gas to solid

During a phase change, energy is exchanged, but the temperature of the substance remains constant until the transition is complete.

Thermodynamics of Phase Changes

• Heat (Enthalpy) of Fusion (ΔHfusion): The energy required to convert a solid to a liquid at constant pressure.

• Heat (Enthalpy) of Vaporization (ΔHvap): The energy required to convert a liquid to a gas at constant pressure.

At equilibrium during a phase change:

• $\Delta G = \Delta H - T\Delta S$

• At equilibrium, $\Delta G = 0$

Heating and Cooling Curves

Heating Curve of Water

A heating curve shows the temperature change of a substance as heat is added. For water, the curve includes plateaus at the melting and boiling points, where phase changes occur and temperature remains constant.

• Region (a): Ice warming from below 0°C to 0°C (solid phase, temperature increases)

• Region (b): Melting of ice at 0°C (phase change, temperature constant)

• Region (c): Water warming from 0°C to 100°C (liquid phase, temperature increases)

• Region (d): Boiling of water at 100°C (phase change, temperature constant)

• Region (e): Steam warming above 100°C (gas phase, temperature increases)

The largest amount of heat is required for vaporization (liquid to gas), as indicated by the length of the plateau at 100°C.

Calculating Heat During Heating and Phase Changes

• Heating/Cooling without Phase Change:

  • Molar Heat Capacity (Cp): The amount of heat required to raise the temperature of 1 mole of a substance by 1°C.

  • Formula: $q = mC_p\Delta T$

  • For moles: $q = nC_{p,m}\Delta T$

• During Phase Change:

  • Heat required: $q = \Delta H \times n$

  • $\Delta H$ is in kJ/mol

• Total Heat Calculation:

  • $q_{total} = q_a + q_b + q_c + \ldots$ (sum of heat for each segment)

Vapor Pressure and Boiling Point

Vapor Pressure Curves

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid at a given temperature. The normal boiling point is the temperature at which the vapor pressure equals 1 atm.

• Substances with higher intermolecular forces have lower vapor pressures and higher boiling points.

• Examples: Diethyl ether, ethanol, water, ethylene glycol (in order of increasing boiling point).

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation relates the vapor pressure of a liquid to its temperature and the enthalpy of vaporization. It is useful for determining the heat of vaporization and predicting boiling points at different pressures.

• Graphical Form: $\ln P_{vap} = -\frac{\Delta H_{vap}}{R} \left( \frac{1}{T} \right) + \ln \beta$

• Two-Point Form: $\ln \left( \frac{P_2}{P_1} \right) = -\frac{\Delta H_{vap}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$

• Gas constant: $R = 8.314$ J/(mol·K)

Plotting $\ln P_{vap}$ versus $1/T$ yields a straight line with a slope of $-\Delta H_{vap}/R$.

Example: Determining Heat of Vaporization

Given vapor pressure data for dichloromethane at various temperatures, the Clausius-Clapeyron equation can be used to determine $\Delta H_{vap}$.

Temperature (K)	Vapor Pressure (torr)
200	0.5
220	4.5
240	21
260	71
280	210
300	391

By plotting $\ln P_{vap}$ vs $1/T$ and determining the slope, $\Delta H_{vap}$ can be calculated:

• Slope = $-\Delta H_{vap}/R$

• Example calculation: Slope = -3773 K, so $\Delta H_{vap} = 31.4$ kJ/mol

Application: Boiling Point at Different Pressures

The Clausius-Clapeyron equation can be used to determine the boiling point of water at pressures other than 1 atm, such as at high altitudes.

• Given: $\Delta H_{vap} = 40.7$ kJ/mol, $P_1 = 760$ mm Hg, $T_1 = 373.1$ K, $P_2 = 407$ mm Hg

• Rearrange the two-point form to solve for $T_2$:

$\ln \left( \frac{P_2}{P_1} \right) = -\frac{\Delta H_{vap}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$

Solving for $T_2$ gives the boiling point at the new pressure.

Summary Table: Key Phase Changes and Terms

Process	Direction	Heat Term
Fusion (Melting)	Solid → Liquid	$\Delta H_{fusion}$
Freezing	Liquid → Solid	$-\Delta H_{fusion}$
Vaporization	Liquid → Gas	$\Delta H_{vap}$
Condensation	Gas → Liquid	$-\Delta H_{vap}$
Sublimation	Solid → Gas	$\Delta H_{sub}$
Deposition	Gas → Solid	$-\Delta H_{sub}$

Key Equations

• Clausius-Clapeyron (Graphical): $\ln P_{vap} = -\frac{\Delta H_{vap}}{R} \left( \frac{1}{T} \right) + \ln \beta$

• Clausius-Clapeyron (Two-Point): $\ln \left( \frac{P_2}{P_1} \right) = -\frac{\Delta H_{vap}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$

• Heat for temperature change: $q = mC_p\Delta T$ or $q = nC_{p,m}\Delta T$

• Heat for phase change: $q = \Delta H \times n$

Additional info:

• During phase changes, temperature remains constant because all energy is used to break or form intermolecular forces.

• The enthalpy of vaporization is always greater than the enthalpy of fusion for the same substance, due to the greater energy required to completely separate molecules into the gas phase.