Triple Point

A pure substance may exist in three principal phases: solid, liquid, and vapour. The equilibrium state of a substance is determined by its thermodynamic variables, particularly pressure \(P\) and temperature \(T\).

A change in pressure or temperature may cause a phase transition. The equilibrium between different phases can be represented by a pressure–temperature (\(P\)–\(T\)) phase diagram.

Consider the phase diagram of water, which contains three phase-equilibrium curves:

1. Solid–liquid equilibrium curve (fusion curve).
2. Liquid–vapour equilibrium curve (vaporization curve).
3. Solid–vapour equilibrium curve (sublimation curve).

Pressure–Temperature Phase Diagram of Water

Fig. 3.10.1: Schematic pressure–temperature phase diagram of water showing the triple point and three phase-equilibrium curves (not to scale).

(i) Solid–Liquid Equilibrium Curve (Fusion Curve)

The solid–liquid equilibrium curve represents the variation of melting temperature with pressure. For water, the melting temperature decreases slightly with increasing pressure because ice has a larger specific volume than liquid water.

Along this curve, solid and liquid phases coexist in thermodynamic equilibrium.

Let \(g_s\) and \(g_l\) denote the molar Gibbs free energies of the solid and liquid phases, respectively.

The condition for phase equilibrium is

\[ \boxed{g_s=g_l} \tag{1} \]

Since the molar Gibbs free energy of a pure substance is equal to its chemical potential, the equilibrium condition may also be written as

\[ \boxed{\mu_s=\mu_l} \]

(ii) Liquid–Vapour Equilibrium Curve

The liquid–vapour equilibrium curve represents the variation of the boiling temperature with pressure.

For water, the boiling temperature increases with increasing pressure.

At every point on this curve, liquid water and water vapour coexist in thermodynamic equilibrium.

Therefore, their molar Gibbs free energies are equal:

\[ \boxed{g_l=g_v} \tag{2} \]

Equivalently,

\[ \boxed{\mu_l=\mu_v} \]

The liquid–vapour equilibrium curve terminates at the critical point, beyond which the distinction between liquid and vapour disappears.

(iii) Solid–Vapour Equilibrium Curve

The solid–vapour equilibrium curve, also called the sublimation curve, represents the pressure and temperature conditions under which solid and vapour phases coexist.

Along this curve, sublimation and deposition occur in equilibrium.

The corresponding equilibrium condition is

\[ \boxed{g_s=g_v} \tag{3} \]

or

\[ \boxed{\mu_s=\mu_v} \]

(iv) Triple Point

The three phase-equilibrium curves intersect at a unique point known as the triple point.

At the triple point, solid, liquid, and vapour phases coexist in thermodynamic equilibrium.

Consequently, the molar Gibbs free energies of all three phases are equal:

\[ \boxed{g_s=g_l=g_v} \tag{4} \]

Equivalently,

\[ \boxed{\mu_s=\mu_l=\mu_v} \]

For pure water, the triple-point temperature and pressure are approximately

\[ \boxed{ T_t=273.16\,\text{K}=0.01^\circ\text{C} } \]

\[ \boxed{ P_t=611.657\,\text{Pa} \approx4.59\,\text{mmHg} } \]

The triple point is an invariant equilibrium state of a pure substance.

According to the Gibbs phase rule,

\[ F=C-\phi+2 \]

where \(C\) is the number of components, \(\phi\) is the number of phases, and \(F\) is the number of degrees of freedom.

For a pure substance at the triple point,

\[ F=1-3+2=0. \]

Thus, neither pressure nor temperature can be varied independently while maintaining three-phase equilibrium.

(v) Phase Diagram of a Normal Substance

For most substances, such as carbon dioxide, the solid phase is denser than the liquid phase. Consequently, the melting temperature increases with increasing pressure.

Their solid–liquid equilibrium curve therefore has a positive slope in the \(P\)–\(T\) diagram.

Fig. 3.10.2: Schematic phase diagram of a substance having a positive slope of the fusion curve.

3.11 Vapour in Equilibrium with a Liquid or Solid

When the vapour of a substance is in thermodynamic equilibrium with its liquid or solid phase at a given temperature, the pressure exerted by the vapour is called the saturated vapour pressure.

The dependence of saturated vapour pressure on temperature is described by the Clapeyron equation.

Derivation of the Clausius–Clapeyron Equation

For two phases in equilibrium, the Clapeyron equation is

\[ \boxed{ \frac{dP}{dT} = \frac{L}{T(V_2-V_1)} } \tag{3.11.1} \]

where

• \(L\) is the molar latent heat of transition.
• \(V_1\) is the molar volume of the condensed phase.
• \(V_2\) is the molar volume of the vapour phase.
• \(T\) is the absolute temperature.

For vaporization or sublimation at sufficiently low pressure, the molar volume of the vapour is much larger than that of the liquid or solid:

\[ V_2\gg V_1. \]

Therefore,

\[ V_2-V_1\approx V_2. \]

Equation (3.11.1) becomes

\[ \frac{dP}{dT} \approx \frac{L}{TV_2}. \tag{3.11.2} \]

Assuming that the vapour behaves as an ideal gas, the equation of state for one mole is

\[ PV_2=RT. \]

Hence,

\[ V_2=\frac{RT}{P}. \tag{3.11.3} \]

Substituting equation (3.11.3) into equation (3.11.2),

\[ \frac{dP}{dT} = \frac{L}{T\left(\frac{RT}{P}\right)}. \]

Simplifying,

\[ \boxed{ \frac{dP}{dT} = \frac{LP}{RT^2} } \tag{3.11.4} \]

Rearranging,

\[ \frac{dP}{P} = \frac{L}{R}\frac{dT}{T^2}. \]

If the latent heat \(L\) is approximately constant over the temperature interval, integration gives

\[ \int\frac{dP}{P} = \frac{L}{R}\int T^{-2}\,dT. \]

Therefore,

\[ \ln P = -\frac{L}{RT}+C, \]

where \(C\) is an integration constant.

Writing \(C=\ln P_0\), where \(P_0\) is a constant having the dimensions of pressure,

\[ \ln P = -\frac{L}{RT}+\ln P_0. \]

Hence,

\[ \ln\left(\frac{P}{P_0}\right) = -\frac{L}{RT}. \]

Taking the exponential of both sides,

\[ \boxed{ P=P_0\exp\left(-\frac{L}{RT}\right) } \tag{3.11.5} \]

This equation gives the approximate dependence of saturated vapour pressure on temperature when the latent heat is constant and the vapour behaves ideally.

Vapour Pressure at Two Temperatures

For two equilibrium states at temperatures \(T_1\) and \(T_2\), with corresponding vapour pressures \(P_1\) and \(P_2\),

\[ \ln P_1=-\frac{L}{RT_1}+C \]

and

\[ \ln P_2=-\frac{L}{RT_2}+C. \]

Subtracting the first equation from the second,

\[ \ln P_2-\ln P_1 = -\frac{L}{RT_2}+\frac{L}{RT_1}. \]

Therefore,

\[ \boxed{ \ln\left(\frac{P_2}{P_1}\right) = \frac{L}{R} \left( \frac{1}{T_1}-\frac{1}{T_2} \right) } \tag{3.11.6} \]

This integrated Clausius–Clapeyron equation is widely used to estimate vapour pressure and latent heat over a limited temperature range.

Physical significance: For a positive latent heat of vaporization or sublimation, the saturated vapour pressure increases with increasing temperature. A plot of \(\ln P\) against \(1/T\) is approximately linear, with slope \(-L/R\), provided \(L\) remains constant.

3.12 Equilibrium Conditions for a System of Fixed Volume in Contact with a Heat Reservoir: Helmholtz Free Energy

Consider a closed thermodynamic system \(A\) in thermal contact with a large heat reservoir \(A'\) maintained at a constant temperature \(T_0\).

The system is enclosed within rigid walls, so its volume remains constant. Only heat is exchanged between the system and the reservoir, and no external work is performed.

Let the system absorb an amount of heat \(Q\) from the reservoir.

Since the reservoir loses the same amount of heat, its entropy change is

\[ \boxed{ \Delta S'=-\frac{Q}{T_0} } \tag{3.12.1} \]

From the first law of thermodynamics,

\[ \Delta U=Q-W. \]

At constant volume, assuming only pressure–volume work,