Phase Equilibrium

Phase Equilibrium

Phase equilibrium refers to the condition where multiple phases of a substance coexist in equilibrium without any net change in their amounts over time. It occurs when the chemical potential of each component is the same in all coexisting phases, ensuring no driving force for phase change.

Basics

In a system involving different phases (solid, liquid, gas), phase equilibrium is established when the rates of phase transitions (such as melting, vaporization, sublimation) between these phases are equal. This results in stable coexistence of phases at certain temperature and pressure conditions.

Chemical Potential and Phase Equilibrium

The key criterion for phase equilibrium is equality of chemical potentials:

${\mu }_i^{(\alpha )}={\mu }_i^{(\beta )}=\cdots$

for component $i$ in all phases $\alpha ,\beta ,\dots$.

Here, ${\mu }_i^{(\alpha )}$ is the chemical potential of component $i$ in phase $\alpha$.

Phase Rule

The Gibbs phase rule governs the degrees of freedom ($F$) in a system at equilibrium:

$F=C-P+2$

where:

• $F$ = number of degrees of freedom (independent intensive variables, e.g., temperature, pressure)
• $C$ = number of components
• $P$ = number of phases present

This rule helps determine how many variables can be changed without disturbing the equilibrium.

Phase Diagrams

Phase equilibrium is often represented graphically in phase diagrams showing the stable phases under different conditions of temperature, pressure, or composition.

- For example, the water phase diagram shows regions where ice, liquid water, and steam coexist. - Lines separating phases are called phase boundaries. - Points where three phases coexist (e.g., ice, liquid water, and vapor) are called triple points.

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation describes the relationship between pressure and temperature along a phase boundary, for example between liquid and vapor phases:

$\frac{dP}{dT}=\frac{L}{T\mathrm{\Delta }V}$

where:

• $L$ = latent heat of the phase transition
• $\mathrm{\Delta }V$ = change in volume during the phase change
• $P$ = pressure
• $T$ = temperature

This equation is fundamental for understanding vapor pressure and boiling point changes with temperature.

Applications

• Designing distillation and separation processes
• Understanding natural phenomena like frost formation and cloud formation
• Studying materials science and metallurgy
• Predicting behavior of mixtures in chemical engineering

References

• Smith, J. M., Van Ness, H. C., & Abbott, M. M. (2005). Introduction to Chemical Engineering Thermodynamics. McGraw-Hill.
• Atkins, P., & de Paula, J. (2010). Physical Chemistry. Oxford University Press.