10.3 Chemical Potential and Phase Stability Criteria

Chemical potential is a key concept in thermodynamics, representing how a substance's energy changes as it moves between phases. It's crucial for understanding phase equilibrium and stability, helping us predict when and how substances will change states or mix together.

Phase stability criteria tell us when a substance will stay in its current state or transform. By comparing chemical potentials, we can determine if a phase is stable, metastable, or unstable. This knowledge is essential for predicting and controlling phase behavior in various systems.

Chemical potential of a component

Definition and interpretation

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• Chemical potential (μ) represents the change in Gibbs free energy of a system when a component is added or removed at constant temperature, pressure, and composition of other components
• It is a measure of the potential for a substance to undergo a change in a system, such as a chemical reaction, phase transition, or transport process
• For a pure substance, the chemical potential is equal to its molar Gibbs free energy at the given temperature and pressure

Factors influencing chemical potential

• The chemical potential of a component in a mixture depends on its concentration, temperature, and pressure, as well as the interactions with other components in the mixture
• In an ideal solution, the chemical potential of a component is related to its mole fraction through the equation: $\mu = \mu^{\circ} + RT \ln(x)$ where $\mu^{\circ}$ is the standard chemical potential, $R$ is the gas constant, $T$ is the absolute temperature, and $x$ is the mole fraction
• Example: In a binary ideal solution of ethanol and water, the chemical potential of ethanol increases as its mole fraction increases, while the chemical potential of water decreases

Conditions for phase stability

Phase stability and metastability

• Phase stability refers to the ability of a phase to maintain its state without transforming into another phase under given conditions of temperature, pressure, and composition
• A phase is stable when its chemical potential is lower than the chemical potential of any other possible phase or combination of phases at the same temperature, pressure, and overall composition
• Metastability occurs when a phase has a higher chemical potential than the stable phase but is kinetically hindered from transforming into the stable phase due to an activation energy barrier
• Example: Diamond is metastable at room temperature and pressure, as it has a higher chemical potential than graphite, but the transformation is kinetically hindered

Derivation and application of stability conditions

• The condition for phase stability can be derived by considering the change in Gibbs free energy ($dG$) when a small amount ($dn$) of a component is transferred from one phase ($\alpha$) to another ($\beta$) at constant temperature, pressure, and composition of other components: $dG = (\mu_{\beta} - \mu_{\alpha})dn$ For stability, $dG$ must be greater than or equal to zero, implying $\mu_{\alpha} \leq \mu_{\beta}$
• The condition for metastability is $\mu_{\alpha} > \mu_{\beta}$, but the transformation from phase $\alpha$ to phase $\beta$ is kinetically hindered
• Example: In a system containing liquid water and water vapor at equilibrium, the chemical potentials of water in both phases are equal, satisfying the condition for phase stability

Phase stability and Gibbs free energy

Gibbs free energy and phase stability

• The Gibbs free energy ($G$) is a thermodynamic potential that determines the stability of phases in a system at constant temperature and pressure
• The stable phase or combination of phases in a system minimizes the total Gibbs free energy at the given conditions
• The Gibbs free energy of a system is a function of temperature, pressure, and composition, and can be expressed as: $G = H - TS$ where $H$ is the enthalpy, $T$ is the absolute temperature, and $S$ is the entropy

Derivatives of Gibbs free energy and stability analysis

• The first derivative of the Gibbs free energy with respect to composition ($\partial G/\partial n$) at constant temperature and pressure gives the chemical potential of a component in the system
• The second derivative of the Gibbs free energy with respect to composition ($\partial^2 G/\partial n^2$) determines the stability of a phase:
  • A positive value indicates stability
  • A negative value indicates instability
  • A zero value indicates a phase boundary or critical point
• The Gibbs phase rule relates the number of degrees of freedom ($F$), the number of components ($C$), and the number of phases ($P$) in a system at equilibrium: $F = C - P + 2$ This helps to analyze the stability and variability of phases
• Example: In a single-component system (e.g., pure water), the Gibbs free energy curve as a function of temperature and pressure can be used to determine the stable phase (solid, liquid, or gas) at given conditions

Chemical potential vs fugacity in equilibrium

Fugacity and its relationship to chemical potential

• Fugacity ($f$) is a measure of the effective partial pressure of a component in a mixture, accounting for non-ideal behavior and intermolecular interactions
• The chemical potential of a component in a mixture can be expressed in terms of its fugacity using the equation: $\mu = \mu^{\circ} + RT \ln(f/f^{\circ})$ where $\mu^{\circ}$ is the standard chemical potential, $R$ is the gas constant, $T$ is the absolute temperature, and $f^{\circ}$ is the standard fugacity (usually chosen as 1 bar)
• For an ideal gas, the fugacity is equal to the partial pressure, and the chemical potential can be expressed as: $\mu = \mu^{\circ} + RT \ln(P/P^{\circ})$ where $P$ is the partial pressure and $P^{\circ}$ is the standard pressure

Phase equilibrium and fugacity equality

• At phase equilibrium, the chemical potential of each component must be equal in all phases, which implies that the fugacity of each component must also be equal in all phases: $f_{\alpha} = f_{\beta} = ... \text{ for each component}$
• The fugacity coefficient ($\phi$) is defined as the ratio of fugacity to partial pressure: $\phi = f/P$ It is a measure of the deviation from ideal gas behavior. For an ideal gas, $\phi = 1$
• The equality of fugacities at equilibrium can be used to derive phase equilibrium conditions, such as the vapor-liquid equilibrium (VLE) equation: $y_i\phi_iP = x_i\gamma_iP_i^*$ where $y_i$ and $x_i$ are the vapor and liquid mole fractions, $\phi_i$ and $\gamma_i$ are the vapor and liquid fugacity coefficients, $P$ is the total pressure, and $P_i^*$ is the vapor pressure of the pure component
• Example: In a vapor-liquid equilibrium system containing a mixture of hydrocarbons, the fugacities of each component in the vapor and liquid phases are equal at equilibrium, allowing the calculation of phase compositions and properties