11.1 Partial molar properties

Partial molar properties are key to understanding how components behave in mixtures. They show how adding a bit of one substance changes the overall properties of the whole mix. This helps predict how mixtures will act under different conditions.

These properties, like partial molar volume and Gibbs free energy, are crucial for calculating changes in mixtures. They're especially useful when dealing with real-world solutions, where components interact in complex ways. Understanding these concepts is vital for solving practical problems in thermodynamics.

Partial Molar Quantities

Definition and Significance

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• Partial molar quantities represent the contribution of each component to the total thermodynamic property of a mixture
• Useful for understanding the behavior and properties of individual components in a mixture
• Allow for the calculation of changes in thermodynamic properties when the composition of a mixture is altered

Types of Partial Molar Quantities

• Partial molar volume
  • Represents the change in volume of a mixture when one mole of a component is added at constant temperature, pressure, and composition
  • Calculated using the equation $\bar{V}_i = (\frac{\partial V}{\partial n_i})_{T,P,n_j}$
• Partial molar Gibbs free energy
  • Represents the change in Gibbs free energy of a mixture when one mole of a component is added at constant temperature, pressure, and composition
  • Equivalent to the chemical potential of the component in the mixture
  • Calculated using the equation $\bar{G}_i = (\frac{\partial G}{\partial n_i})_{T,P,n_j}$
• Partial molar enthalpy
  • Represents the change in enthalpy of a mixture when one mole of a component is added at constant temperature, pressure, and composition
  • Calculated using the equation $\bar{H}_i = (\frac{\partial H}{\partial n_i})_{T,P,n_j}$

Chemical Potential

• Chemical potential is the partial molar Gibbs free energy of a component in a mixture
• Represents the change in Gibbs free energy when one mole of a component is added to a mixture at constant temperature, pressure, and composition
• Determines the direction of chemical reactions and phase transitions in a mixture
• Equals the molar Gibbs free energy for pure substances

Thermodynamic Relationships

Gibbs-Duhem Equation

• Relates changes in chemical potentials of components in a mixture to changes in temperature and pressure
• Constrains the values of partial molar quantities in a mixture
• Expressed as $\sum_{i=1}^{n} x_i d\mu_i = -SdT + VdP$, where $x_i$ is the mole fraction, $\mu_i$ is the chemical potential, $S$ is the entropy, $T$ is the temperature, $V$ is the volume, and $P$ is the pressure

Relationship between Chemical Potential and Partial Molar Gibbs Free Energy

• Chemical potential and partial molar Gibbs free energy are equivalent for a component in a mixture
• $\mu_i = \bar{G}_i = (\frac{\partial G}{\partial n_i})_{T,P,n_j}$
• Allows for the calculation of chemical potentials from Gibbs free energy data and vice versa

Applications of Thermodynamic Relationships

• Predicting the behavior of mixtures under different conditions (temperature, pressure, composition)
• Calculating changes in thermodynamic properties during mixing or separation processes
• Determining the equilibrium composition of a mixture based on chemical potentials

Composition Variables

Molality

• Molality (m) is the number of moles of solute per kilogram of solvent
• Expressed as $m = \frac{n_{solute}}{m_{solvent}}$, where $n_{solute}$ is the number of moles of solute and $m_{solvent}$ is the mass of solvent in kilograms
• Molality is independent of temperature and pressure, making it a useful composition variable for thermodynamic calculations

Mole Fraction

• Mole fraction (x) is the ratio of the number of moles of a component to the total number of moles in a mixture
• Expressed as $x_i = \frac{n_i}{\sum_{i=1}^{n} n_i}$, where $n_i$ is the number of moles of component i and $\sum_{i=1}^{n} n_i$ is the total number of moles in the mixture
• Mole fractions are dimensionless and sum to unity for all components in a mixture

Relationship between Composition Variables and Chemical Potential

• Chemical potential of a component in a mixture depends on its composition
• For ideal solutions, the chemical potential is related to the mole fraction by $\mu_i = \mu_i^0 + RT \ln x_i$, where $\mu_i^0$ is the standard chemical potential, $R$ is the gas constant, and $T$ is the temperature
• Composition variables allow for the calculation of chemical potentials and the prediction of mixture behavior based on composition changes