9.1 Chemical potential and Gibbs-Duhem equation

Chemical potential and Gibbs free energy are key concepts in thermodynamics. They help us understand how energy changes in chemical systems, driving reactions and phase transitions. These ideas are crucial for predicting stability and equilibrium in various processes.

The Gibbs-Duhem equation shows how chemical potentials in a system are connected. It's a powerful tool for analyzing mixtures and solutions, helping us grasp how changes in one component affect others. This equation has wide-ranging applications in thermodynamics and chemistry.

Chemical Potential and Gibbs Free Energy

Chemical potential and Gibbs free energy

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• Chemical potential ($\mu_i$) partial molar Gibbs free energy represents change in Gibbs free energy when one mole of component $i$ added to system at constant temperature, pressure, and composition of other components
• Mathematically defined as $\mu_i = \left(\frac{\partial G}{\partial n_i}\right)_{T,P,n_{j \neq i}}$ where $G$ Gibbs free energy, $n_i$ number of moles of component $i$
• In multi-component system, total Gibbs free energy sum of chemical potentials of each component multiplied by their respective mole numbers $G = \sum_{i} \mu_i n_i$
• Useful for understanding driving forces behind chemical reactions and phase transitions (melting, boiling)
• Helps predict stability and equilibrium of chemical systems (solubility, vapor pressure)

Gibbs-Duhem Equation and Its Applications

Derivation of Gibbs-Duhem equation

• Derived from total differential of Gibbs free energy $dG = -SdT + VdP + \sum_{i} \mu_i dn_i$
• At constant temperature and pressure, $dG = \sum_{i} \mu_i dn_i$
• Dividing by total number of moles ($n_t$) yields $\frac{dG}{n_t} = \sum_{i} \mu_i dx_i$ where $x_i$ mole fraction of component $i$
• Since $\sum_{i} x_i = 1$, differentiating gives $\sum_{i} dx_i = 0$
• Combining equations results in Gibbs-Duhem equation $\sum_{i} x_i d\mu_i = 0$
• Shows chemical potentials in system are interdependent change in chemical potential of one component must be balanced by changes in chemical potentials of other components

Applications of Gibbs-Duhem equation

• For binary system (components 1 and 2) at constant temperature and pressure, Gibbs-Duhem equation simplifies to $x_1 d\mu_1 + x_2 d\mu_2 = 0$ rearranged to $d\mu_2 = -\frac{x_1}{x_2} d\mu_1$
• By integrating, change in chemical potential of component 2 calculated from change in chemical potential of component 1 and composition of system
• Can relate changes in chemical potentials to changes in activity coefficients or partial pressures in non-ideal systems
• Used to derive other important thermodynamic relations (Clausius-Clapeyron equation, Duhem-Margules equation)
• Helps understand behavior of mixtures and solutions (azeotropes, colligative properties)

Factors affecting chemical potential

• Effect of temperature given by $\left(\frac{\partial \mu_i}{\partial T}\right)_{P,n_j} = -S_i$ where $S_i$ partial molar entropy of component $i$
  • Increase in temperature generally decreases chemical potential
• Effect of pressure given by $\left(\frac{\partial \mu_i}{\partial P}\right)_{T,n_j} = V_i$ where $V_i$ partial molar volume of component $i$
  • Increase in pressure generally increases chemical potential
• Effect of composition depends on specific system and interactions between components
  • In ideal systems, chemical potential of component related to its mole fraction by $\mu_i = \mu_i^0 + RT \ln x_i$ where $\mu_i^0$ standard chemical potential and $R$ gas constant
  • In non-ideal systems, activity coefficients or fugacities used to account for deviations from ideal behavior (intermolecular forces, hydrogen bonding)