12.3 Calculation of equilibrium compositions

Chemical reactions don't always go to completion. Equilibrium composition is when forward and reverse reaction rates balance out. This topic dives into calculating these compositions, a key skill for understanding real-world chemical processes.

We'll explore factors like temperature and pressure that influence equilibrium. We'll also learn about partial pressures, mole fractions, and iterative techniques for solving complex equilibrium problems. These tools are essential for predicting and controlling chemical reactions.

Reaction Equilibrium

Equilibrium State and Composition

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• Equilibrium composition represents the concentrations or partial pressures of reactants and products at which the forward and reverse reaction rates are equal
• Stoichiometric coefficients in a balanced chemical equation determine the relative amounts of reactants consumed and products formed as the reaction proceeds towards equilibrium
• Extent of reaction ($\xi$) quantifies the progress of a reaction from the initial state to equilibrium
  • Defined as the number of moles of a reactant consumed or a product formed divided by its stoichiometric coefficient
• Equilibrium yield measures the fraction of a limiting reactant that is converted to products at equilibrium
  • Calculated as the ratio of the actual amount of product formed to the theoretical maximum amount based on the limiting reactant (ethanol production from glucose fermentation)

Factors Influencing Equilibrium

• Temperature changes affect the equilibrium composition by shifting the equilibrium position according to Le Chatelier's principle
  • Increasing temperature favors the endothermic direction, while decreasing temperature favors the exothermic direction (Haber process for ammonia synthesis)
• Pressure changes influence the equilibrium composition for gaseous reactions where there is a change in the total number of moles between reactants and products
  • Increasing pressure favors the side with fewer moles of gas, while decreasing pressure favors the side with more moles of gas (nitrogen dioxide dimerization)

Composition Measures

Partial Pressures and Mole Fractions

• Partial pressures represent the individual pressures exerted by each gas in a mixture, as if it occupied the entire volume alone
  • Calculated using the ideal gas equation: $P_i = \frac{n_i RT}{V}$, where $P_i$ is the partial pressure of gas $i$, $n_i$ is the number of moles of gas $i$, $R$ is the universal gas constant, $T$ is the temperature, and $V$ is the volume
• Mole fractions express the composition of a mixture as the ratio of the number of moles of each component to the total number of moles in the mixture
  • Defined as $x_i = \frac{n_i}{n_{total}}$, where $x_i$ is the mole fraction of component $i$, $n_i$ is the number of moles of component $i$, and $n_{total}$ is the total number of moles in the mixture
• Partial pressures and mole fractions are related by Dalton's law of partial pressures: $P_i = x_i P_{total}$, where $P_{total}$ is the total pressure of the mixture (air composition: 78% nitrogen, 21% oxygen, 1% argon)

Calculation Methods

Iterative Techniques for Equilibrium Composition

• Iterative methods are used to solve for the equilibrium composition when the reaction involves multiple species and the equilibrium constants are known
• The basic steps involve assuming an initial composition, calculating the equilibrium constants using the assumed composition, comparing the calculated constants with the known values, and iteratively adjusting the composition until convergence is achieved
  • Convergence is reached when the calculated equilibrium constants match the known values within a specified tolerance (water-gas shift reaction: $\ce{CO + H2O <=> CO2 + H2}$)
• Iterative techniques can be implemented using numerical methods such as the Newton-Raphson method or the bisection method
  • These methods systematically refine the assumed composition by minimizing the difference between the calculated and known equilibrium constants (methanol synthesis from syngas: $\ce{CO + 2H2 <=> CH3OH}$)