🪫Chemical Process Balances Unit 8 – Chemical Equilibrium in Reactive Processes

Key Concepts and Definitions

• Chemical equilibrium occurs when the rates of the forward and reverse reactions are equal, resulting in no net change in the concentrations of reactants and products over time
• Dynamic equilibrium maintains constant concentrations of reactants and products, but the reactions continue to occur in both directions at equal rates
• Equilibrium constant ($K$) expresses the relationship between the concentrations of reactants and products at equilibrium, indicating the extent to which a reaction proceeds
• Homogeneous equilibrium involves reactants and products in the same phase (gas or liquid), while heterogeneous equilibrium involves different phases (solid, liquid, or gas)
• Law of mass action states that the rate of a chemical reaction is proportional to the product of the concentrations of the reactants, each raised to a power equal to its stoichiometric coefficient
• Reaction quotient ($Q$) represents the ratio of product concentrations to reactant concentrations at any given point in the reaction, not necessarily at equilibrium
• Equilibrium position refers to the relative amounts of reactants and products present at equilibrium, which can be shifted by changing conditions such as temperature, pressure, or concentration

Equilibrium Constants and Reaction Quotients

• The equilibrium constant ($K$) is calculated using the equilibrium concentrations of reactants and products, raised to their stoichiometric coefficients
  • For a general reaction $aA + bB \rightleftharpoons cC + dD$, the equilibrium constant is expressed as $K = \frac{[C]^c[D]^d}{[A]^a[B]^b}$
• The value of $K$ depends on the specific reaction and temperature but is independent of the initial concentrations of reactants and products
• Larger $K$ values indicate that the equilibrium favors the products, while smaller $K$ values indicate that the equilibrium favors the reactants
• The reaction quotient ($Q$) is calculated using the same formula as the equilibrium constant but with the concentrations of reactants and products at any given point in the reaction
• Comparing $Q$ to $K$ determines the direction in which the reaction will proceed to reach equilibrium
  • If $Q < K$, the reaction will proceed in the forward direction to form more products
  • If $Q > K$, the reaction will proceed in the reverse direction to form more reactants
  • If $Q = K$, the reaction is at equilibrium, and no net change in concentrations will occur
• Equilibrium constants can also be expressed in terms of partial pressures ($K_p$) for gas-phase reactions or in terms of activities ($K_a$) for reactions involving non-ideal solutions

Factors Affecting Chemical Equilibrium

• Temperature changes affect the equilibrium position by altering the equilibrium constant ($K$)
  • Increasing temperature shifts the equilibrium in the endothermic direction (absorbs heat), while decreasing temperature shifts the equilibrium in the exothermic direction (releases heat)
• Pressure changes affect the equilibrium position of gas-phase reactions by altering the partial pressures of the reactants and products
  • Increasing pressure shifts the equilibrium towards the side with fewer moles of gas, while decreasing pressure shifts the equilibrium towards the side with more moles of gas
• Concentration changes affect the equilibrium position by altering the reaction quotient ($Q$)
  • Adding reactants or removing products shifts the equilibrium towards the products, while removing reactants or adding products shifts the equilibrium towards the reactants
• Catalysts accelerate the rates of both forward and reverse reactions equally, reaching equilibrium faster without changing the equilibrium position or the value of the equilibrium constant ($K$)
• Inert gases do not participate in the reaction but can affect the equilibrium position of gas-phase reactions by altering the total pressure and partial pressures of the reactants and products
• Volume changes in a gas-phase reaction affect the equilibrium position by altering the partial pressures of the reactants and products, similar to the effect of pressure changes

Le Chatelier's Principle

• Le Chatelier's principle states that when a system at equilibrium is subjected to a stress or change in conditions, the equilibrium will shift in the direction that minimizes the stress or counteracts the change
• Stresses or changes can include temperature, pressure, concentration, or volume
• Applying Le Chatelier's principle helps predict the direction of the equilibrium shift in response to various disturbances
  • Increasing temperature shifts the equilibrium in the endothermic direction, while decreasing temperature shifts the equilibrium in the exothermic direction
  • Increasing pressure or decreasing volume shifts the equilibrium towards the side with fewer moles of gas, while decreasing pressure or increasing volume shifts the equilibrium towards the side with more moles of gas
  • Adding reactants or removing products shifts the equilibrium towards the products, while removing reactants or adding products shifts the equilibrium towards the reactants
• Le Chatelier's principle is a qualitative tool for understanding equilibrium shifts and does not provide quantitative information about the extent of the shift
• The principle assumes that the system is at equilibrium before the stress or change is applied and that the stress or change is relatively small
• In some cases, applying multiple stresses simultaneously can lead to more complex equilibrium shifts that may not be easily predicted using Le Chatelier's principle alone

Equilibrium Calculations

• Equilibrium calculations involve determining the concentrations of reactants and products at equilibrium, given the initial concentrations and the equilibrium constant ($K$)
• The ICE table method (Initial, Change, Equilibrium) is a systematic approach to solving equilibrium problems
  • Initial concentrations are the starting concentrations of reactants and products before the reaction proceeds
  • Change in concentrations is determined using the stoichiometric coefficients and a variable (e.g., $x$) to represent the change in the limiting reactant
  • Equilibrium concentrations are the sum of the initial concentrations and the change in concentrations
• The equilibrium concentrations are substituted into the equilibrium constant expression, forming an equation that can be solved for the variable ($x$)
• The calculated value of $x$ is then used to determine the equilibrium concentrations of all reactants and products
• For gas-phase reactions, the ideal gas law ($PV = nRT$) can be used to relate the partial pressures of reactants and products to their concentrations
• Quadratic equations may arise in equilibrium calculations when the variable ($x$) appears in multiple terms of the equilibrium constant expression
  • The quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, can be used to solve for $x$ in these cases
• Simplifying assumptions, such as the small $x$ approximation, may be used in some cases to avoid solving quadratic equations, but the validity of the assumption must be checked after the calculation

Applications in Industrial Processes

• Haber-Bosch process for ammonia synthesis ($\ce{N2(g) + 3H2(g) <=> 2NH3(g)}$) relies on high pressure and moderate temperature to shift the equilibrium towards the production of ammonia
  • Iron catalysts are used to accelerate the reaction and improve the yield of ammonia
• Contact process for sulfuric acid production ($\ce{2SO2(g) + O2(g) <=> 2SO3(g)}$) employs vanadium pentoxide catalyst and optimized temperature and pressure conditions to maximize the conversion of sulfur dioxide to sulfur trioxide
• Ostwald process for nitric acid production ($\ce{4NH3(g) + 5O2(g) <=> 4NO(g) + 6H2O(g)}$, followed by $\ce{2NO(g) + O2(g) <=> 2NO2(g)}$ and $\ce{3NO2(g) + H2O(l) <=> 2HNO3(aq) + NO(g)}$) uses platinum-rhodium catalysts and carefully controlled conditions to optimize the yield and selectivity of nitric acid
• Steam reforming of methane ($\ce{CH4(g) + H2O(g) <=> CO(g) + 3H2(g)}$) is a key process in hydrogen production and syngas generation, using nickel catalysts and high temperatures to drive the equilibrium towards the products
• Water-gas shift reaction ($\ce{CO(g) + H2O(g) <=> CO2(g) + H2(g)}$) is used to adjust the composition of syngas and produce additional hydrogen, with iron-chromium or copper-zinc catalysts and temperature control to optimize the equilibrium
• Ethanol fermentation ($\ce{C6H12O6(aq) -> 2C2H5OH(aq) + 2CO2(g)}$) is a biological process that relies on yeast enzymes and controlled conditions (temperature, pH, and substrate concentration) to maximize the production of ethanol at equilibrium

Common Pitfalls and Misconceptions

• Confusing the equilibrium constant ($K$) with the reaction quotient ($Q$) can lead to incorrect predictions about the direction of the equilibrium shift
• Neglecting to consider the stoichiometric coefficients when setting up the equilibrium constant expression or the ICE table can result in incorrect calculations
• Assuming that the equilibrium constant ($K$) changes with the initial concentrations of reactants and products, rather than understanding that $K$ is a constant at a given temperature
• Misinterpreting Le Chatelier's principle by assuming that the equilibrium always shifts to the side with the smallest number of moles, rather than considering the specific stress or change applied to the system
• Forgetting to check the validity of simplifying assumptions, such as the small $x$ approximation, after performing equilibrium calculations
• Incorrectly applying the ideal gas law to non-ideal systems or solutions, leading to inaccurate equilibrium calculations
• Neglecting to consider the effect of catalysts on the rate of the reaction, while understanding that catalysts do not change the equilibrium position or the value of the equilibrium constant ($K$)
• Confusing the concepts of kinetics and thermodynamics, such as assuming that a reaction with a large equilibrium constant ($K$) will always proceed quickly, or that a reaction with a small $K$ will not occur at all

Practice Problems and Examples

• For the reaction $\ce{2NOCl(g) <=> 2NO(g) + Cl2(g)}$, the equilibrium constant $K_c$ is $1.6 \times 10^{-5}$ at $430^\circ C$. If the initial concentration of $\ce{NOCl}$ is $0.1 M$, calculate the equilibrium concentrations of all species.
• The equilibrium constant $K_p$ for the reaction $\ce{N2(g) + 3H2(g) <=> 2NH3(g)}$ is $1.6 \times 10^{-4}$ at $400^\circ C$. If the initial partial pressures of $\ce{N2}$, $\ce{H2}$, and $\ce{NH3}$ are $1.2 atm$, $0.8 atm$, and $0.5 atm$, respectively, determine the direction in which the reaction will proceed to reach equilibrium.
• For the reaction $\ce{CO(g) + H2O(g) <=> CO2(g) + H2(g)}$, the equilibrium constant $K_c$ is $4.2$ at $1200 K$. If the equilibrium concentrations of $\ce{CO}$, $\ce{H2O}$, and $\ce{CO2}$ are $0.02 M$, $0.01 M$, and $0.03 M$, respectively, calculate the equilibrium concentration of $\ce{H2}$.
• The decomposition of $\ce{PCl5(g)}$ occurs according to the reaction $\ce{PCl5(g) <=> PCl3(g) + Cl2(g)}$. At $250^\circ C$, the equilibrium constant $K_c$ is $2.0$. If the initial concentration of $\ce{PCl5}$ is $0.5 M$, and no $\ce{PCl3}$ or $\ce{Cl2}$ are present initially, determine the equilibrium concentrations of all species.
• For the reaction $\ce{2SO2(g) + O2(g) <=> 2SO3(g)}$, the equilibrium constant $K_c$ is $280$ at $727^\circ C$. If the initial concentrations of $\ce{SO2}$ and $\ce{O2}$ are $0.1 M$ and $0.05 M$, respectively, and no $\ce{SO3}$ is present initially, calculate the equilibrium concentrations of all species.