🧤Physical Chemistry I Unit 7 – Free Energy and Chemical Equilibrium

Key Concepts and Definitions

• Gibbs free energy ($\Delta G$) represents the maximum amount of non-expansion work that can be extracted from a thermodynamically closed system
• Standard state refers to a reference point where the pressure is 1 atm, the concentration of all species is 1 M, and the temperature is usually 298 K
• Chemical potential ($\mu_i$) describes the change in free energy when the amount of a substance changes in a system at constant temperature and pressure
• Equilibrium constant ($K$) quantifies the ratio of product concentrations to reactant concentrations at equilibrium
• Le Chatelier's principle states that when a system at equilibrium is disturbed, it will shift to counteract the disturbance and re-establish equilibrium
• Thermodynamic equilibrium occurs when a system reaches a state of maximum entropy and minimum free energy, with no net change in macroscopic properties over time
• Spontaneous processes occur without external intervention and are characterized by a decrease in Gibbs free energy ($\Delta G < 0$)

Thermodynamic Foundations

• The first law of thermodynamics states that energy cannot be created or destroyed, only converted from one form to another
  • It introduces the concept of internal energy ($U$) and its relation to heat ($q$) and work ($w$)
• The second law of thermodynamics introduces the concept of entropy ($S$), a measure of the disorder or randomness of a system
  • It states that the entropy of an isolated system always increases over time
• Enthalpy ($H$) is a state function that represents the total heat content of a system at constant pressure
  • It is defined as $H = U + PV$, where $U$ is the internal energy, $P$ is the pressure, and $V$ is the volume
• Gibbs free energy ($G$) is a thermodynamic potential that combines enthalpy and entropy to determine the spontaneity of a process at constant temperature and pressure
  • It is defined as $G = H - TS$, where $H$ is the enthalpy, $T$ is the temperature, and $S$ is the entropy
• The fundamental equation of thermodynamics relates the change in internal energy to the changes in entropy, volume, and composition of a system
  • $dU = TdS - PdV + \sum_i \mu_i dN_i$, where $\mu_i$ is the chemical potential of species $i$ and $N_i$ is the number of moles of species $i$

Free Energy and Spontaneity

• The change in Gibbs free energy ($\Delta G$) determines the spontaneity of a process at constant temperature and pressure
  • If $\Delta G < 0$, the process is spontaneous and will occur without external intervention
  • If $\Delta G > 0$, the process is non-spontaneous and requires external energy input to occur
  • If $\Delta G = 0$, the system is at equilibrium, and no net change occurs
• The standard Gibbs free energy change ($\Delta G^\circ$) is the change in free energy when reactants and products are in their standard states
  • It is related to the equilibrium constant ($K$) by the equation $\Delta G^\circ = -RT \ln K$, where $R$ is the gas constant and $T$ is the temperature
• The Gibbs free energy change ($\Delta G$) under non-standard conditions is given by $\Delta G = \Delta G^\circ + RT \ln Q$, where $Q$ is the reaction quotient
• The van 't Hoff equation relates the change in equilibrium constant with temperature: $\ln \frac{K_2}{K_1} = -\frac{\Delta H^\circ}{R} (\frac{1}{T_2} - \frac{1}{T_1})$
  • It allows for the prediction of equilibrium constants at different temperatures

Chemical Equilibrium Basics

• Chemical equilibrium is a dynamic state in which the forward and reverse reactions proceed at equal rates, resulting in no net change in concentrations over time
• The law of mass action states that the rate of a reaction is proportional to the product of the concentrations of the reactants raised to their stoichiometric coefficients
• The equilibrium constant ($K$) is the ratio of the product of the equilibrium concentrations of the products raised to their stoichiometric coefficients to the product of the equilibrium concentrations of the reactants raised to their stoichiometric coefficients
  • For a general reaction $aA + bB \rightleftharpoons cC + dD$, the equilibrium constant is $K = \frac{[C]^c[D]^d}{[A]^a[B]^b}$
• The reaction quotient ($Q$) has the same form as the equilibrium constant but is calculated using the actual concentrations at any given time
  • Comparing $Q$ to $K$ allows for the determination of the direction in which a reaction will proceed to reach equilibrium
• Equilibrium constants can be expressed in terms of concentrations ($K_c$), partial pressures ($K_p$), or activities ($K_a$) depending on the phase of the reactants and products

Equilibrium Constants and Their Significance

• The magnitude of the equilibrium constant indicates the extent of a reaction at equilibrium
  • A large $K$ value (> 1000) suggests that the products are favored, and the reaction proceeds mostly to completion
  • A small $K$ value (< 0.001) suggests that the reactants are favored, and the reaction proceeds only to a small extent
• Equilibrium constants are dimensionless, as the concentration or pressure terms are divided by their respective standard state values
• The value of the equilibrium constant depends on the form of the balanced chemical equation
  • Reversing the equation reciprocates the equilibrium constant: $K_\text{reverse} = \frac{1}{K_\text{forward}}$
  • Multiplying the equation by a factor $n$ raises the equilibrium constant to the power $n$: $K_\text{new} = (K_\text{original})^n$
• Equilibrium constants are temperature-dependent but are independent of the initial concentrations of reactants and products
• The solubility product constant ($K_{sp}$) is a special case of the equilibrium constant that describes the equilibrium between a solid solute and its dissolved ions in a saturated solution

Factors Affecting Equilibrium

• Le Chatelier's principle states that when a system at equilibrium is disturbed, it will shift to counteract the disturbance and re-establish equilibrium
• Changes in concentration of reactants or products will shift the equilibrium position
  • Adding reactants or removing products will shift the equilibrium to the right (towards products)
  • Removing reactants or adding products will shift the equilibrium to the left (towards reactants)
• Changes in pressure affect equilibrium only if there is a change in the total number of moles of gas 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
• Changes in temperature affect the equilibrium position based on the sign of the enthalpy change of the reaction ($\Delta H$)
  • For endothermic reactions ($\Delta H > 0$), increasing temperature shifts the equilibrium to the right (towards products)
  • For exothermic reactions ($\Delta H < 0$), increasing temperature shifts the equilibrium to the left (towards reactants)
• The presence of a catalyst does not affect the equilibrium position, as it accelerates both the forward and reverse reactions equally

Applications in Real-World Systems

• Haber-Bosch process for ammonia synthesis: $\ce{N2(g) + 3H2(g) <=> 2NH3(g)}$
  • High pressure and moderate temperature are used to maximize the yield of ammonia
• Solvay process for sodium carbonate production: $\ce{NaCl(aq) + NH3(aq) + CO2(g) + H2O(l) <=> NaHCO3(s) + NH4Cl(aq)}$
  • Ammonia is used to shift the equilibrium towards the products
• Hemoglobin-oxygen binding in blood: $\ce{Hb(aq) + O2(aq) <=> HbO2(aq)}$
  • The partial pressure of oxygen affects the equilibrium position and the oxygen-carrying capacity of hemoglobin
• Ocean acidification due to increased atmospheric CO2: $\ce{CO2(g) + H2O(l) <=> H2CO3(aq) <=> H+(aq) + HCO3-(aq)}$
  • The increased dissolution of CO2 in ocean water shifts the equilibrium towards the products, lowering the pH
• Buffers in biological systems (e.g., carbonic acid-bicarbonate buffer): $\ce{H2CO3(aq) <=> H+(aq) + HCO3-(aq)}$
  • Buffers minimize pH changes by absorbing or releasing H+ ions when acids or bases are added to the system

Problem-Solving Strategies

• Identify the type of problem (e.g., calculating equilibrium concentrations, determining the direction of shift, or finding the equilibrium constant)
• Write a balanced chemical equation for the reaction and express the equilibrium constant in terms of the appropriate concentration, partial pressure, or activity terms
• If given initial concentrations and one equilibrium concentration, use an ICE table (Initial, Change, Equilibrium) to solve for the remaining equilibrium concentrations
  • Define the change in concentration for each species based on the reaction stoichiometry and the variable $x$
  • Substitute the equilibrium concentrations into the equilibrium constant expression and solve for $x$
• When dealing with gases, use the ideal gas law ($PV = nRT$) to convert between concentration and partial pressure, if necessary
• For problems involving the direction of shift, compare the reaction quotient ($Q$) to the equilibrium constant ($K$)
  • If $Q < K$, the reaction will proceed to the right to reach equilibrium
  • If $Q > K$, the reaction will proceed to the left to reach equilibrium
  • If $Q = K$, the system is already at equilibrium, and no net change will occur
• Apply Le Chatelier's principle to predict the direction of shift when a system at equilibrium is disturbed by changes in concentration, pressure, or temperature
• Remember to consider the assumptions made in the problem (e.g., ideal gas behavior, constant temperature, or constant pressure) and the limitations of the models used