Chemical Equilibrium

Equilibrium In Chemical Processes – Dynamic Equilibrium

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Chemical Equilibrium: Chemical equilibrium is a state in a reversible reaction where the rate of the forward reaction is equal to the rate of the reverse reaction. At this point, the net concentrations of reactants and products remain constant, even though the forward and reverse reactions are still occurring.

Reversible Reactions: Reactions that can proceed in both the forward (reactants to products) and reverse (products to reactants) directions are called reversible reactions. They are often represented by double arrows ($\rightleftharpoons$).

Dynamic Equilibrium: Chemical equilibrium is a dynamic process. This means that the reactions have not stopped; rather, the forward and reverse reactions are occurring at exactly the same rate. Imagine a busy store where customers enter and leave at the same rate – the number of people inside remains constant, but the individuals are changing.

• Forward Reaction Rate = Reverse Reaction Rate

Characteristics of Equilibrium:

• Constant Macroscopic Properties: Observable properties like concentration, pressure, color, and density remain constant.
• Achieved from Either Direction: Equilibrium can be reached starting with only reactants, only products, or any initial mixture.
• Requires a Closed System: For equilibrium to be established and maintained, the system must be closed, meaning no matter can enter or leave.
• Dynamic Nature: Both forward and reverse reactions continue at the molecular level.

Law Of Chemical Equilibrium And Equilibrium Constant

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Law of Chemical Equilibrium: For a general reversible reaction occurring at a given temperature:

$$aA + bB \rightleftharpoons cC + dD$$

The law of chemical equilibrium states that at equilibrium, the ratio of the product of the concentrations (or partial pressures) of the products raised to their stoichiometric coefficients to the product of the concentrations (or partial pressures) of the reactants raised to their stoichiometric coefficients is a constant. This constant is called the equilibrium constant.

Equilibrium Constant ($K$):

1. Equilibrium Constant in Terms of Molar Concentrations ($K_c$):

For the general reaction:

$$aA + bB \rightleftharpoons cC + dD$$

The equilibrium constant expression in terms of molar concentrations is:

$$K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b}$$

Where $[X]$ denotes the molar concentration of species X at equilibrium.

Units of $K_c$: The units of $K_c$ depend on the stoichiometry of the reaction (specifically, the difference between the sum of product coefficients and the sum of reactant coefficients).

2. Equilibrium Constant in Terms of Partial Pressures ($K_p$):

This is applicable for reactions involving gases. For the same general reaction:

$$aA(g) + bB(g) \rightleftharpoons cC(g) + dD(g)$$

The equilibrium constant expression in terms of partial pressures is:

$$K_p = \frac{(P_C)^c (P_D)^d}{(P_A)^a (P_B)^b}$$

Where $P_X$ denotes the partial pressure of gaseous species X at equilibrium.

Units of $K_p$: The units of $K_p$ also depend on the stoichiometry.

Relationship Between $K_c$ and $K_p$:

For reactions involving gases, we can relate $K_p$ and $K_c$ using the ideal gas law ($PV=nRT \Rightarrow P = \frac{n}{V}RT = [ ]RT$).

For the reaction $aA(g) + bB(g) \rightleftharpoons cC(g) + dD(g)$:

$$K_p = \frac{(P_C)^c (P_D)^d}{(P_A)^a (P_B)^b} = \frac{([C]RT)^c ([D]RT)^d}{([A]RT)^a ([B]RT)^b}$$ $$K_p = \frac{[C]^c [D]^d}{[A]^a [B]^b} \times \frac{(RT)^{c+d}}{(RT)^{a+b}}$$ $$K_p = K_c (RT)^{(c+d) - (a+b)}$$

Let $\Delta n_g = (c+d) - (a+b)$, the change in the number of moles of gas.

$$K_p = K_c (RT)^{\Delta n_g}$$

Characteristics of the Equilibrium Constant:

• Temperature Dependent: $K$ is constant for a given reaction only at a specific temperature. If the temperature changes, $K$ changes.
• Independent of Initial Concentrations: The value of $K$ is the same regardless of the starting concentrations of reactants and products, as long as equilibrium is reached.
• Independent of Stoichiometric Coefficients (for a given reaction): While the expression for $K$ uses coefficients, changing the overall balanced equation (e.g., doubling it) changes the value of $K$ (it becomes $K^2$).
• Units: $K_c$ and $K_p$ can have units, but often they are treated as dimensionless, especially when activities are used instead of concentrations/pressures.

Relationship Between Equilibrium Constant K, Reaction Quotient Q And Gibbs Energy G

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Reaction Quotient ($Q$): The reaction quotient ($Q$) has the same mathematical form as the equilibrium constant ($K$), but it is calculated using the concentrations or partial pressures of reactants and products at any point in time during the reaction, not just at equilibrium.

For the reaction $aA + bB \rightleftharpoons cC + dD$:

$$Q_c = \frac{[C]^c [D]^d}{[A]^a [B]^b} \quad \text{or} \quad Q_p = \frac{(P_C)^c (P_D)^d}{(P_A)^a (P_B)^b}$$

Comparing $Q$ and $K$: The comparison of $Q$ with $K$ indicates the direction in which the reaction must proceed to reach equilibrium:

• If $Q < K$: The ratio of products to reactants is too low. The reaction must proceed in the forward direction (towards products) to reach equilibrium.
• If $Q > K$: The ratio of products to reactants is too high. The reaction must proceed in the reverse direction (towards reactants) to reach equilibrium.
• If $Q = K$: The system is already at equilibrium, and there is no net change.

Relationship Between $\Delta G$, $\Delta G^\circ$, $Q$, and $K$:

The change in Gibbs free energy ($\Delta G$) under any conditions (not necessarily equilibrium) is related to the standard Gibbs free energy change ($\Delta G^\circ$) and the reaction quotient ($Q$) by the equation:

$$\Delta G = \Delta G^\circ + RT \ln Q$$

Connecting to Equilibrium:

At equilibrium, $\Delta G = 0$ and $Q = K$. Substituting these into the above equation yields:

$$0 = \Delta G^\circ + RT \ln K$$

Rearranging gives the fundamental relationship:

$$\Delta G^\circ = -RT \ln K$$

This equation is central to chemical thermodynamics and equilibrium, as it links the thermodynamic driving force for a reaction under standard conditions ($\Delta G^\circ$) to the position of equilibrium ($K$).

Interpretation:

• $\Delta G^\circ < 0$ (spontaneous under standard conditions): $\ln K > 0$, so $K > 1$. Equilibrium favors products.
• $\Delta G^\circ > 0$ (non-spontaneous under standard conditions): $\ln K < 0$, so $K < 1$. Equilibrium favors reactants.
• $\Delta G^\circ = 0$ (at equilibrium under standard conditions): $\ln K = 0$, so $K = 1$. Reactants and products are equally favored at equilibrium.

Homogeneous Equilibria

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Homogeneous Equilibrium: A homogeneous equilibrium is an equilibrium in a system where all reactants and products are in the same phase.

Examples:

• Gas Phase: $N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)$
• Solution Phase: $CH_3COOH(aq) + H_2O(l) \rightleftharpoons H_3O^+(aq) + CH_3COO^-(aq)$

Equilibrium Constant In Gaseous Systems

Use of $K_p$ and $K_c$: For gaseous homogeneous equilibria, we can express the equilibrium constant using either partial pressures ($K_p$) or molar concentrations ($K_c$).

$K_c$ Expression: Based on molar concentrations.

$K_p$ Expression: Based on partial pressures.

Relationship between $K_p$ and $K_c$:

$$K_p = K_c (RT)^{\Delta n_g}$$

Where $\Delta n_g = (\text{moles of gaseous products}) - (\text{moles of gaseous reactants})$.

• If $\Delta n_g = 0$ (e.g., $H_2(g) + I_2(g) \rightleftharpoons 2HI(g)$), then $K_p = K_c$.
• If $\Delta n_g > 0$ (e.g., $N_2(g) + O_2(g) \rightleftharpoons 2NO(g)$), then $K_p > K_c$.
• If $\Delta n_g < 0$ (e.g., $N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)$), then $K_p < K_c$.

Example: For the reaction $N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)$, if $K_c = 6.0 \times 10^{-2}$ at 500°C (773 K), calculate $K_p$. Assume $R = 0.0821 \text{ L atm/mol K}$.

Answer:

Heterogeneous Equilibria

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Heterogeneous Equilibrium: A heterogeneous equilibrium is an equilibrium in a system where reactants and products are in different phases.

Examples:

• Solid and Gas: $CaCO_3(s) \rightleftharpoons CaO(s) + CO_2(g)$
• Liquid and Gas: $H_2O(l) \rightleftharpoons H_2O(g)$
• Solid and Aqueous: $AgCl(s) \rightleftharpoons Ag^+(aq) + Cl^-(aq)$

Equilibrium Constant Expressions for Heterogeneous Equilibria:

The concentrations or partial pressures of pure solids and pure liquids are considered constant and are therefore omitted from the equilibrium constant expression. Only species in the gaseous or aqueous phases are included.

Examples of $K_c$ Expressions:

• For $CaCO_3(s) \rightleftharpoons CaO(s) + CO_2(g)$:
$$K_c = [CO_2]$$

(Concentrations of $CaCO_3(s)$ and $CaO(s)$ are omitted).

• For $H_2O(l) \rightleftharpoons H_2O(g)$:
$$K_c = [H_2O(g)] \quad \text{(This is essentially the vapor pressure}$$ $$K_p = P_{H_2O(g)}) \quad \text{in terms of partial pressure}$$
• For $AgCl(s) \rightleftharpoons Ag^+(aq) + Cl^-(aq)$:
$$K_c = [Ag^+][Cl^-]$$

(Concentration of $AgCl(s)$ is omitted).

Implications: The position of a heterogeneous equilibrium is not affected by the amounts of pure solids or liquids present, as long as some of each phase is present.

Applications Of Equilibrium Constants

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Equilibrium constants ($K$) are fundamental in predicting and understanding chemical reactions. They quantify the extent and direction of a reaction.

Predicting The Extent Of A Reaction

The magnitude of $K$ indicates how far a reaction proceeds towards products at equilibrium:

• Large $K$ (e.g., $K >> 1$): The equilibrium lies far to the right. Products are strongly favored over reactants. The reaction proceeds nearly to completion.
• Small $K$ (e.g., $K << 1$): The equilibrium lies far to the left. Reactants are strongly favored over products. The reaction hardly proceeds in the forward direction.
• $K \approx 1$: The equilibrium lies in the middle. Significant amounts of both reactants and products are present at equilibrium.

Predicting The Direction Of The Reaction

By comparing the reaction quotient ($Q$) with the equilibrium constant ($K$), we can determine the direction in which a reaction will proceed to reach equilibrium:

• If $Q < K$: The ratio of products to reactants is less than it would be at equilibrium. The reaction will proceed in the forward direction (towards products) to reach equilibrium.
• If $Q > K$: The ratio of products to reactants is greater than it would be at equilibrium. The reaction will proceed in the reverse direction (towards reactants) to reach equilibrium.
• If $Q = K$: The system is at equilibrium, and there will be no net change.

Example: For the reaction $N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)$, $K_c = 6.0 \times 10^{-2}$ at 500°C. If a mixture has $[NH_3]=0.1$ M, $[N_2]=1.0$ M, and $[H_2]=1.0$ M, in which direction will the reaction proceed?

Answer:

Calculating Equilibrium Concentrations

If the equilibrium constant ($K$) and the initial concentrations of reactants and products are known, we can calculate the concentrations of all species at equilibrium.

Steps:

1. Write the balanced chemical equation for the reaction.
2. Write the expression for the equilibrium constant ($K_c$ or $K_p$).
3. Calculate the reaction quotient ($Q$) using the initial concentrations/pressures. Compare $Q$ with $K$ to determine the direction of reaction.
4. Set up an ICE (Initial, Change, Equilibrium) table:
   • I (Initial): List the initial concentrations or partial pressures of all reactants and products.
   • C (Change): Define the change in concentration/pressure in terms of a variable (e.g., $x$). If the reaction proceeds forward, reactants decrease by their stoichiometric coefficients times $x$, and products increase by their stoichiometric coefficients times $x$.
   • E (Equilibrium): The equilibrium concentrations/pressures are the sum of the initial and change values.
5. Substitute the equilibrium concentrations/pressures into the equilibrium constant expression.
6. Solve the resulting algebraic equation for $x$.
7. Calculate the equilibrium concentrations of all species by substituting the value of $x$.

Example: For the synthesis of ammonia, $N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)$, $K_c = 0.590$ at 400°C. If initial concentrations are $[N_2]_0 = 1.000$ M, $[H_2]_0 = 2.000$ M, and $[NH_3]_0 = 0.500$ M, calculate the equilibrium concentrations.

Answer:

Factors Affecting Equilibria

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The equilibrium constant ($K$) is only affected by temperature. However, the position of equilibrium (i.e., the actual equilibrium concentrations or partial pressures) can be shifted by changes in concentration, pressure, or temperature. These effects are described by Le Chatelier's Principle.

Le Chatelier's Principle: If a change of condition (e.g., concentration, temperature, pressure) is applied to a system in equilibrium, the system will shift in a direction that relieves the stress.

Effect Of Concentration Change

Principle: If the concentration of a reactant or product is changed, the system will shift to counteract this change.

• Adding Reactant: The equilibrium shifts to the right (towards products) to consume the added reactant.
• Removing Reactant: The equilibrium shifts to the left (towards reactants) to replenish the removed reactant.
• Adding Product: The equilibrium shifts to the left (towards reactants) to consume the added product.
• Removing Product: The equilibrium shifts to the right (towards products) to replenish the removed product.

Effect on $K$: Changes in concentration do not affect the value of the equilibrium constant $K$. They only shift the position of equilibrium.

Effect Of Pressure Change

Pressure changes primarily affect equilibria involving gases, especially when there is a change in the total number of moles of gas in the reaction.

Principle: If the pressure of a system at equilibrium is changed, the system will shift to counteract this change. This is achieved by shifting in the direction that produces fewer moles of gas (if pressure is increased) or more moles of gas (if pressure is decreased).

• Increasing Pressure (by decreasing volume): The equilibrium shifts towards the side with fewer moles of gas.
• Decreasing Pressure (by increasing volume): The equilibrium shifts towards the side with more moles of gas.

No Change: If the number of moles of gas is the same on both sides of the equation (i.e., $\Delta n_g = 0$), pressure changes have no effect on the position of equilibrium. For example, $H_2(g) + I_2(g) \rightleftharpoons 2HI(g)$ ($\Delta n_g = 0$).

Effect on $K$: Pressure changes do not affect the value of the equilibrium constant $K$ (for gaseous systems, $K_p$ remains constant if temperature is constant, and $K_c$ might change slightly if the volume change alters partial pressures non-linearly, but the primary effect is on the position). For the relationship $K_p = K_c(RT)^{\Delta n_g}$, if $\Delta n_g \neq 0$, changes in $T$ can affect the ratio $K_p/K_c$, but the effect of pressure is directly on the partial pressures themselves, shifting the equilibrium.

Effect Of Inert Gas Addition

At Constant Volume: If an inert gas (a gas that does not participate in the reaction) is added to a system at equilibrium at constant volume, the partial pressures of the reacting gases remain unchanged. Therefore, the equilibrium position is unaffected.

At Constant Pressure: If an inert gas is added to a system at equilibrium at constant pressure, the total volume of the system must increase to maintain the constant pressure. This causes a decrease in the partial pressures of all reacting gases. The system will shift towards the side with more moles of gas to counteract this decrease in partial pressures, thus shifting the equilibrium position.

Effect on $K$: The addition of an inert gas does not affect the value of the equilibrium constant $K$.

Effect Of Temperature Change

Principle: Temperature changes affect the equilibrium constant $K$ itself.

• Endothermic Reaction ($\Delta H > 0$): An increase in temperature favors the endothermic direction (forward reaction). This means $K$ increases with increasing temperature. Heat can be thought of as a "reactant" in an endothermic reaction: $Reactants + Heat \rightleftharpoons Products$.
• Exothermic Reaction ($\Delta H < 0$): An increase in temperature favors the exothermic direction (reverse reaction). This means $K$ decreases with increasing temperature. Heat can be thought of as a "product" in an exothermic reaction: $Reactants \rightleftharpoons Products + Heat$.

Van't Hoff Equation: This equation quantifies the temperature dependence of $K$:

$$\ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H^\circ}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)$$

Where $K_1$ is the equilibrium constant at temperature $T_1$, and $K_2$ is the equilibrium constant at temperature $T_2$.

Effect Of A Catalyst

Definition: A catalyst is a substance that increases the rate of a chemical reaction by providing an alternative reaction pathway with a lower activation energy.

Effect on Equilibrium:

• A catalyst increases the rate of both the forward and reverse reactions equally.
• Therefore, a catalyst helps the system reach equilibrium faster.
• However, a catalyst does not change the position of equilibrium or the value of the equilibrium constant ($K$). It does not alter the equilibrium concentrations of reactants and products.