Equilibrium

Calculating K_c from Equilibrium Concentrations

The equilibrium constant, K_c, quantifies the ratio of products to reactants at equilibrium for a reversible reaction. To calculate K_c, the equilibrium concentrations of all reactants and products are substituted into the equilibrium expression. For instance, consider the reaction: CO_(g) + H₂O_(g) ⇌ H_2(g) + CO_2(g).
The equilibrium expression for this reaction is K_c = [H₂][CO₂] / [CO][H₂O]. If the equilibrium concentrations are given as [CO] = 0.150 mol dm^-3, [H₂] = 0.200 mol dm^-3, [H₂O] = 0.0145 mol dm^-3, and [CO₂] = 0.0200 mol dm^-3, then K_c can be calculated by substituting these values:
K_c = (0.200)(0.0200) / (0.150)(0.0145).

Introduction to ICE Tables

ICE tables are a powerful tool used to determine equilibrium constants (K_c) or equilibrium concentrations when not all equilibrium concentrations of reactants and/or products are initially known. The acronym "ICE" stands for Initial, Change, and Equilibrium, representing the concentrations of species at different stages of a reaction. It is crucial to ensure that the precision of any calculated answer aligns with the significant figures provided in the initial data.

Guidelines for Constructing ICE Tables

To effectively use an ICE table, begin by writing a balanced chemical equation for the reaction. Below this equation, create three rows labeled "Initial," "Change," and "Equilibrium."

• Initial: This row represents the concentrations of all reactants and products at the beginning of the reaction, before any significant reaction has occurred. Unless explicitly stated otherwise, it is generally assumed that the initial concentrations of products are zero.
• Change: This row quantifies the change in concentration for each species as the reaction proceeds from its initial state to equilibrium. A minus sign indicates a decrease in concentration (typically for reactants), while a plus sign indicates an increase (typically for products). The changes are determined by the stoichiometry of the balanced chemical equation, often represented by a variable 'x'.
• Equilibrium: This row represents the concentrations of all species once the system has reached equilibrium. The equilibrium concentration for each substance is calculated by adding or subtracting the "Change" from the "Initial" concentration (Equilibrium concentration = Initial concentration ± Change in concentration).

Once the equilibrium concentrations are determined, the equilibrium expression can be written, and K_c can be calculated by substituting these equilibrium values.

Calculating K_c from Initial and Equilibrium Concentrations using an ICE Table

Consider the reaction: PCl_3(g) + Cl_2(g) ⇌ PCl_5(g). If we start with 0.20 mol of PCl₃ and 0.10 mol of Cl₂ in a 1.0 dm³ flask, and at equilibrium, the amount of PCl₃ is 0.12 mol, we can use an ICE table to find K_c. First, convert amounts to concentrations by dividing by the volume (1.0 dm³). So, initial [PCl₃] = 0.20 mol dm^-3, initial [Cl₂] = 0.10 mol dm^-3, and equilibrium [PCl₃] = 0.12 mol dm^-3. The initial concentration of PCl₅ is 0.00 mol dm^-3.

	PCl3(g)	Cl2(g)	PCl5(g)
Initial (mol dm-3)	0.20	0.10	0.00
Change (mol dm-3)	-0.08	-0.08	+0.08
Equilibrium (mol dm-3)	0.12	0.02	0.08

From the equilibrium concentration of PCl₃ (0.12 mol dm^-3) and its initial concentration (0.20 mol dm^-3), the change in [PCl₃] is 0.12 - 0.20 = -0.08 mol dm^-3. Due to the 1:1 stoichiometric ratio between PCl₃ and Cl₂, the change in [Cl₂] is also -0.08 mol dm^-3. Therefore, the equilibrium [Cl₂] = 0.10 - 0.08 = 0.02 mol dm^-3. Similarly, the change in [PCl₅] is +0.08 mol dm^-3, making its equilibrium concentration 0.08 mol dm^-3. Now, K_c can be calculated: K_c = [PCl₅] / ([PCl₃][Cl₂]) = 0.08 / (0.12)(0.02) = 33.

Calculating Equilibrium Concentrations from the Equilibrium Constant

When the equilibrium constant (K_c) is known, it is possible to calculate the equilibrium concentrations of reactants and products. For the reaction CO_(g) + 2H_2(g) ⇌ CH₃OH_(g), if K_c = 0.500 at 350K, and the equilibrium concentrations of CO and H₂ are 0.200 mol dm^-3 and 0.155 mol dm^-3 respectively, we can find the equilibrium concentration of CH₃OH. The equilibrium expression is K_c = [CH₃OH] / ([CO][H₂]²). Substituting the known values: 0.500 = [CH₃OH] / ((0.200)(0.155)²). Solving for [CH₃OH] yields [CH₃OH] = 0.500 * (0.200)(0.155)² = 0.00240 mol dm^-3, or 2.40 x 10^-3 mol dm^-3.

Calculating Equilibrium Concentrations from Initial Concentrations and K_c

In cases where only initial concentrations and K_c are given, an ICE table combined with algebraic manipulation is necessary to determine equilibrium concentrations. Consider the reaction SO_3(g) + NO_(g) ⇌ NO_2(g) + SO_2(g), with K_c = 6.78 at a specified temperature. If the initial concentrations of both NO and SO₃ are 0.0300 mol dm^-3, and initial concentrations of products are zero, we can set up an ICE table. Let 'x' represent the change in concentration.

	SO3(g)	NO(g)	NO2(g)	SO2(g)
Initial (mol dm-3)	0.0300	0.0300	0.00	0.00
Change (mol dm-3)	-x	-x	+x	+x
Equilibrium (mol dm-3)	0.0300-x	0.0300-x	x	x

Since the stoichiometric ratios are 1:1 for all species, the change in concentration for NO and SO₃ is -x, and for NO₂ and SO₂ is +x. The equilibrium expression is K_c = [NO₂][SO₂] / ([SO₃][NO]).
Substituting the equilibrium concentrations: 6.78 = (x)(x) / ((0.0300-x)(0.0300-x)) = x² / (0.0300-x)².
Taking the square root of both sides: √6.78 = x / (0.0300-x), which simplifies to 2.60 = x / (0.0300-x). Solving for x: 2.60(0.0300-x) = x → 0.0780 - 2.60x = x → 0.0780 = 3.60x → x = 0.0217.
Therefore, at equilibrium: [SO₃] = [NO] = 0.0300 - 0.0217 = 0.00830 mol dm^-3, and [NO₂] = [SO₂] = 0.0217 mol dm^-3.

Approximations for Very Small K_c Values

When the equilibrium constant K_c is very small (typically less than 10^-3), it indicates that the reaction proceeds to a very limited extent, meaning very little product is formed at equilibrium. In such cases, the equilibrium mixture consists predominantly of reactants. Consequently, the change in reactant concentrations (represented by 'x' in an ICE table) is negligible compared to their initial concentrations. This allows for a simplifying assumption: the initial reactant concentrations are approximately equal to their equilibrium concentrations. This approximation significantly simplifies the algebra required to solve for 'x'.

Calculating Equilibrium Concentrations with a Very Small K_c

Let's consider the thermal decomposition of water: 2H₂O_(g) ⇌ 2H_2(g) + O_2(g). This reaction has a very small K_c = 7.3 x 10^-18 at 1000°C. If the initial concentration of H₂O is 0.10 mol dm^-3, we can calculate the equilibrium [H₂].

	2H2O(g)	2H2(g)	O2(g)
Initial (mol dm-3)	0.10	0.00	0.00
Change (mol dm-3)	-2x	+2x	+x
Equilibrium (mol dm-3)	0.10-2x	2x	x

The equilibrium expression is K_c = [H₂]²[O₂] / [H₂O]². Substituting the equilibrium concentrations:
7.3 x 10^-18 = (2x)²(x) / (0.10-2x)².
Since K_c is very small, we can assume that 0.10 - 2x ≈ 0.10.
This simplifies the equation to: 7.3 x 10^-18 = (4x²)(x) / (0.10)² = 4x³ / 0.01.
Rearranging to solve for x: 4x³ = (7.3 x 10^-18)(0.10)² = 7.3 x 10^-20.
Therefore, x³ = 7.3 x 10^-20 / 4 = 1.825 x 10^-20.
Taking the cube root, x = 2.632 x 10^-7.
Finally, the equilibrium concentration of H₂ is 2x = 2(2.632 x 10^-7) = 5.3 x 10^-7 mol dm^-3.

Free Energy and Equilibrium: Driving Reaction Direction

The extent to which a reaction proceeds towards products or reactants at equilibrium is fundamentally linked to the Gibbs free energy, denoted as ΔG. This thermodynamic quantity represents the maximum amount of work available from a system at constant temperature and pressure. A larger amount of available Gibbs free energy indicates a greater tendency for the reaction to proceed in the forward direction, favoring product formation. As a reaction progresses, the free energy of the reactants decreases while the free energy of the products increases. Equilibrium is achieved when the total free energies of the reactants and products become equal, meaning ΔG_products = ΔG_reactants. At this point, the system is in its lowest possible energy state. A more negative value of ΔG signifies more work available for the reaction, indicating a greater driving force towards product formation.

Understanding Gibbs Free Energy and Spontaneity

Gibbs free energy change (ΔG) for a reaction is defined as the difference between the free energy of the products and the free energy of the reactants (ΔG = ΔG_products - ΔG_reactants). A negative value for ΔG (ΔG < 0) indicates that a reaction will proceed spontaneously in the forward direction. Conversely, a positive value for ΔG (ΔG > 0) signifies that the forward reaction is non-spontaneous, meaning the reverse reaction is spontaneous.

Summary of Reaction Direction and Spontaneity Based on ΔG

The relationship between the sign of ΔG, the spontaneity of a reaction, and the composition of the equilibrium mixture can be summarized as follows:

Reaction Direction	Spontaneous or Non-spontaneous	Equilibrium mixture
negative	forward direction	spontaneous	mostly products
positive	reverse direction	non-spontaneous	mostly reactants
zero	equilibrium	-	-

Calculating the Equilibrium Constant (K_c) from Thermodynamic Data

The equilibrium constant (K_c) is directly related to the Gibbs free energy change (ΔG) through the following fundamental thermodynamic equation: ΔG = -RTlnK. In this equation, ΔG represents the standard free energy change of the reaction, R is the ideal gas constant (8.31 J K^-1 mol^-1), T is the absolute temperature in Kelvin, and lnK is the natural logarithm of the equilibrium constant (K_c). This equation allows for the calculation of the equilibrium constant from thermodynamic data, or vice versa.

Relationship Between ΔG and K_c: Predicting Equilibrium Composition

The relationship ΔG = -RTlnK provides a powerful tool for predicting the composition of a reaction mixture at equilibrium. The sign and magnitude of ΔG directly correlate with the value of K and, consequently, the relative amounts of reactants and products at equilibrium.

ΔG	lnK	K	Equilibrium mixture
negative	positive	K>1	Mainly products
positive	negative	K<1	Mainly reactants
zero	zero	K=1	Appreciable amounts of both reactants and products

Practical Applications of Free Energy Calculations

The equation ΔG = -RTlnK is particularly useful in situations where the equilibrium constant cannot be accurately measured experimentally. This can occur if a reaction is too slow to reach equilibrium within a reasonable timeframe, or if the amounts of reactants and products at equilibrium are too small to be measured directly. For example, consider the esterification reaction that produces ethyl ethanoate: CH₃COOH (aq) + C₂H₅OH (aq) ⇌ CH₃COOC₂H₅(aq) + H₂O(aq)
 If the free energy change for this reaction (ΔG) is -4.38 kJ mol^-1, we can calculate the value of the equilibrium constant (K_c) at 298K using the equation ΔG = -RTlnK.

Introduction to Kinetics and Equilibrium

Chemical kinetics is the study of reaction rates, investigating how fast a reaction proceeds and the factors that influence this speed. In contrast, chemical equilibrium describes the state where the rates of the forward and reverse reactions are equal, leading to no net change in reactant and product concentrations. These two fundamental concepts are distinct yet interconnected in understanding chemical processes.

Distinguishing Kinetics from Equilibrium

It is crucial to understand that the magnitude of the equilibrium constant provides no information about the rate at which a reaction occurs. For instance, consider the rusting of iron, a reaction with a highly favorable Gibbs free energy change (ΔG = -1490 x 10⁶ J) and an exceptionally large equilibrium constant (K_c = 10²⁶¹). The large K_c indicates that the reaction is thermodynamically favorable and will proceed extensively towards product formation, while the large negative ΔG confirms its spontaneity. However, despite these thermodynamic indicators, rusting is a notoriously slow process that can take years to complete, demonstrating that a thermodynamically favorable reaction is not necessarily a fast one.

The Relationship Between Rate and Equilibrium Constants

The rate constants of the forward and reverse reactions offer insights into the nature of equilibrium. For a reversible reaction such as A + B ⇌ C + D, the rate of the forward reaction can be expressed as
Rate_forward = k[A][B], where k is the forward rate constant.
Similarly, the rate of the reverse reaction is
Rate_reverse = k'[C][D], with k' being the reverse rate constant.
 At equilibrium, the rates of the forward and reverse reactions are equal: k[A][B] = k'[C][D]
 Rearranging this equation allows us to establish a direct relationship between the rate constants and the equilibrium constant, K_c: k/{k' = [C][D]/[A][B]
Since K_c is defined as [C][D]/[A][B], it follows that: 
This relationship reveals that if the forward rate constant (k) is significantly larger than the reverse rate constant (k'), then K_c will be large, indicating that the reaction strongly favors product formation at equilibrium. Conversely, if k' is much larger than k, K_c will be small, meaning reactants are favored at equilibrium.