Question

For a typical equilibrium problem, the value of \(K\) and the initial reaction conditions are given for a specific reaction, and you are asked to calculate the equilibrium concentrations. Many of these calculations involve solving a quadratic or cubic equation. What can you do to avoid solving a quadratic or cubic equation and still come up with reasonable equilibrium concentrations?

Short answer

To calculate equilibrium concentrations without solving a quadratic or cubic equation, apply the 5% approximation method. First, set up an ICE table based on the initial reaction conditions. Write the equilibrium constant expression and apply the 5% approximation by assuming the change in concentration, x, is small. This simplifies the expression. Check if the 5% approximation is valid by ensuring x is less than 5% of the smallest initial concentration. If valid, solve for x and calculate the equilibrium concentrations using the ICE table values.

Step-by-step solution

Understand the equilibrium constant (K) and the reaction

The equilibrium constant (K) is a measure of how far a reaction goes to completion. This constant, in combination with initial reaction conditions, helps to determine the final concentrations of reactants and products.

Write the reaction and set up an ICE table

Write the balanced chemical equation for the reaction and set up an Initial, Change, and Equilibrium (I.C.E.) table using the given initial reaction conditions. Example reaction: \(aA + bB \rightleftharpoons cC + dD\) ICE table: | | A | B | C | D | |---------|---------|---------|---------|-----------| | Initial | [A]₀ | [B]₀ | [C]₀ | [D]₀ | | Change | -ax | -bx | +cx | +dx | | Equilibrium | [A]₀-ax | [B]₀-bx | [C]₀+cx | [D]₀+dx |

Write the equilibrium expression and apply 5% approximation

Express the equilibrium constant in terms of the concentrations of the reactants and products at equilibrium using the balanced chemical equation. Apply the 5% approximation, by assuming that the change, x, is small relative to the initial concentrations of the reactants. Example: Equilibrium expression: \(K = \frac{([C]^c \cdot [D]^d)}{([A]^a \cdot [B]^b)}\) Using the 5% approximation: \(K \approx \frac{([C]_0 + cx)^c \cdot ([D]_0 + dx)^d}{([A]_0 - ax)^a \cdot ([B]_0 - bx)^b}\)

Determine the conditions for applying 5% approximation

In order for the 5% approximation to be accurate, the value of x should be less than or equal to 5% of the smallest initial concentration. Once the equilibrium concentrations have been found, double-check if the 5% approximation is valid. If x is more than 5% of the smallest concentration, the approximation should not be used, and the quadratic or cubic equation must be solved. Example: For 5% approximation to be accurate: \(x \leq 0.05 \cdot \min ([A]_0, [B]_0, [C]_0, [D]_0)\)

Calculate equilibrium concentrations

Using the 5% approximation in the K expression, solve for x and then calculate the equilibrium concentrations of the reactants and products using the ice table. Example: 1. Solve for x in the K expression with the 5% approximation. 2. Find equilibrium concentrations: [A] = [A]₀-ax, [B] = [B]₀-bx, [C] = [C]₀+cx, and [D] = [D]₀+dx By following these steps and applying the 5% approximation, we can calculate the equilibrium concentrations without having to solve a quadratic or cubic equation for most typical equilibrium problems.

Key concepts

Chemical Equilibrium

Chemical equilibrium is a state in a chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction. This balance means that the concentration of the reactants and products will remain constant over time, not necessarily equal.

At equilibrium, even though the reactants and products remain in a constant ratio, the reaction hasn't necessarily stopped; rather, it continually occurs in both directions at an equal rate. Recognizing the dynamic nature of equilibrium is crucial; molecules are always in motion, and reactions are proceeding, just that there is no net change in concentration of the substances involved.

It's important to remember that equilibrium can only be achieved in a closed system where the reactants and products cannot escape. Environmental changes like temperature, pressure, and the presence of catalysts can also shift the position of the equilibrium, changing the concentrations of reactants and products accordingly.

Equilibrium Constant (K)

The equilibrium constant, represented by the symbol K, quantifies the ratio of the concentrations of products to reactants at chemical equilibrium. The value of K is specific to a particular reaction at a given temperature. If K is much greater than 1, the products are favored; if K is much less than 1, the reactants are favored.

For a balanced chemical equation, the equilibrium constant expression is written using the concentrations (or partial pressures for gases) of the reactants and products raised to the power of their stoichiometric coefficients. This relation provides a key insight into the quantitative behavior of equilibrium systems and allows for predictions about how the system will respond to changes in conditions.

ICE Table

An ICE table — which stands for Initial, Change, and Equilibrium — is a helpful tool for organizing and calculating the concentrations of reactants and products at equilibrium. It starts with the initial concentrations of reactants and products, often with products being zero for a completely reactant-started reaction.

The 'Change' row is used to represent the changes in concentration as the system moves towards equilibrium, expressed in terms of a variable, usually 'x', corresponding to the stoichiometry of the reaction. The 'Equilibrium' row then combines the 'Initial' and 'Change' rows to express the final concentrations at equilibrium. This graphical organizer simplifies the process of calculating the equilibrium state, especially for complex reactions.

5% Approximation

The 5% approximation is a mathematical simplification used when the change in concentration of reactants or products (x) is very small relative to the initial concentrations. If x is less than 5% of the initial concentration, the approximation can be made that the concentration has not changed significantly.

This assumption allows us to simplify the algebra involved in solving equilibrium concentration problems. When applying the 5% approximation, it's important to check after calculating the equilibrium concentrations that the approximation was valid—that x is indeed less than or equal to 5% of the smallest initial concentration. If the approximation holds, it can save a lot of time, avoiding the complexity of solving quadratic or cubic equations.