# Lecture 4: Equilibrium Calculations and Le Chatelier's Principle

## General Chemistry II

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## Learning Objectives

By the end of this lecture, students will be able to:

1. Perform ICE table calculations to determine equilibrium concentrations
2. Use the small-x approximation and determine when it is valid
3. State Le Chatelier's principle and predict the direction of equilibrium shifts
4. Analyze the effects of changes in concentration, pressure, volume, and temperature on equilibrium
5. Explain why a catalyst does not shift equilibrium
6. Apply Le Chatelier's principle to industrial processes

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## Lecture Content

### I. The ICE Table Method

ICE stands for Initial, Change, Equilibrium, and the ICE table is a systematic framework for solving equilibrium problems. To use this method, begin by writing the balanced equation and the corresponding equilibrium expression. Set up the ICE table beneath the balanced equation, entering known initial concentrations in the I row. Express the changes in concentration in terms of a variable x in the C row, using stoichiometric ratios to relate the changes for each species. Products increase by a positive multiple of x, while reactants decrease by a negative multiple of x. The equilibrium concentrations in the E row are then the sum of the initial concentration and the change. Substitute these equilibrium expressions into the K expression, solve for x, and calculate the final equilibrium concentrations.

### II. Solving Equilibrium Problems: Examples

Equilibrium problems generally fall into a few common types. In the most frequent type, you are given K and initial concentrations and asked to find the equilibrium concentrations. After setting up the ICE table and substituting into the K expression, you may need to solve a quadratic equation of the form ax^2 + bx + c = 0 using the quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / (2a). Always select the root that gives physically meaningful, positive concentrations.

In a second type of problem, you are given equilibrium concentrations and asked to find K. This requires simply substituting the equilibrium values directly into the K expression. A third type provides K along with one equilibrium concentration and asks you to find the remaining concentrations using the K expression and stoichiometric relationships.

### III. The Small-x Approximation

When K is very small (roughly K < 10^-3) and the initial concentrations are not too small, the change x will be negligible compared to the initial concentrations. In this situation, the approximation [A]_0 - x is approximately equal to [A]_0 dramatically simplifies the algebra by eliminating the need to solve a quadratic equation.

The validity of this approximation is checked using the 5% rule: if x / [A]_0 < 0.05, the approximation is acceptable. If x / [A]_0 exceeds 5%, you must solve the equation exactly using the quadratic formula. When the approximation is close to the borderline, the approximate value of x can serve as a starting point for successive approximation (iteration) to converge on the exact answer.

<image>A side-by-side comparison of solving an equilibrium problem with and without the small-x approximation. Left panel: Full ICE table setup for N2O4 <=> 2NO2 with K = 4.6 x 10^-3 and initial [N2O4] = 0.100 M. The quadratic equation is shown and solved exactly. Right panel: The same problem using the approximation 0.100 - x approximately equals 0.100, showing the simplified algebra. Both panels arrive at the same answer (within 5%), and the validity check (x/0.100 = 2.1% < 5%) is highlighted with a checkmark.</image>

### IV. Le Chatelier's Principle

Le Chatelier's principle states that when a system at equilibrium is subjected to a stress, the system will shift in the direction that partially counteracts that stress and re-establishes equilibrium. The key word is "partially": the system does not fully undo the stress, but only offsets it to some degree. The types of stress that can disturb an equilibrium include changes in concentration, changes in pressure or volume, and changes in temperature.

### V. Effect of Concentration Changes

Adding a reactant or removing a product causes Q to become less than K, so the system shifts to the right, producing more products. Conversely, removing a reactant or adding a product makes Q greater than K, causing the system to shift to the left, regenerating reactants. Crucially, altering concentrations does not change the value of K itself, because K depends only on temperature. Adding an inert gas at constant volume has no effect on the equilibrium because the partial pressures of the reacting gases remain unchanged.

### VI. Effect of Pressure and Volume Changes

Pressure and volume changes affect gaseous equilibria in which the total number of moles of gas differs between the two sides of the equation (Delta n(gas) is not zero). Decreasing the volume, which increases the pressure, causes the system to shift toward the side with fewer moles of gas. Increasing the volume, which decreases the pressure, shifts the equilibrium toward the side with more moles of gas. If Delta n(gas) = 0, changes in pressure or volume have no effect on the equilibrium position.

Adding an inert gas at constant volume does not affect the equilibrium because it does not change the partial pressures of the reacting species. However, adding an inert gas at constant pressure causes the total volume to increase, effectively decreasing the partial pressures of all gases, and the system shifts toward the side with more moles of gas.

<image>A diagram illustrating Le Chatelier's principle for the reaction N2(g) + 3H2(g) <=> 2NH3(g). Three scenarios are shown as before/after piston-cylinder setups. Panel A: Volume is decreased (piston pushed down), arrow shows equilibrium shifts right (toward 2 moles of gas from 4 moles of gas), resulting in increased [NH3]. Panel B: H2 is added to the system, arrow shows equilibrium shifts right to consume the added H2. Panel C: Temperature is increased (exothermic forward reaction), arrow shows equilibrium shifts left, producing more N2 and H2. Each panel labels Delta n(gas) or Delta H as appropriate.</image>

### VII. Effect of Temperature Changes

Temperature changes are unique among equilibrium perturbations because they actually alter the value of K, unlike concentration or pressure changes, which leave K unchanged. For exothermic reactions (Delta H < 0), increasing the temperature shifts the equilibrium to the left and K decreases, while decreasing the temperature shifts it to the right and K increases. A helpful way to think about this is to treat heat as a product: A + B <=> C + D + heat.

For endothermic reactions (Delta H > 0), the opposite is true. Increasing the temperature shifts the equilibrium to the right and K increases, while decreasing the temperature shifts it to the left and K decreases. Here, heat functions as a reactant: heat + A + B <=> C + D. The van't Hoff equation, ln(K2/K1) = -(Delta H^0/R)(1/T2 - 1/T1), quantifies this temperature dependence.

### VIII. Effect of a Catalyst

A catalyst does not shift the position of equilibrium. Because a catalyst accelerates both the forward and reverse reactions equally, the system reaches equilibrium faster, but the equilibrium concentrations remain the same as they would be without the catalyst. The value of K is unchanged by a catalyst.

### IX. Applications: The Haber Process

The industrial synthesis of ammonia, N2(g) + 3H2(g) <=> 2NH3(g) with Delta H = -92 kJ/mol, beautifully illustrates the application of Le Chatelier's principle to optimize a real chemical process. High pressure (200-300 atm) is used because it favors the product side, which has fewer moles of gas (2 versus 4). The temperature is set at a moderate 400-500 C as a compromise between thermodynamic and kinetic considerations: lower temperatures favor ammonia production thermodynamically but slow the reaction kinetically to impractical rates. An iron catalyst speeds the approach to equilibrium without altering its position. Finally, ammonia is continuously removed as it forms, shifting the equilibrium to the right and maximizing overall yield.

<image>A flow diagram of the Haber process showing a reactor vessel with N2 and H2 entering, an iron catalyst bed inside, and a condenser that removes liquid NH3. Unreacted N2 and H2 are recycled back to the reactor. Annotations indicate the conditions: T = 400-500 degrees C, P = 200-300 atm. Side notes explain the Le Chatelier reasoning for each condition choice: high pressure favors fewer moles (products), moderate temperature is a kinetic-thermodynamic compromise, and continuous removal of NH3 shifts equilibrium right.</image>

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