Question

Consider the decomposition of ammonium chloride at a certain temperature: $$\mathrm{NH}_{4} \mathrm{Cl}(s) \rightleftarrows \mathrm{NH}_{3}(g)+\mathrm{HCl}(g)$$ Calculate the equilibrium constant \(K_{P}\) if the total pressure is \(2.2 \mathrm{~atm}\) at that temperature.

Short answer

The equilibrium constant \(K_P\) is 1.21.

Step-by-step solution

Identify the Reaction

The decomposition reaction involves solid ammonium chloride separating into ammonia and hydrogen chloride gases: \(\mathrm{NH}_{4} \mathrm{Cl}(s) \rightleftharpoons \mathrm{NH}_{3}(g) + \mathrm{HCl}(g)\).

Express Partial Pressures

Assume the partial pressure of \(\mathrm{NH}_3\) is \(P\) and the partial pressure of \(\mathrm{HCl}\) is also \(P\), since they are produced in a 1:1 molar ratio. Thus, the total pressure \(P_{\text{total}} = P_{\mathrm{NH}_3} + P_{\mathrm{HCl}} = 2P\).

Calculate Individual Partial Pressures

Given the total pressure is \(2.2\,\mathrm{atm}\), and \(P_{\text{total}} = 2P\), solve for \(P\): \(2P = 2.2\,\mathrm{atm}\), so \(P = 1.1\,\mathrm{atm}\).

Write the Expression for \(K_P\)

The equilibrium constant \(K_P\) is expressed as: \(K_P = P_{\mathrm{NH}_3} \cdot P_{\mathrm{HCl}} = (1.1)\times(1.1)\).

Solve for \(K_P\)

Calculate \(K_P\): \(K_P = 1.1^2 = 1.21\).

Key concepts

Equilibrium Constant (Kp)

The equilibrium constant, denoted as \( K_p \), is a crucial concept in chemical equilibrium involving gases. It relates to the balance of products and reactants at a specific temperature when a reaction reaches equilibrium in terms of partial pressures.
In the case of the decomposition of ammonium chloride, the expression for \( K_p \) involves the partial pressures of the gaseous products, ammonia \( (\text{NH}_3) \) and hydrogen chloride \( (\text{HCl}) \).
The expression can be written as:

• \( K_p = P_{\text{NH}_3} \cdot P_{\text{HCl}} \).

At equilibrium, \( K_p \) is only affected by temperature changes and remains constant under these conditions. Understanding how to calculate and apply \( K_p \) helps in predicting the direction of the gas phase reaction and determining the amounts of product and reactant at equilibrium.

Partial Pressure

Partial pressure refers to the pressure that each gas in a mixture would exert if it alone occupied the entire volume. For reactions involving gases, each component's partial pressure contributes to the total pressure of the system.
In the ammonium chloride decomposition reaction, both ammonia and hydrogen chloride are formed in a 1:1 molar ratio. This means their partial pressures should be equal. Assume each has a partial pressure \( P \), leading to a total pressure of:

• \( P_{\text{total}} = P_{\text{NH}_3} + P_{\text{HCl}} = 2P \).

Given a total pressure of \( 2.2 \) atm, you can solve for each partial pressure:

• \( 2P = 2.2 \) atm \( \Rightarrow P = 1.1 \) atm.

Partial pressures are vital for determining the equilibrium constant \( K_p \) and understanding how changes in system conditions affect the equilibrium.

Ammonium Chloride Decomposition

The decomposition of ammonium chloride is an interesting chemical reaction because it involves a solid turning directly into gases. When \( \text{NH}_4\text{Cl} \) decomposes, it produces ammonia \( \text{NH}_3 \) and hydrogen chloride \( \text{HCl} \), which are both gases. This reaction can be represented as follows:

• \( \text{NH}_4\text{Cl}(s) \rightleftarrows \text{NH}_3(g) + \text{HCl}(g) \).

The forward and reverse rates of this reaction will equalize once equilibrium is reached.
This equilibrium can be affected by various factors like temperature or pressure and is commonly used to illustrate concepts in chemical equilibrium. The reaction is particularly useful in studying how solids can form gases and how these gases behave once released into the environment.

Gas Equilibrium

Gas equilibrium refers to the state in which the rate of a forward chemical reaction equals the rate of its reverse reaction, specifically involving gaseous reactants and products. In the example of ammonium chloride decomposition, reaching gas equilibrium means the formation rates of \( \text{NH}_3 \) and \( \text{HCl} \) gases balance with their recombination rate back into solid \( \text{NH}_4\text{Cl} \).
Analyzing gas equilibrium allows us to understand many important chemical processes, predict system behavior under various conditions, and calculate equilibrium constants like \( K_p \). Different conditions such as temperature or pressure modifications can shift the equilibrium position based on Le Chatelier’s Principle, simultaneously affecting the amounts of each species present. Understanding this helps in designing controlled environments for reactions and manipulating outputs in industrial applications.