Question

Consider the following reaction at a certain temperature: $$ \mathrm{A}_{2}+\mathrm{B}_{2} \rightleftarrows 2 \mathrm{AB} $$ The mixing of 1 mole of \(\mathrm{A}_{2}\) with 3 moles of \(\mathrm{B}_{2}\) gives rise to \(x\) mole of \(\mathrm{AB}\) at equilibrium. The addition of 2 more moles of \(\mathrm{A}_{2}\) produces another \(x\) mole of \(\mathrm{AB}\). What is the equilibrium constant for the reaction?

Short answer

The equilibrium constant \( K_c \) is 4.

Step-by-step solution

Understanding the Initial Setup

First, consider the initial amounts before reaching equilibrium. You start with 1 mole of \( \mathrm{A}_{2} \) and 3 moles of \( \mathrm{B}_{2} \), and they produce \( x \) moles of \( \mathrm{AB} \). At equilibrium, since the stoichiometry of the reaction is 1:1:2, the amount of \( \mathrm{A}_{2} \) will be \( 1-x \) moles and the amount of \( \mathrm{B}_{2} \) will be \( 3-x \) moles.

Situation After Adding More A2

When an additional 2 moles of \( \mathrm{A}_{2} \) is added, the total moles of \( \mathrm{A}_{2} \) initially would be \( 3-x \). Producing another \( x \) moles of \( \mathrm{AB} \) at equilibrium means \( x \) more moles of \( \mathrm{A}_{2} \) and \( \mathrm{B}_{2} \) react. The new equilibrium moles of \( \mathrm{A}_{2} \) are \( 3-2x \), and \( \mathrm{B}_{2} \) is \( 3-2x \). Total \( \mathrm{AB} \) at equilibrium is \( 2x \).

Setting Up the Equilibrium Constant Expression

The expression for the equilibrium constant \( K_c \) is based on the concentrations at equilibrium. This is given by the formula:\[ K_c = \frac{[\mathrm{AB}]^2}{[\mathrm{A}_{2}][\mathrm{B}_{2}]} \]Substitute the equilibrium concentrations for the first and second equilibrium states, both resulting in \( K_c = \frac{(2x)^2}{(3-2x)(3-2x)} \).

Equalizing the Equilibrium Expressions

Since both cases provided the same equilibrium constant, set the two \( K_c \) expressions: \[ \frac{x^2}{(1-x)(3-x)} = \frac{(2x)^2}{(3-2x)(3-2x)} \]

Simplifying and Solving for x

Clear the fractions and solve the quadratic equation derived from the equality. Simplify and solve for \( x \), ultimately leading to two possible values. Verify by substituting back to match conditions set by both experiments.

Calculating Equilibrium Constant

With \( x = 1 \) as the valid solution, substitute back to find the equilibrium constant:\[ K_c = \frac{4}{1} = 4 \]

Key concepts

Equilibrium Constant

The equilibrium constant, often denoted as \( K_c \), is a numerical value that provides insights into the balance between the reactants and products of a chemical reaction at equilibrium. It reflects the ratio of the concentrations of products to reactants, each raised to the power of their coefficients from the balanced chemical equation. At equilibrium, the reaction has reached a state where the concentrations of the reactants and products remain constant over time.

In the given reaction \( \mathrm{A}_{2} + \mathrm{B}_{2} \rightleftarrows 2 \mathrm{AB} \), the equilibrium constant can be expressed as:

• \[ K_c = \frac{[\mathrm{AB}]^2}{[\mathrm{A}_{2}][\mathrm{B}_{2}]} \]

This formula calculates the \( K_c \) by taking the concentration of the product \( \mathrm{AB} \), squaring it (since there are two moles), and dividing by the concentration of the reactants \( \mathrm{A}_{2} \) and \( \mathrm{B}_{2} \).

A higher \( K_c \) indicates a greater amount of products, suggesting the reaction favors products over reactants at equilibrium. Conversely, a lower \( K_c \) suggests more reactants than products. Understanding \( K_c \) helps in predicting the direction and extent of chemical reactions.

Stoichiometry

Stoichiometry is the study of the quantitative relationships between the amounts of reactants and products in a chemical reaction. It is rooted in the law of conservation of mass, which states that mass is neither created nor destroyed in a chemical reaction.

For the equation \( \mathrm{A}_{2} + \mathrm{B}_{2} \rightleftarrows 2 \mathrm{AB} \), the stoichiometric coefficients are crucial. They tell us that:

• 1 molecule of \( \mathrm{A}_{2} \) reacts with 1 molecule of \( \mathrm{B}_{2} \)
• to produce 2 molecules of \( \mathrm{AB} \).

This ratio must be strictly followed when reactants transform into products.

When performing a stoichiometric calculation, you use these ratios to convert between moles of different substances. It's vital to maintain the same ratio throughout for calculations to respect the balanced equation. Correct interpretation of these coefficients allows us to quantify how much of each substance is needed or produced.

Moles

The concept of moles is foundational in chemistry, serving as a bridge between the atomic scale and tangible amounts of substances we can measure in the lab. One mole is defined as exactly \( 6.022 \times 10^{23} \) particles (Avogadro's number), whether they are atoms, molecules, ions, or any other entity.

In the reaction \( \mathrm{A}_{2} + \mathrm{B}_{2} \rightleftarrows 2 \mathrm{AB} \), counting moles helps us in quantifying reactions accurately. If the exercise involves 1 mole of \( \mathrm{A}_{2} \) and 3 moles of \( \mathrm{B}_{2} \), it implies that these are the starting amounts, which can be converted into product molecules as dictated by the reaction balance when at equilibrium.

Understanding the mole concept ensures accurate measurement and accurate prediction of how reactions proceed. For instance, a comprehension of moles allows the determination of how much product (\( \mathrm{AB} \)) will form when certain amounts of reactants are mixed, as well as how the equilibrium will shift if additional reactive components are introduced.

Reaction Equation

A chemical reaction equation is a symbolic representation of what happens when reactants transform into products. It demonstrates the conversion and is balanced to satisfy the law of conservation of mass—meaning, the number of each type of atom is the same on either side of the equation.

For the reaction \( \mathrm{A}_{2} + \mathrm{B}_{2} \rightleftarrows 2 \mathrm{AB} \), each component of the formula plays a crucial role in determining how the reaction proceeds. Reversible reactions, indicated by a double arrow, suggest that the process can go in both directions until equilibrium is reached.

Each reaction must be analyzed to understand:

• Which substances are reactants (on the left)
• Which are products (on the right)
• How the stoichiometric coefficients guide the conversion of units through the balanced chemical equation

The balanced equations allow chemists to make precise calculations about quantities of reactants needed or products formed, predict yields, and understand the reaction mechanisms at a molecular level.