Question

If the value of equllibrium constant \(K_{r}\) for the reaction, \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}\) is \(7 .\) The equilibrium constant for the reaction \(2 \mathrm{~N}_{2}+6 \mathrm{H}_{2} \rightleftharpoons 4 \mathrm{NH}_{3}\) will be (a) 49 (b) 7 (c) 14 (d) 28

Short answer

The equilibrium constant for the reaction \(2 \text{N}_{2}+6 \text{H}_{2} \rightleftharpoons 4 \text{NH}_{3}\) will be 49.

Step-by-step solution

Understand the Relationship of Equilibrium Constants

The equilibrium constant expression for a chemical reaction is dependent on the stoichiometry of the reaction. If an equation is multiplied by a factor, the equilibrium constant for the new equation is the original equilibrium constant raised to the power of that factor.

Identify the Multiplication Factor

Comparing the original reaction with the new reaction, we can see that the coefficients in the second reaction are exactly twice those in the first. Therefore, the multiplication factor is 2.

Apply the Multiplication Factor to the Equilibrium Constant

Since the original equilibrium constant (\(K_{r}\)) for the first reaction is given as 7, the equilibrium constant for the new reaction will be \(K_{r}(new) = K_{r}^{factor} = 7^{2} = 49\).

Key concepts

Chemical Equilibrium

Understanding chemical equilibrium is essential for a thorough grasp of how reactions occur. It represents the state of a reaction when the rates of the forward and reverse reactions are equal, resulting in no net change in the concentrations of the reactants and products. This dynamic state might seem static, but it is actually a result of continuous and opposing processes.

To quantify this state, we use the equilibrium constant, denoted as \( K \). It is a mathematical ratio of the concentrations of the products to the concentrations of the reactants, each raised to the power of their respective coefficients from the balanced chemical equation. For example, for the reaction \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}\), the equilibrium constant (\(K_{r}\)) can be written as:\[K_{r} = \frac{[\mathrm{NH}_{3}]^{2}}{[\mathrm{N}_{2}][\mathrm{H}_{2}]^{3}}\]
where the square brackets signify concentration.

Reaction Stoichiometry

Reaction stoichiometry involves the quantitative relationship between the amounts of reactants and products in a chemical reaction. It's similar to a recipe, but instead of cups and teaspoons, chemists use moles to measure the reactants and products. When dealing with equilibrium, the stoichiometry of the reaction is particularly crucial because the power to which the concentration of each substance is raised in the equilibrium expression is based on the stoichiometric coefficients from the balanced equation.

For example, if we consider the reaction \( \mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}\), and we double all the coefficients to get \( 2\mathrm{N}_{2}+6 \mathrm{H}_{2} \rightleftharpoons 4 \mathrm{NH}_{3}\), we are effectively altering the ratio of reactants to products. The new equilibrium constant will be the old constant raised to the power equal to the factor of change in the stoichiometric coefficients, which in our case, is 2: \(K_{r}(new) = K_{r}^{2} = 7^{2} = 49\).

Le Chatelier's Principle

Le Chatelier's principle provides insight into how a system at equilibrium responds to a change in conditions. If an external stress, such as a change in concentration, pressure, or temperature, is applied to a system in equilibrium, the system will adjust to counteract the change and restore a new equilibrium.

For instance, if we increase the concentration of \(\mathrm{N}_{2}\) in the reaction \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}\), according to Le Chatelier's principle, the system will shift to the right to reduce the concentration of \(\mathrm{N}_{2}\) by producing more \(\mathrm{NH}_{3}\). This behavior helps to understand and predict the effects of different changes on the system and is a fundamental concept in chemical reactions and processes.