Question

If the equilibrium constant for the reaction, \(2 X Y \rightleftharpoons X_{2}+Y_{2}\) is 81 what is the value of equilibrium constant for the reaction $$ X Y \rightleftharpoons \frac{1}{2} X_{2}+\frac{1}{2} Y_{2} $$ (a) 81 (b) 9 (c) 6561 (d) \(40.5\)

Short answer

The equilibrium constant for the new reaction is 9 (option b).

Step-by-step solution

Identify the Relationship Between the Two Reactions

The original reaction is doubled to get the second reaction, thus the original equilibrium constant, K, will change accordingly when the reaction is halved.

Determine the New Equilibrium Constant

For the reaction where coefficients are halved, the equilibrium constant for the new reaction, K', will be the square root of the original equilibrium constant, since equilibrium constants are raised to the power of their stoichiometric coefficients in the balanced chemical equation.

Calculate the New Equilibrium Constant

Take the square root of the original equilibrium constant, K = 81, to find the equilibrium constant for the new reaction. K' = \( \sqrt{81} \) = 9.

Choose the Correct Answer

The calculated equilibrium constant for the new reaction is 9. Therefore, option (b) is the correct answer.

Key concepts

Chemical Equilibrium

Chemical equilibrium is a dynamic state where the rate of the forward reaction equals the rate of the reverse reaction, resulting in no overall change in the concentrations of reactants and products over time. In a balanced chemical equation, the point at which this equilibrium is achieved can be quantified using the equilibrium constant, denoted as K.

For the reaction \(2 XY \rightleftharpoons X_{2} + Y_{2}\), the equilibrium constant (K) is provided as 81. This means that, at equilibrium, the concentration ratio of the products to the reactants, raised to the power of their stoichiometric coefficients, is 81. Being a constant, K remains the same at a given temperature, regardless of the initial concentrations of reactants and products.

Reaction Stoichiometry

Reaction stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It is based on the conservation of mass, where the number of atoms of each element remains the same before and after a reaction. The coefficients in a chemical equation, such as \(2\) in \(2 XY \rightleftharpoons X_{2} + Y_{2}\), represent the stochiometric coefficients, and they tell us the ratio in which chemical species react and are formed.

When adjusting the coefficients, the equilibrium constant's value changes depending on the stoichiometric coefficients' powers. For instance, halving the coefficients in the reaction equation results in the square root of the original equilibrium constant, which reflects the mathematical relationship between the stoichiometry and the equilibrium constant.