Question

Consider the following gaseous equilibria given below : (I) \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}\); Eqm. Constant \(=K_{1}\) (II) \(\mathrm{N}_{2}+\mathrm{O}_{2} \rightleftharpoons 2 \mathrm{NO} ; \quad\) Eqm. Constant \(=K_{2}\) (III) \(\mathrm{H}_{2}+\frac{1}{2} \mathrm{O}_{2} \rightleftharpoons \mathrm{H}_{2} \mathrm{O} ;\) Eqm. Constant \(=K_{3}\) The equilibrium constant for the reaction, \(2 \mathrm{NH}_{3}+\frac{5}{2} \mathrm{O}_{2} \rightleftharpoons 2 \mathrm{NO}+3 \mathrm{H}_{2} \mathrm{O}\) in terms of \(K_{1}, K_{2}\) and \(K_{3}\) will be : \(\begin{array}{llll}\text { (a) } K_{1} K_{2} K_{3} & \text { (b) } \frac{K_{1} K_{2}}{K_{3}} & \text { (c) } \frac{K_{1} K_{3}^{2}}{K_{2}} & \text { (d) } \frac{K_{2} K_{3}^{3}}{K_{1}}\end{array}\)

Short answer

\(\frac{K_{2}^{2} \cdot K_{3}^{3}}{K_{1}}\)

Step-by-step solution

Identify the Target Equation

The target equilibrium reaction we are analyzing is: \(2 \mathrm{NH}_{3} + \frac{5}{2} \mathrm{O}_{2} \rightleftharpoons 2 \mathrm{NO} + 3 \mathrm{H}_{2}O\). We need to write this equation in terms of the given three equilibria I, II, and III.

Write Each Given Reaction

Write down each of the given equilibrium reactions and their respective equilibrium constants:\(\mathrm{N}_{2} + 3\mathrm{H}_{2} \rightleftharpoons 2\mathrm{NH}_{3}, K_{1}\)\(\mathrm{N}_{2}+\mathrm{O}_{2} \rightleftharpoons 2\mathrm{NO}, K_{2}\)\(\mathrm{H}_{2}+\frac{1}{2}\mathrm{O}_{2} \rightleftharpoons \mathrm{H}_{2} O, K_{3}\)

Reverse Reaction I

Reverse reaction I in order to produce \(\mathrm{NH}_{3}\) as a reactant instead of a product:\(2 \mathrm{NH}_{3} \rightleftharpoons \mathrm{N}_{2} + 3\mathrm{H}_{2}\).\(K') = \frac{1}{K_1}\)

Combine Reactions II and III

Combine reactions II and III appropriately to align with the products and reactants in the target reaction. We need \(2\) moles of NO and \(3\) moles of \(H_{2}O\) in the products, so we write:\(2(\mathrm{N}_{2}+\mathrm{O}_{2} \rightleftharpoons 2\mathrm{NO})\), so the equilibrium constant becomes \(K_{2}^2\).\(3(\mathrm{H}_{2}+\frac{1}{2}\mathrm{O}_{2} \rightleftharpoons \mathrm{H}_{2}O)\), so the equilibrium constant becomes \(K_{3}^3\).

Combine the Modified Reactions

Now add the modified version of reaction I from step 3 with those in step 4, making sure the coefficients of all molecules match those in the target reaction:\(2 \mathrm{NH}_{3} \rightleftharpoons \mathrm{N}_{2} + 3 \mathrm{H}_{2}\)\(2(\mathrm{N}_{2}+\mathrm{O}_{2} \rightleftharpoons 2\mathrm{NO})\)\(3(\mathrm{H}_{2}+\frac{1}{2}\mathrm{O}_{2} \rightleftharpoons \mathrm{H}_{2}O)\)

Determine the Equilibrium Constant for the Target Equation

The overall equilibrium constant for the target reaction is the product of the constants for the reactions used, considering their stoichiometry and direction:\(K = \frac{K_{2}^{2} \cdot K_{3}^{3}}{K_{1}}\). Thus, the equilibrium constant in terms of \(K_{1}, K_{2}\), and \(K_{3}\) for the target reaction is \(\frac{K_{2}^{2} \cdot K_{3}^{3}}{K_{1}}\).

Key concepts

Equilibrium Constant

An equilibrium constant (\(K\textsubscript{eq}\)) is a numerical value that illustrates the ratio of the concentration of products to the concentration of reactants at equilibrium for a reversible chemical reaction. This constant is unique for every chemical reaction at a given temperature. Understanding this concept is essential as it gives insight into the extent of a reaction: whether it favors the formation of products or reactants. For instance, in the provided exercise, the gaseous equilibria of three reactions are expressed through their equilibrium constants (\(K\textsubscript{1}\), \(K\textsubscript{2}\), and \(K\textsubscript{3}\)).

The value of the equilibrium constant is determined by the balanced chemical equation. A high value of \(K\textsubscript{eq}\) indicates the reaction favors the production of the products, while a low value denotes that the reactants are favored. When reactions are combined or reversed, as in the exercise, the resulting equilibrium constant for the overall reaction is the product (or quotient) of the individual constants. This reflects how interconnected equilibria can be manipulated to find a new equilibrium position for a different chemical reaction.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It describes the ratio in which substances react and form products, which is based on the balanced chemical equation. Understanding stoichiometry is crucial for determining the quantity of reactants needed and the amount of products that will be formed.

When examining the stoichiometry of the equilibrium reactions in the exercise, attention must be paid to the coefficients of each substance. These coefficients become exponents when determining the combined equilibrium constant of a derived reaction. For example, since we are combining two instances of Reaction II and three instances of Reaction III (as seen in Step 4 of the solution), we must square \(K\textsubscript{2}\) and cube \(K\textsubscript{3}\) respectively. Thus, stoichiometry not only allows us to balance chemical equations but also plays a key role in determining the relationships between different equilibrium constants.

Chemical Reaction Reversal

The direction in which a chemical reaction proceeds significantly influences its equilibrium constant. When a chemical reaction is reversed, the equilibrium constant for the new, reversed reaction is the inverse of the original. This concept becomes particularly important when dealing with complex systems where reactions can proceed in multiple directions.

In our problem, to find the equilibrium constant for the target reaction, we first had to reverse Reaction I, which switches the reactants and products (as seen in Step 3 of the solution). In this case, the equilibrium constant for the reversed reaction, \(K'\textsubscript{1}\), becomes \frac{1}{K\textsubscript{1}}. This illustrates that the relationship between forward and reverse reactions is reciprocal in terms of their equilibrium constants. This understanding is essential when manipulating multiple reactions to determine the equilibrium constant for a subsequent reaction not explicitly provided, as shown in the exercise.