Question

The equilibrium constant for the following reaction will be \(3 \mathrm{~A}+2 \mathrm{~B} \rightleftharpoons \mathrm{C}\) : (a) \(\frac{[3 \mathrm{~A}][2 \mathrm{~B}]}{[\mathrm{C}]}\) (b) \(\frac{[\mathrm{C}]}{[3 \mathrm{~A}][2 \mathrm{~B}]}\) (c) \(\frac{[\mathrm{C}]}{[\mathrm{A}]^{2}[\mathrm{~B}]^{2}}\) (d) \(\frac{[\mathrm{C}]}{[\mathrm{A}]^{3}[\mathrm{~B}]^{2}}\)

Short answer

Option (d) \(\frac{[\mathrm{C}]}{[\mathrm{A}]^3[\mathrm{B}]^2}\) is correct.

Step-by-step solution

Understand the reaction and equilibrium constant

The chemical equation is given as \(3A + 2B \rightleftharpoons C\). The equilibrium constant \(K\) for this reaction is expressed in terms of the concentrations of the products and reactants at equilibrium.

Formulate the equilibrium expression

In general, the equilibrium constant expression for a reaction \(aA + bB \rightleftharpoons cC\) is given by \([C]^c / ([A]^a[B]^b)\). For this problem, \(c = 1\), \(a = 3\), and \(b = 2\). Substitute these values into the expression to find the equilibrium constant.

Write the equilibrium constant for the given reaction

Substitute \(c = 1\), \(a = 3\), and \(b = 2\) into the equilibrium constant expression: \(K = \frac{[C]^1}{[A]^3[B]^2}\). This gives the equilibrium constant expression as \(\frac{[\mathrm{C}]}{[\mathrm{A}]^3[\mathrm{B}]^2}\).

Match the option with the correct expression

Compare the derived expression \(\frac{[\mathrm{C}]}{[\mathrm{A}]^3[\mathrm{B}]^2}\) with the given options. Option (d) \(\frac{[\mathrm{C}]}{[\mathrm{A}]^3[\mathrm{B}]^2}\) matches our expression exactly.

Key concepts

Equilibrium Constant

In the realm of chemical reactions, the equilibrium constant, denoted by \( K \), serves as a pivotal gauge for the state of a chemical reaction once it has reached equilibrium. This constant doesn't change over time if the system is maintained at a constant temperature. The equilibrium constant is derived from the concentrations of the products and reactants at equilibrium.
The general equation to determine \( K \) is expressed as \( \frac{[\text{Products}]}{[\text{Reactants}]} \), where the concentrations are raised to the power of their stoichiometric coefficients. For the given reaction \( 3A + 2B \rightleftharpoons C \), this translates to the equation \( \frac{[C]}{[A]^3 [B]^2} \). This tells us how much of each substance we have at equilibrium.

• If \( K \) is large (\( K >> 1 \)), it favors the formation of products.
• If \( K \) is small (\( K << 1 \)), it favors the formation of reactants.

Understanding the equilibrium constant helps chemists predict how a reaction mixture will behave under different conditions.

Reaction Quotient

The reaction quotient, represented as \( Q \), is an indispensable tool in determining the current status of a chemical reaction in comparison to its equilibrium state. Unlike the equilibrium constant, \( Q \) can be computed at any given point during a reaction.
Just like the equilibrium constant, the reaction quotient is calculated using the expression \( \frac{[\text{Products}]}{[\text{Reactants}]} \), with the concentrations raised to their stoichiometric coefficients. When applied to our reaction \( 3A + 2B \rightleftharpoons C \), \( Q \) would similarly be derived from \( \frac{[C]}{[A]^3 [B]^2} \).

• If \( Q = K \), the system is at equilibrium.
• If \( Q < K \), the reaction will proceed forward, producing more products to reach equilibrium.
• If \( Q > K \), the reaction will go in reverse, producing more reactants to reach equilibrium.

By understanding \( Q \), chemists can take steps to adjust a reaction's conditions to drive it towards equilibrium if desired.

Stoichiometry

Stoichiometry is the core principle that allows us to quantitatively analyze chemical reactions. It involves using balanced chemical equations to predict the amounts of reactants and products involved during a chemical process.
In the balanced chemical equation \( 3A + 2B \rightleftharpoons C \), the stoichiometric coefficients are pivotal. Specifically, the numbers \( 3 \) and \( 2 \) denote how many molecules of \( A \) and \( B \) combine to form one molecule of \( C \). These coefficients are essential, as they help us form the equilibrium constant expression \( \frac{[C]}{[A]^3 [B]^2} \).

• Stoichiometry ensures that the law of conservation of mass is upheld in chemical reactions.
• These coefficients allow us to scale reactions up or down quantitatively to predict the outcome of reactions.

Grasping stoichiometry is crucial for anyone studying chemistry, as it lays the groundwork for more advanced concepts such as reaction yields and limiting reagents.