Question

For the reaction, \(\mathrm{A}+\mathrm{B} \rightleftharpoons \mathrm{C}+\mathrm{D}\), the rate constants for the forward and backward reactions are found to be \(4.2 \times 10^{-2}\) and \(3.36 \times 10^{-3} \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}\) respectively. What is the equilibrium constant for the reaction: (a) \(11.5\) (b) \(12.5\) (c) \(8.0\) (d) \(6.0\)

Short answer

The equilibrium constant is approximately 12.5, which is option (b).

Step-by-step solution

Identify Given Values

We are given the rate constant for the forward reaction, \(k_f = 4.2 \times 10^{-2} \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}\), and the rate constant for the backward reaction, \(k_r = 3.36 \times 10^{-3} \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}\).

Formula for Equilibrium Constant

The equilibrium constant \(K\) for the reaction is given by the ratio of the forward and reverse rate constants. The formula to calculate \(K\) is: \[ K = \frac{k_f}{k_r} \]

Calculate the Equilibrium Constant

Substitute the given values into the formula: \[ K = \frac{4.2 \times 10^{-2}}{3.36 \times 10^{-3}} \] Calculate to find: \[ K \approx 12.5 \]

Conclusion

The calculated equilibrium constant \(K\) is approximately \(12.5\). Therefore, the correct answer is option (b) \(12.5\).

Key concepts

Rate Constants

In chemical reactions, we often talk about how fast products are formed from reactants, which is where the concept of rate constants comes in.
Rate constants are crucial for understanding chemical kinetics. They are denoted by the letter 'k' and help quantify the speed of a reaction.
For a given reaction like \( \mathrm{A} + \mathrm{B} \rightleftharpoons \mathrm{C} + \mathrm{D} \), we have rate constants for both the forward \( (k_f) \) and backward \( (k_r) \) reactions. This implies that reactions can occur in both directions, producing a state of dynamic equilibrium where the rate of the forward reaction equals the rate of the backward reaction.

• For a forward reaction, a higher rate constant suggests the reaction proceeds more quickly towards forming products.
• The backward rate constant shows the reverse as products convert back to reactants.

Understanding these constants helps us predict which way a reaction is more likely to proceed and how fast it'll reach equilibrium.

Chemical Equilibrium

The concept of chemical equilibrium is a fundamental part of chemistry and describes a state where the concentrations of reactants and products remain constant over time.
It doesn't mean that the reactions have stopped, but rather that they occur at the same rate in both directions.
When you measure the proportion of product to reactant at this equilibrium state, you derive the equilibrium constant, \( K \). This constant provides valuable insights into the reaction's characteristics:

• If \( K > 1 \), the products are favored at equilibrium.
• If \( K < 1 \), the reactants are favored.

This ratio of the rate constants \( (K = \frac{k_f}{k_r}) \) shows us how likely a reaction is to favor the formation of substances on one side of the equation.

Reaction Kinetics

Reaction kinetics is the study of rates at which chemical processes occur. Determining the speed of a reaction involves understanding the reaction mechanism and the conditions that affect rate. One key aspect is understanding how the rate of a reaction changes with the concentration of reactants and products.
Factors like temperature, pressure, and the presence of catalysts can play a significant role.

• High temperatures generally increase reaction rates.
• Pressure can affect reactions involving gases.
• Catalysts effectively lower the activation energy, making reactions proceed more rapidly without being consumed in the process.

By studying these conditions and how they affect the rate constants, chemists can manipulate the rates at which reactions occur, which is crucial in numerous industrial and laboratory processes.