Question

For the reaction $$ \mathrm{N}_{2}(g)+3 \mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{NCl}_{3}(g) $$ an analysis of an equilibrium mixture at a particular temperature is performed. It is found that \(\left[\mathrm{NCl}_{3}\right]=1.9 \times 10^{-1} M,\left[\mathrm{~N}_{2}\right]=1.4 \times 10^{-3} M,\) and \(\left[\mathrm{Cl}_{2}\right]=4.3 \times 10^{-4} M\) Calculate \(K\) for the reaction.

Short answer

The equilibrium constant (K) for the reaction \( \mathrm{N}_{2}(g)+3 \mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{NCl}_{3}(g) \) is \( 3.25 \times 10^{12} \).

Step-by-step solution

Write down the balanced chemical equation

We are provided with the balanced chemical equation: \( \mathrm{N}_{2}(g)+3 \mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{NCl}_{3}(g) \)

Write the expression for the equilibrium constant (K)

The equilibrium constant (K) for the given reaction can be expressed as: \( K = \frac{\left[\mathrm{NCl}_{3}\right]^2}{\left[\mathrm{N}_{2}\right] \left[\mathrm{Cl}_{2}\right]^3} \)

Substitute the given concentration values

To calculate the value of K, substitute the equilibrium concentrations given: \( [\mathrm{NCl}_{3}] = 1.9 \times 10^{-1} \: M \) \( [\mathrm{N}_{2}] = 1.4 \times 10^{-3} \: M \) \( [\mathrm{Cl}_{2}] = 4.3 \times 10^{-4} \: M \)

Calculate the value of K

Plug the concentration values into the K expression: \( K = \frac{(1.9 \times 10^{-1})^2}{(1.4 \times 10^{-3})(4.3 \times 10^{-4})^3} \)

Simplify and find the value of K

Perform the arithmetic operations to find the value of K: \( K = \frac{(3.61 \times 10^{-2})}{(1.4 \times 10^{-3})(7.9249 \times 10^{-12})} \) \( K = \frac{3.61 \times 10^{-2}}{1.10949 \times 10^{-14}} \) \( K = 3.25 \times 10^{12} \) The equilibrium constant for this reaction is \( 3.25 \times 10^{12} \).

Key concepts

Chemical Equilibrium

Chemical equilibrium refers to the state in a reversible reaction where the rate of the forward reaction equals the rate of the backward reaction.
This means that the concentrations of the reactants and products remain constant over time, although the reactions are still occurring.
Let's imagine a seesaw where both sides are perfectly balanced; that's the essence of chemical equilibrium in a reaction.

• Each component of the reaction retains its concentration at equilibrium, forming a stable system.
• The concept of equilibrium constant, denoted as \( K \), quantifies the ratio of the concentrations of products to reactants at equilibrium.

When analyzing chemical equilibrium, it's important to understand that this constant, \( K \), is specific to each reaction and changes solely with temperature.
Thus, knowing \( K \) provides insights into which side of the reaction is favored — toward products or reactants — under specific conditions.

Reaction Kinetics

Reaction kinetics deals with the speed or rate at which chemical reactions occur.
While chemical equilibrium focuses on the state of balance in a reaction, kinetics studies how quickly this balance is reached.

• The rate of a reaction can be influenced by factors such as temperature, concentration of reactants, surface area, and the presence of catalysts.
• The equation for the equilibrium constant is developed from the rates of the forward and reverse reactions being equal at equilibrium.

Through the lens of reaction kinetics, we can learn how to manipulate conditions to speed up or slow down a reaction.
This can be particularly useful in industrial processes where controlling the rate of reaction can impact efficiency and productivity.
Furthermore, by understanding the underlying kinetics, chemists can propose mechanisms for the way reactions proceed and how intermediates form before reaching equilibrium.

Stoichiometry

Stoichiometry is an integral part of chemistry that involves calculating the quantities of reactants and products in chemical reactions.
In the context of equilibrium, it helps to understand the proportional relationship between the reactants and products in a balanced chemical equation.

• The balanced equation \( \mathrm{N}_{2}(g)+3 \mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{NCl}_{3}(g) \) gives the stoichiometric coefficients indicating the ratio in which substances react and are produced.
• Stoichiometry helps calculate how changes in the concentration of one component at equilibrium affect others, using the principle of conservation of mass.

Using stoichiometry along with the defined equilibrium constant, we can predict the concentrations of unknown species at equilibrium for given initial concentrations.
This is critical not only in academic problems but also in fields like pharmacology and environmental science, where precise calculations are necessary for formulation and monitoring of chemical reactions.