Question

The equilibrium constant for the reaction $$ 2 \mathrm{NOCl}(g) \rightleftharpoons 2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) $$ has the value \(9.2 \times 10^{-6}\) at a particular temperature. The system is analyzed at equilibrium, and it is found that the concentrations of \(\mathrm{NOCl}(g)\) and \(\mathrm{NO}(g)\) are \(0.44 \mathrm{M}\) and \(1.5 \times 10^{-3} M,\) respectively. What is the concentration of \(\mathrm{Cl}_{2}(g)\) in the equilibrium system under these conditions?

Short answer

The equilibrium concentration of Cl2 in the given system under the provided conditions is approximately \(7.29 \times 10^{-3} M\).

Step-by-step solution

Write the equilibrium expression for the given reaction

The equilibrium expression for the reaction can be written as: \[K=\frac{[\mathrm{NO}]^2 [\mathrm{Cl}_2]}{[\mathrm{NOCl}]^2}\] Where K is the equilibrium constant, and the concentrations of the species are denoted by square brackets.

Substitute the given values into the equilibrium expression

We are given the value of K as \(9.2 \times 10^{-6}\). We are also given the equilibrium concentrations of NOCl and NO as 0.44 M and \(1.5 \times 10^{-3} M\), respectively. We can now substitute these values in the equilibrium expression: \(9.2 \times 10^{-6} = \frac{[1.5 \times 10^{-3}]^2 [\mathrm{Cl}_2]}{[0.44]^2}\)

Solve for the concentration of Cl2

Simplify the equation and solve for the concentration of Cl2: \([\mathrm{Cl}_2] = \frac{9.2 \times 10^{-6} [0.44]^2}{[1.5 \times 10^{-3}]^2}\) \([\mathrm{Cl}_2] = \frac{9.2 \times 10^{-6} \times (0.44)^2}{(1.5 \times 10^{-3})^2}\) Calculating this expression, we get: \([\mathrm{Cl}_2] = 7.29 \times 10^{-3}\)

Final Answer

The equilibrium concentration of Cl2 in the given system under the provided conditions is approximately \(7.29 \times 10^{-3} M\).

Key concepts

Chemical Equilibrium

Chemical equilibrium is a state in a chemical reaction where the rates of the forward reaction and the reverse reaction are equal, leading to no overall change in the concentrations of the reactants and products over time. It's vital to note that this does not mean the reactants and products are present in equal amounts; rather, their ratios are constant at equilibrium.

An understanding of chemical equilibrium is immensely important in various fields, including chemistry, biology, environmental science, and many industrial processes. Equilibrium doesn't imply that a reaction has stopped; both the forward and reverse reactions are still occurring, just at the same rate, resulting in a dynamic but stable system.

Equilibrium Expression

The equilibrium expression for a chemical reaction relates the concentrations of the reactants and products at equilibrium. Derived from the law of mass action, it takes the form of the equilibrium constant, K, which is unique for every reaction at a given temperature.

For the general reaction $$aA + bB \rightleftharpoons cC + dD$$the equilibrium expression would be$$K = \frac{[C]^c[D]^d}{[A]^a[B]^b}$$where the concentrations of products and reactants are raised to the power of their respective stoichiometric coefficients. In the solved exercise, the equilibrium expression is about finding the ratio between the products and reactants concentrations squared, as determined by the balanced chemical equation.

Concentration Calculation

Concentration calculation plays a crucial role in determining the amounts of reactants and products in equilibrium reactions. Using the value of the equilibrium constant along with the known concentrations of other species in the reaction, one can compute the unknown concentrations.

In the context of our exercise, the known equilibrium constant and the concentrations of NOCl and NO allowed us to solve for the concentration of Cl₂, showcasing how stoichiometry and equilibrium principles work together to provide a complete picture of the system's state. This sort of calculation is fundamental in predicting the outcome of reactions and is widely applied in chemical industries and research.

Le Chatelier's Principle

Le Chatelier's Principle is an essential concept when studying chemical equilibria. It predicts how a system at equilibrium will respond to a change in concentration, temperature, or pressure. Essentially, if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to counteract the change. This can result in the increased or decreased production of products or reactants depending on the direction of the shift.

When a system is at equilibrium and a reactant is added, the system shifts to produce more products. Conversely, if a product is removed, more reactants are converted into products to maintain the equilibrium. In terms of concentration, an increase in a reactant's concentration shifts the equilibrium towards the products, whereas an increase in a product's concentration shifts it towards the reactants.