Question

The equilibrium constant, \(K\), for the reaction $$ 2 \mathrm{NOCl}(\mathrm{g}) \rightleftarrows 2 \mathrm{NO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g}) $$ is \(3.9 \times 10^{-3}\) at \(300^{\circ} \mathrm{C} .\) A mixture contains the gases at the following concentrations: \([\mathrm{NOCl}]=5.0 \times 10^{-8} \mathrm{mol} / \mathrm{L}\) \([\mathrm{NO}]=2.5 \times 10^{-3} \mathrm{mol} / \mathrm{L},\) and \(\left[\mathrm{Cl}_{2}\right]=2.0 \times 10^{-3} \mathrm{mol} / \mathrm{L}\) Is the reaction at equilibrium at \(300^{\circ} \mathrm{C}\) ? If not, in which direction does the reaction proceed to come to equilibrium?

Short answer

The reaction is not at equilibrium. It will shift to the left to form more NOCl.

Step-by-step solution

Write the Expression for the Reaction Quotient, Q

For the reaction \(2 \mathrm{NOCl}(\mathrm{g}) \rightleftarrows 2 \mathrm{NO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g})\), the expression for the reaction quotient \(Q\) is formulated similarly to the equilibrium constant, \(K\). The expression is: \[ Q = \frac{[\mathrm{NO}]^2[\mathrm{Cl}_2]}{[\mathrm{NOCl}]^2} \] where the concentrations of the gases are substituted into this formula.

Substitute the Concentrations into the Expression for Q

Insert the given concentrations into the expression for \(Q\): \[ Q = \frac{(2.5 \times 10^{-3})^2 (2.0 \times 10^{-3})}{(5.0 \times 10^{-8})^2} \]Calculate each component: \([\mathrm{NO}] = 2.5 \times 10^{-3}\), \([\mathrm{Cl}_2] = 2.0 \times 10^{-3}\), \([\mathrm{NOCl}] = 5.0 \times 10^{-8}\).

Calculate the Reaction Quotient, Q

First calculate the numerator:\[(2.5 \times 10^{-3})^2 (2.0 \times 10^{-3}) = 6.25 \times 10^{-6} \times 2.0 \times 10^{-3} = 1.25 \times 10^{-8} \].Next, calculate the denominator: \((5.0 \times 10^{-8})^2 = 2.5 \times 10^{-15} \).Therefore, \[ Q = \frac{1.25 \times 10^{-8}}{2.5 \times 10^{-15}} = 5.0 \times 10^{6} \].

Compare Q to the Equilibrium Constant K

With \( Q = 5.0 \times 10^{6} \) and \( K = 3.9 \times 10^{-3} \), note that \( Q \) is much greater than \( K \). This means the reaction quotient indicates a greater than expected amount of products compared to reactants at equilibrium.

Determine the Direction of Reaction Shift

Since \( Q > K \), the reaction will shift to the left to reach equilibrium, meaning more reactants (\(\mathrm{NOCl}\)) will form and the amount of products (\(\mathrm{NO}\) and \(\mathrm{Cl}_2\)) will decrease.

Key concepts

Equilibrium Constant

In a chemical reaction, the equilibrium constant, denoted as \( K \), is a vital measure. It quantifies the ratio of product concentrations to reactant concentrations at equilibrium. A low \( K \) value, like \(3.9 \times 10^{-3}\), suggests the reactant side is favored at equilibrium. This is because the products' concentrations are smaller compared to those of the reactants when the system is balanced. Calculating \( K \) involves using balanced chemical equations where the concentrations of products are raised to their stoichiometric coefficients and divided by the concentrations of reactants similarly raised. This value varies only with temperature changes, making \( K \) a useful tool for predicting the behavior of reactions under different conditions.

Reaction Quotient

The reaction quotient, \( Q \), serves a similar purpose to \( K \) but applies to any moment in time, not just equilibrium. It helps in determining whether a system has reached equilibrium or predicts the direction of the reaction to reach equilibrium. For the given reaction, \( Q \) is calculated using the same expression as \( K \): the concentrations of products over reactants, each raised to the power of their stoichiometric coefficients. By comparing \( Q \) to \( K \), one can tell if the reaction is at equilibrium or which way it needs to shift. If \( Q > K \), as in this scenario, it means there are too many products relative to the reactants compared to the ideal state, prompting a shift in proportion.

Le Chatelier's Principle

Le Chatelier's Principle helps us understand how a system at equilibrium responds to changes in concentration, pressure, or temperature. According to this principle, if a change disturbs the equilibrium, the system will adjust itself to counteract or reduce that change. In the example of the reactions, because \( Q > K \), the equilibrium will shift to the left, increasing the concentration of \( \mathrm{NOCl} \) and reducing \( \mathrm{NO} \) and \( \mathrm{Cl}_2 \) to reach equilibrium again. This natural counteraction helps maintain balance in a dynamic chemical system and is a key understanding in chemical equilibrium studies to predict how systems will adjust to alterations.

Direction of Reaction

The direction of a chemical reaction can be predicted by comparing \( Q \) and \( K \). If \( Q < K \), the reaction proceeds to the right, forming more products. Conversely, if \( Q > K \), the reaction moves to the left, producing more reactants. In the problem discussed, \( Q \) was found to be greater than \( K \), indicating an excess of products present. To reach equilibrium, the reaction shifts left, converting products back into reactants. Understanding the directionality of reactions is important for controlling chemical processes, maximizing yields, and maintaining desired conditions within industrial and laboratory settings.