Question

At \(35^{\circ} \mathrm{C}, K=1.6 \times 10^{-5}\) for the reaction $$ 2 \mathrm{NOCl}(g) \rightleftharpoons 2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) $$ If \(2.0 \mathrm{~mol} \mathrm{NO}\) and \(1.0 \mathrm{~mol} \mathrm{Cl}_{2}\) are placed into a \(1.0\) - \(\mathrm{L}\) flask, calculate the equilibrium concentrations of all species.

Short answer

The equilibrium concentrations of the species are: [NOCl] ≈ 0.0056 M, [NO] ≈ 1.9944 M, and [Cl₂] ≈ 0.9972 M.

Step-by-step solution

Write the balanced equation and the equilibrium expression

The balanced equation for the given reaction is: \(2 NOCl(g) \rightleftharpoons 2 NO(g) + Cl_{2}(g)\) The equilibrium constant (K) expression for the reaction is: \(K = \frac{[NO]^2 \times [Cl₂]}{[NOCl]^2}\)

Set up the ICE (Initial, Change, Equilibrium) table

To begin with, we'll create an ICE table to illustrate the initial concentrations, the changes that occur, and the equilibrium concentrations: | Species: | NOCl | NO | Cl₂ | |-----------|------|-----|-----| | Initial: | 0 | 2.0 | 1.0 | | Change: | +2x | -2x | -x | | Equilibrium: | 2x | 2-2x|1-x | Here, x represents the change in concentration of NO and Cl₂ as the reaction proceeds towards equilibrium.

Substitute the equilibrium concentrations into the equilibrium expression

From the ICE table, we can substitute the equilibrium concentrations from the last row into the expression for K: \(1.6 \times 10^{-5} = \frac{(2-2x)^2 \times (1-x) }{(2x)^2}\)

Solve for x

This is a quadratic equation which can be quite difficult to solve algebraically. However, since K is very small, the change in the concentrations of reactants and products as the reaction proceeds towards equilibrium will be insignificant. Therefore, we can assume x ≈ 0 and (2-2x) ≈ 2 and (1-x) ≈ 1. This simplifies our equation to: \(1.6 \times 10^{-5} = \frac{2^2 \times 1}{(2x)^2}\) Solving for x: \(x = \sqrt{\frac{2^2 \times 1}{1.6 \times 10^{-5}}}\) \(x \approx 0.0028\)

Calculate the equilibrium concentrations

Now that we know the value of x, we can find the equilibrium concentrations of all species from the ICE table: [NOCl] = 2x = 2(0.0028) ≈ 0.0056 M [NO] = 2-2x ≈2 - 2(0.0028) ≈ 1.9944 M [Cl₂] = 1-x ≈ 1 - 0.0028 ≈ 0.9972 M So, the equilibrium concentrations of the species are: [NOCl] ≈ 0.0056 M, [NO] ≈ 1.9944 M, and [Cl₂] ≈ 0.9972 M

Key concepts

Chemical Equilibrium

Chemical equilibrium represents a state of a chemical reaction where the rates of the forward and reverse reactions are equal, resulting in no overall change in the concentrations of reactants and products. It's crucial to understand that this doesn't mean the reactants and products are in equal concentrations, but rather that their concentrations have stabilized to constant values.

For the given reaction \(2 \mathrm{NOCl}(g) \rightleftharpoons 2 \mathrm{NO}(g) + \mathrm{Cl}_{2}(g)\), equilibrium will be reached when the rate at which \(\mathrm{NOCl}\) decomposes into \(\mathrm{NO}\) and \(\mathrm{Cl}_{2}\) is equal to the rate at which \(\mathrm{NO}\) and \(\mathrm{Cl}_{2}\) combine to form \(\mathrm{NOCl}\). At that point, the observable properties of the system become constant. Understanding equilibrium allows chemists to predict how changing conditions like temperature and pressure will affect the composition of the system.

ICE Table

The ICE (Initial, Change, Equilibrium) table is an incredibly valuable tool used to track the changes in concentrations or pressures of reactants and products as a chemical reaction approaches equilibrium. The acronym ICE stands for:

• Initial: The initial concentrations or pressures of reactants and products before the reaction takes place.
• Change: The amount by which reactants decrease and products increase as the system moves towards equilibrium. Changes are typically represented by a variable (x) that is defined based on the stoichiometry of the balanced chemical equation.
• Equilibrium: The concentrations or pressures of reactants and products when the reaction has reached equilibrium. These are calculated by combining the initial amounts with the change.

By setting up an ICE table, not only can we visualize what happens to a reaction over time, but we also create a system of equations that can be solved to find the equilibrium concentrations of all chemical species involved. This table is crucial when solving equilibrium problems, as it organizes the data and calculations in a clear and logical manner.

Equilibrium Constant

The equilibrium constant (\(K\)) is a number that provides a quantitative measure of the position of equilibrium for a reversible chemical reaction at a given temperature. The value of \(K\) is derived from the ratio of the concentrations of products to the concentrations of reactants, each raised to the power of their coefficients in the balanced chemical equation.

For reaction \(2 \mathrm{NOCl}(g) \rightleftharpoons 2 \mathrm{NO}(g) + \mathrm{Cl}_{2}(g)\), the equilibrium expression is \(K = \frac{[NO]^2 \times [Cl₂]}{[NOCl]^2}\). The equilibrium constant is dimensionless and provides insights into the extent of the reaction: a large \(K\) indicates a reaction that favors the formation of products, whereas a small \(K\) suggests a reaction that favors reactants.

Knowing the equilibrium constant is essential for calculating the equilibrium concentrations of reactants and products. By rearranging the equilibrium expression, we can solve for unknowns given an equilibrium constant and a set of initial conditions. This allows us to predict how the system will respond to changes and to calculate the concentrations of all species at equilibrium, as illustrated in the step-by-step solution of the exercise provided.