Question

The equilibrium constant is \(0.0900\) at \(25^{\circ} \mathrm{C}\) for the reaction $$ \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{Cl}_{2} \mathrm{O}(g) \rightleftharpoons 2 \mathrm{HOCl}(g) $$ For which of the following sets of conditions is the system at equilibrium? For those that are not at equilibrium, in which direction will the system shift? a. A 1.0-L flask contains \(1.0\) mole of \(\mathrm{HOCl}, 0.10\) mole of \(\mathrm{Cl}_{2} \mathrm{O}\), and \(0.10\) mole of \(\mathrm{H}_{2} \mathrm{O}\). b. A \(2.0\) - \(\mathrm{L}\) flask contains \(0.084\) mole of \(\mathrm{HOCl}, 0.080\) mole of \(\mathrm{Cl}_{2} \mathrm{O}\), and \(0.98 \mathrm{~mole}\) of \(\mathrm{H}_{2} \mathrm{O}\). c. A 3.0-L flask contains \(0.25\) mole of HOCl, \(0.0010\) mole of \(\mathrm{Cl}_{2} \mathrm{O}\), and \(0.56 \mathrm{~mole}\) of \(\mathrm{H}_{2} \mathrm{O}\).

Short answer

In summary: a. The system is not at equilibrium, and the reaction shifts to the left. b. The system is not at equilibrium, and the reaction shifts to the right. c. The system is not at equilibrium, and the reaction shifts to the left.

Step-by-step solution

Write the expression for Q

The reaction quotient (Q) can be written in terms of the concentrations of the reactants and products. For this reaction, the expression for Q is: \[Q = \frac{[\mathrm{HOCl}]^2}{[\mathrm{H}_2\mathrm{O}][\mathrm{Cl}_2\mathrm{O}]}\] Now let's analyze each case and determine if the system is at equilibrium, and if not, the direction of the shift.

Calculate Q for case a

In case a, the given concentrations are given as moles in a 1.0-L flask, so we can directly use these values as molar concentrations. \[[\mathrm{HOCl}] = 1.0 \,M, \, [\mathrm{Cl}_2\mathrm{O}] = 0.10 \,M, \, [\mathrm{H}_2\mathrm{O}] = 0.10 \,M\] Calculate Q using these values: \[Q = \frac{(1.0)^2}{(0.10)(0.10)} = 100\] Now compare Q to K: \[Q > K \Rightarrow 100 > 0.0900\] Hence, the reaction will shift to the left (toward the reactants).

Calculate Q for case b

In case b, the given moles are in a 2.0-L flask, so we need to calculate the molar concentrations. \[[\mathrm{HOCl}] = \frac{0.084}{2.0} = 0.042 \,M, \, [\mathrm{Cl}_2\mathrm{O}] = \frac{0.080}{2.0} = 0.040 \,M, \, [\mathrm{H}_2\mathrm{O}] = \frac{0.98}{2.0} = 0.49 \,M\] Calculate Q using these values: \[Q = \frac{(0.042)^2}{(0.040)(0.49)} = 0.0716\] Now compare Q to K: \[Q < K \Rightarrow 0.0716 < 0.0900\] Hence, the reaction will shift to the right (toward the products).

Calculate Q for case c

In case c, the given moles are in a 3.0-L flask, so we need to calculate the molar concentrations. \[[\mathrm{HOCl}] = \frac{0.25}{3.0} = 0.0833 \,M, \, [\mathrm{Cl}_2\mathrm{O}] = \frac{0.0010}{3.0} = 0.000333 \,M, \, [\mathrm{H}_2\mathrm{O}] = \frac{0.56}{3.0} = 0.1867 \,M\] Calculate Q using these values: \[Q = \frac{(0.0833)^2}{(0.000333)(0.1867)} = 108.7\] Now compare Q to K: \[Q > K \Rightarrow 108.7 > 0.0900\] Hence, the reaction will shift to the left (toward the reactants). In summary, a. The system is not at equilibrium, and the reaction shifts to the left. b. The system is not at equilibrium, and the reaction shifts to the right. c. The system is not at equilibrium, and the reaction shifts to the left.

Key concepts

Chemical Equilibrium

Chemical equilibrium is a state in which the rate of the forward reaction equals the rate of the reverse reaction, meaning that the concentrations of the reactants and products remain constant over time. At this point, no net change is taking place in the composition of the mixture, though individual molecules continue to react with each other. An important aspect to note is that equilibrium does not mean that the reactants and products are present in equal amounts, but rather that their ratios do not change.

To use a simple analogy, imagine chemical equilibrium as a busy street where the number of cars going in each direction is the same. Even though cars are constantly moving, the overall picture looks the same from moment to moment.

Reaction Quotient Q

The reaction quotient, or Q, is a measure calculated similarly to the equilibrium constant, but for any mixture of reactants and products, not necessarily at equilibrium. It is determined by plugging the molar concentrations of the reactants and products into the equilibrium expression. If Q equals the equilibrium constant (K), the system is at equilibrium. If Q does not equal K, the reaction will shift in the direction that brings Q closer to the value of K.

Think of Q as a snapshot of where the reaction currently stands in comparison to where it wants to be (at equilibrium). This concept is crucial when predicting the direction in which a reaction will proceed to reach equilibrium.

Le Châtelier’s Principle

Le Châtelier’s principle provides a way to predict how a change in conditions will affect a system at chemical equilibrium. If an external stress, such as changes in concentration, temperature, or pressure, is applied to a system at equilibrium, the system adjusts in such a way as to counteract the imposed change and re-establish equilibrium. This can mean shifting to the right (toward products) or to the left (toward reactants), depending on the specific change made.

It's akin to balancing on a seesaw: if more weight is added on one side, the other side lifts to accommodate and find a new balance point.

Molar Concentration

Molar concentration, often represented as molarity and denoted by the unit M (mole/liter), is a measure of the concentration of a solute in a solution. It indicates the number of moles of a substance per liter of solution. Molar concentration is critical in the calculation of the reaction quotient and the equilibrium constant because it allows us to relate the amount of substance to the volume of the solution, giving us an insight into the 'intensity' or 'strength' of the reaction.

For instance, having a solution with a high molarity of reactants suggests a stronger possibility of reactant molecules colliding and reacting, compared to a solution with a lower molarity of reactants.