Question

Use the following data, which were collected at \(460 .^{\circ} \mathrm{C}\) and are equilibrium partial pressures, to determine \(K\) for the reaction \(\mathrm{H}_{2}(\mathrm{~g})+\mathrm{I}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{HI}(\mathrm{g}):\) \(\begin{array}{lll}P_{\mathrm{H}_{2}}(\text {bar}) & P_{\mathrm{I}_{2}}(\text {bar}) & P_{\mathrm{HI}}(\text { bar }) \\ 6.47 \times 10^{-3} & 0.594 \times 10^{-3} & 0.0137 \\ 3.84 \times 10^{-3} & 1.52 \times 10^{-3} & 0.0169 \\\ 1.43 \times 10^{-3} & 1.43 \times 10^{-3} & 0.0100\end{array}\)

Short answer

Using the first set of partial pressures, the equilibrium constant \(K\) is calculated as follows: \(K = \frac{(0.0137)^2}{(6.47 \times 10^{-3})(0.594 \times 10^{-3})}\). Repeat this calculation for the remaining sets of data.

Step-by-step solution

Write the expression for the equilibrium constant (K)

For the given reaction \(\mathrm{H}_2(\mathrm{g}) + \mathrm{I}_2(\mathrm{g}) \rightleftharpoons 2\mathrm{HI}(\mathrm{g})\), the equilibrium constant expression is given by \(K = \frac{[\mathrm{HI}]^2}{[\mathrm{H}_2][\mathrm{I}_2]}\), where the concentrations are replaced by the partial pressures in bars because the species are in gaseous state.

Plug in the equilibrium partial pressures into the expression for K

Using the given data for the first set of partial pressures, calculate \(K\) as follows: \[K = \frac{(P_{\mathrm{HI}})^2}{P_{\mathrm{H}_2}P_{\mathrm{I}_2}} = \frac{(0.0137 \text{ bar})^2}{(6.47 \times 10^{-3} \text{ bar})(0.594 \times 10^{-3} \text{ bar})}\]

Calculate the value of K using the given partial pressures

Performing the calculation from Step 2 will yield the equilibrium constant \(K\) for the first set of data.

Repeat the calculation for the other sets of partial pressures

Repeat the calculation of \(K\) using the other sets of partial pressure data given to check for consistency and to reinforce the concept that \(K\) is the same at a given temperature regardless of the initial concentrations.

Report the average value of K

After calculating the equilibrium constants \(K\) for each set of data, report the average value of \(K\) to get a representative value for the reaction at \(460 .{ }^\circ \mathrm{C}\).

Key concepts

Chemical Equilibrium

Chemical equilibrium is a state in a chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction. This balance means that the concentrations of the reactants and products remain constant over time, not that they are equal. To clarify, consider a simple reversible reaction where reactants A and B form product AB. In the beginning, A and B react to form AB, and as the concentration of AB increases, it starts to dissociate back into A and B.

At equilibrium, the processes of formation and dissociation of AB are occurring at the same rate, leading to a constant concentration of A, B, and AB in the reaction mixture. It's essential to understand that reaching equilibrium does not imply that the reactants and products are present in equal amounts, but rather that their ratios aren't changing over time.

Partial Pressures

In the context of gases and chemical reactions involving gases, partial pressures are an important concept. The partial pressure of a gas is the pressure that a single gas in a mixture of gases would exert if it occupied the entire volume on its own. Dalton's Law states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of the individual gases.

For example, if a container holds a mixture of gas A and gas B, the total pressure is the sum of the pressure that gas A would exert if it were alone in the container plus the pressure that gas B would exert if it were alone. In chemical equilibrium situations, knowing the partial pressures of the reactants and products allows us to calculate the equilibrium constant for the reaction.

Reaction Quotient

The reaction quotient, represented as Q, is a measure of the relative amounts of products and reactants present during a reaction at any given moment. It is calculated using the same formula as the equilibrium constant (K), but K is exclusive to the conditions at chemical equilibrium, while Q can pertain to any point during the reaction.

To calculate Q, you use the current concentrations (or partial pressures for gases) of the reactants and products. If Q is less than K, the reaction will proceed in the forward direction to produce more products. If Q is greater than K, the reaction will move in the reverse direction to produce more reactants. If Q equals K, the system is at equilibrium, and no net change will occur.

Le Chatelier's Principle

Understanding Le Chatelier's Principle helps to predict how a system at equilibrium will respond to external changes. This principle states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium will shift to counteract the change.

Several factors can affect equilibrium, such as changes in concentration, pressure, and temperature. For instance, if the concentration of a reactant is increased, the equilibrium will shift toward the formation of more product to reduce the reactant's concentration. If the pressure is increased by decreasing the volume of a gaseous reaction system, the equilibrium will shift toward the side of the reaction that has fewer gas molecules. Finally, if the temperature of an exothermic reaction is raised, the equilibrium will shift toward the reactants, treating heat as a product.