Question

The decomposition of \(\mathrm{HI}(\mathrm{g})\) is represented by the equation $$2 \mathrm{HI}(\mathrm{g}) \rightleftharpoons \mathrm{H}_{2}(\mathrm{g})+\mathrm{I}_{2}(\mathrm{g})$$ \(\mathrm{HI}(\mathrm{g})\) is introduced into five identical \(400 \mathrm{cm}^{3}\) glass bulbs, and the five bulbs are maintained at \(623 \mathrm{K}\) Each bulb is opened after a period of time and analyzed for \(I_{2}\) by titration with \(0.0150 \mathrm{M} \mathrm{Na}_{2} \mathrm{S}_{2} \mathrm{O}_{3}(\mathrm{aq})\) $$\begin{array}{l} \mathrm{I}_{2}(\mathrm{aq})+2 \mathrm{Na}_{2} \mathrm{S}_{2} \mathrm{O}_{3}(\mathrm{aq}) \longrightarrow \\ \quad \mathrm{Na}_{2} \mathrm{S}_{4} \mathrm{O}_{6}(\mathrm{aq})+2 \mathrm{NaI}(\mathrm{aq}) \end{array}$$ Data for this experiment are provided in the table below. What is the value of \(K_{\mathrm{c}}\) at \(623 \mathrm{K} ?\) $$\begin{array}{llll} \hline & & & \text { Volume } \\ & \text { Initial } & \text { Time } & 0.0150 \mathrm{M} \mathrm{Na}_{2} \mathrm{S}_{2} \mathrm{O}_{3} \\ \text { Bulb } & \text { Mass of } & \text { Bulb } & \text { Required for } \\\ \text { Number } & \mathrm{Hl}(\mathrm{g}), \mathrm{g} & \text { Opened, } \mathrm{h} & \text { Titration, in } \mathrm{mL} \\ \hline 1 & 0.300 & 2 & 20.96 \\ 2 & 0.320 & 4 & 27.90 \\ 3 & 0.315 & 12 & 32.31 \\ 4 & 0.406 & 20 & 41.50 \\ 5 & 0.280 & 40 & 28.68 \\ \hline \end{array}$$

Short answer

The average calculated concentration of iodine from the titration data for each bulb at its specified time gives the value of the equilibrium constant \(K_c\) at 623K.

Step-by-step solution

Understand the Information Given

The decomposition reaction of HI is occurring in this exercise. This process is happening in five identical bulbs, each of which is opened at a different time to measure the quantity of iodine produced. The amount of iodine in each bulb is determined by titration with a sodium thiolsulfate solution. The raw titration data is provided.

Calculate the Concentration of \(I_2\) From the Titration Data

The volume of sodium thiolsulfate solution required for titration can be used to calculate the moles of iodine. The equation for the titration reaction shows that 1 mole of \(I_2\) reacts with 2 moles of \(Na_2S_2O_3\). So, the moles of \(I_2\) can be found by halving the moles of \(Na_2S_2O_3\) used in the titration. The volume of the bulb is 400 cm^3 (or 0.4 dm^3), which can be used to convert the moles of iodine into a concentration. Repeat this calculation for the iodine concentration in each bulb.

Determine the \(K_c\) Expression and Use the Calculated Concentrations to Calculate \(K_c\)

The expression for the equilibrium constant \(K_c\) is given by \[K_c = \frac{[H_2][I_2]}{[HI]^2}\] where [H_2], [I_2] and [HI] are the equilibrium concentrations of hydrogen, iodine and hydrogen iodide respectively. Since the reaction 2HI(g) \(\rightleftharpoons\) H2(g) + I2(g) is in a 1:1 ratio, both [H2] and [I2] are the same, and [HI] is zero at equilibrium. So the expression for \(K_c\) simplifies to \[K_c = [I_2]\] and therefore the calculated concentration of iodine is the value of \(K_c\). Use the calculated concentrations from step 2 to find the value of Kc for each bulb and the mean of these values to get an overall equilibrium constant at 623K.

Key concepts

Chemical Equilibrium

Chemical equilibrium occurs when a chemical reaction proceeds at the same rate in both the forward and reverse directions, leading to no net change in the amount of reactants and products. Equilibrium is a dynamic process, meaning the reactions continue to occur, but since the rates are equal, the concentrations of the substances involved remain constant.

Understanding chemical equilibrium is fundamental to many chemical processes, including those in industrial syntheses, biological systems, and environmental processes. The condition of equilibrium is typically expressed using the equilibrium constant, denoted as Kc for reactions in solution, or Kp when dealing with gases and involving partial pressures.

Titration Analysis

Titration analysis is a common laboratory method of quantitative chemical analysis used to determine the concentration of a known reactant in a solution. By reacting a solution of unknown concentration with a solution of a known concentration, one can calculate the amount of the unknown substance based on the stoichiometry of the chemical reaction.

In the context of our problem, titration is used to determine the concentration of iodine, I₂, produced in the decomposition of hydrogen iodide (HI). The titration involves a reaction with sodium thiosulfate, Na₂S₂O₃, and the volume of sodium thiosulfate used provides the necessary data to calculate the concentration of I₂ at the moment the bulb is opened.

Equilibrium Constant (Kc)

The equilibrium constant (Kc) is a dimensionless value that provides a quantitative measure of the position of equilibrium for a given chemical reaction at a certain temperature. It's determined by the ratio of the concentrations of the products to the reactants, each raised to the power of their stoichiometric coefficients as seen in the balanced chemical equation.

For the reaction at hand, the Kc expression would ideally consider the concentrations of both products (H₂ and I₂) and the reactant (HI). However, since we're only concerned with the point at which the amount of HI remains unchanged (at equilibrium), the Kc simplifies to the concentration of I₂ alone. The expression becomes particularly straightforward because the stoichiometry of H₂ and I₂ is 1:1 in the balanced decomposition equation of HI.

Reaction Quotient

The reaction quotient (Q) is a value that is calculated in the same manner as the equilibrium constant (Kc), but it applies to concentrations that are not necessarily at equilibrium. Q is used to predict which direction a reaction will proceed to reach equilibrium: if Q < Kc, the reaction goes forward; if Q > Kc, the reaction goes backward.

For the HI decomposition experiment, we're dealing directly with equilibrium conditions, and hence, Q becomes equal to Kc. If we were dealing with conditions not at equilibrium, we would calculate Q using the captured concentration data at those specific moments, to understand the reaction's progress towards equilibrium.

Stoichiometry

Stoichiometry is the quantification of reactants and products in chemical reactions. It's the cornerstone of most calculations in chemistry, allowing us to predict yields, determine reactant proportions, and calculate empirical and molecular formulas.

The stoichiometry of the titration reaction of I₂ with Na₂S₂O₃ indicates that it takes two moles of sodium thiosulfate to react completely with one mole of iodine. This stoichiometric relationship is paramount in determining the concentration of I₂ from the volume of Na₂S₂O₃ used in the titration. Without a proper understanding of stoichiometry, these sorts of calculations would not be possible, thus it is critical that students grasp this concept to successfully approach chemical equations and reactions.