Problem 90 - Chemistry The Central Science In Si Units

Question

Hydrogen sulfide \(\left(\mathrm{H}_{2} \mathrm{~S}\right)\) is a common and troublesome pollutant in industrial wastewaters. One way to remove \(\mathrm{H}_{2} \mathrm{~S}\) is to treat the water with chlorine, in which case the following reaction occurs: $$ \mathrm{H}_{2} \mathrm{~S}(a q)+\mathrm{Cl}_{2}(a q) \longrightarrow \mathrm{S}(s)+2 \mathrm{H}^{+}(a q)+2 \mathrm{Cl}^{-}(a q) $$ The rate of this reaction is first order in each reactant. The rate constant for the disappearance of \(\mathrm{H}_{2} \mathrm{~S}\) at \(30^{\circ} \mathrm{C}\) is \(4.0 \times 10^{-2} \mathrm{M}^{-1} \mathrm{~s}^{-1}\). If at a given time the concentration of \(\mathrm{H}_{2} \mathrm{~S}\) is \(2.5 \times 10^{-4} \mathrm{M}\) and that of \(\mathrm{Cl}_{2}\) is \(2.0 \times 10^{-2} \mathrm{M},\) what is the rate of formation of \(\mathrm{H}^{+} ?\)

Step-by-Step Solution

Answer

The rate of formation of \( \mathrm{H}^{+} \) is \( 4.0 \times 10^{-7} \text{ M/s} \).

Step 1: Understanding Reaction Orders

The reaction is first-order in each reactant: hydrogen sulfide (H₂S) and chlorine (Cl₂). This means the reaction rate is directly proportional to the product of the concentrations of H₂S and Cl₂.

Step 2: Rate Law Definition

The rate of the reaction can be expressed using the rate law: \[ \text{Rate} = k [\mathrm{H}_{2} \mathrm{~S}][\mathrm{Cl}_{2}] \] where \( k \) is the rate constant, \([\mathrm{H}_{2} \mathrm{~S}]\) is the concentration of hydrogen sulfide, and \([\mathrm{Cl}_{2}]\) is the concentration of chlorine.

Step 3: Substitute Values Into Rate Law

Substitute the given values into the rate law: - \( k = 4.0 \times 10^{-2} \; \text{M}^{-1} \text{s}^{-1} \)- \([\mathrm{H}_{2} \mathrm{~S}] = 2.5 \times 10^{-4} \; \mathrm{M} \)- \([\mathrm{Cl}_{2}] = 2.0 \times 10^{-2} \; \mathrm{M} \)This gives: \[ \text{Rate} = (4.0 \times 10^{-2} \; \text{M}^{-1} \text{s}^{-1})(2.5 \times 10^{-4} \; \mathrm{M})(2.0 \times 10^{-2} \; \mathrm{M}) \]

Step 4: Calculate the Reaction Rate

Calculate the reaction rate using the substituted values: \[ \text{Rate} = 4.0 \times 10^{-2} \times 2.5 \times 10^{-4} \times 2.0 \times 10^{-2} \]Simplify the calculations: - First, multiply the concentrations: \[ 2.5 \times 10^{-4} \times 2.0 \times 10^{-2} = 5.0 \times 10^{-6} \]- Then multiply this with the rate constant: \[ \text{Rate} = 4.0 \times 10^{-2} \times 5.0 \times 10^{-6} \] \[ \text{Rate} = 2.0 \times 10^{-7} \text{ M/s} \]

Step 5: Determine the Rate of H⁺ Formation

The stoichiometry of the reaction shows that for every mole of H₂S that reacts, two moles of H⁺ ions are formed. Thus, the rate of formation of H⁺ is twice the rate of disappearance of H₂S, which is equal to the computed rate.Therefore, the rate of formation of H⁺ is: \[ 2 \times 2.0 \times 10^{-7} = 4.0 \times 10^{-7} \, \text{M/s} \]

Key Concepts

Rate Law

In chemical kinetics, the rate law is a mathematical equation that relates the rate of a chemical reaction to the concentration of its reactants. It helps us understand how fast a reaction progresses under certain conditions. For the reaction between hydrogen sulfide (H₂S) and chlorine (Cl₂), the rate law is expressed as:

• \[ \text{Rate} = k [\mathrm{H}_{2} \mathrm{~S}][\mathrm{Cl}_{2}] \]
• Where \( k \) is the rate constant.

The concentrations of H₂S and Cl₂ appear as products in this equation, which means the reaction rate depends on their amounts present in the reaction mixture. This expression is derived from experimental observations and fits the given reaction mechanism. By substituting the values of the rate constant and concentrations, the rate at which chemicals react can be determined.

Reaction Order

The concept of reaction order helps us identify how the concentration of reactants affects the reaction rate. In this exercise, the reaction is first-order in hydrogen sulfide ( H₂S) and first-order in chlorine ( Cl₂). This means:

• Each reactant's concentration affects the rate proportionally.
• If the concentration of either H₂S or Cl₂ doubles, the reaction rate doubles.

The overall reaction order is the sum of the individual orders, which in this case is two (first-order for H₂S plus first-order for Cl₂).
This information is crucial for predicting how changes in concentration can speed up or slow down the reaction. Reaction order must be determined through experiments and cannot merely be deduced from a reaction equation.

Rate Constant

The rate constant, symbolized as \( k \), is a crucial component of the rate law equation and dictates the speed of a reaction at a given temperature. For our reaction:

• \( k = 4.0 \times 10^{-2} \; \text{M}^{-1} \text{s}^{-1} \)
• It shows the relationship between reactant concentrations and the reaction rate.

Its units vary depending on the overall order of the reaction it pertains to. Here, the units \( \text{M}^{-1} \text{s}^{-1} \) correspond to a second-order reaction. A higher rate constant suggests a faster reaction at the same conditions. As the temperature changes, so does the rate constant, emphasizing the need to specify it along with particular conditions. Calculating the rate constant involves using experimental data to fit the rate of reaction to observed changes in concentration over time.

Stoichiometry

Stoichiometry involves the quantitative relationships between reactants and products in a chemical reaction. It gives us a ratio of reactants to products, using balanced chemical equations. In the given reaction:

• \(\mathrm{H}_{2} \mathrm{~S}(a q)+\mathrm{Cl}_{2}(a q) \longrightarrow\mathrm{S}(s)+2 \mathrm{H}^{+}(a q)+2 \mathrm{Cl}^{-}(a q)\)
• 1 mole of H₂S reacts with 1 mole of Cl₂ to produce 2 moles of H⁺ and 2 moles of Cl⁻.

The stoichiometry of this reaction reveals that two moles of hydronium ions (H⁺) are produced per mole of hydrogen sulfide. This understanding is key to determining the rate of formation of products like H⁺ from the reaction rate. By knowing the stoichiometry, we can predict how much of each product is formed, given a certain amount of reactants. This foundational concept allows chemists to calculate precise amounts needed for reactions and interpret reaction outcomes effectively.