Question

The reaction $$ \mathrm{ICl}(g)+\frac{1}{2} \mathrm{H}_{2}(g) \longrightarrow \frac{1}{2} \mathrm{I}_{2}(g)+\mathrm{HCl}(g) $$ is first-order in both reactants. At a certain temperature, the rate of the reaction is \(2.86 \times 10^{-5} \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}\) when \([\mathrm{ICl}]=\) $$ 0.0500 \mathrm{M} \text { and }\left[\mathrm{H}_{2}\right]=0.0150 \mathrm{M} $$ (a) What is the value of \(k\) at that temperature? (b) At what concentration of hydrogen is the rate $$ 2.66 \times 10^{-4} \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s} \text { and }[\mathrm{ICl}]=0.100 \mathrm{M} ? $$ (c) What is [ICl] when the rate is \(0.0715 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}\) and \(\mathrm{H}_{2}=3[\mathrm{ICl}] ?\)

Short answer

Question: Determine the following for a first-order chemical reaction involving ICl and H2: (a) the value of the rate constant, k, when the rate is \(2.86 \times 10^{-5} \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}\), [ICl] = 0.0500 M, and [H2] = 0.0150 M (b) the concentration of hydrogen when the rate is \(2.66 \times 10^{-4} \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}\) and [ICl] = 0.100 M, and (c) the concentration of ICl when the rate is \(0.0715 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}\) and [H2] = 3[ICl]. Answer: (a) The rate constant, k, is \(3.80 \times 10^{-2} \mathrm{~s}^{-1}\). (b) The concentration of hydrogen when the rate is \(2.66 \times 10^{-4} \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}\) and [ICl] = 0.100 M is 5.60×10^-3 M. (c) The concentration of ICl when the rate is \(0.0715 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}\) and [H2] = 3[ICl] is 0.30 M.

Step-by-step solution

(a) Determine the value of k

Given that the reaction is first-order in both reactants, the rate law equation is: Rate = k[ICl][H2]. We are given the rate (\(2.86 \times 10^{-5} \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}\)) and the concentrations of ICl (0.0500 M) and H2 (0.0150 M). Now, we can solve for k: Rate = k[ICl][H2] \(2.86 \times 10^{-5} \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}=k(0.0500\mathrm{M})(0.0150 \mathrm{M})\) To find the value of k, divide both sides of the equation by (0.0500)(0.0150): \(k=\frac{2.86 \times 10^{-5} \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}}{(0.0500\mathrm{M})(0.0150 \mathrm{M})}\) Now, calculate the value of k: k = \(3.80 \times 10^{-2} \mathrm{~s}^{-1}\)

(b) Determine the concentration of hydrogen

Now, we need to find the concentration of hydrogen when the rate is \(2.66 \times 10^{-4} \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}\) and the concentration of ICl is 0.100 M. Based on the rate law equation, we have: Rate = k[ICl][H2] Solving for [H2], \([H_{2}]=\frac{\mathrm{Rate}}{k[\mathrm{ICl}]}=\frac{2.66 \times 10^{-4} \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}}{(3.80 \times 10^{-2} \mathrm{~s}^{-1})(0.100\mathrm{M})}\) Now, calculate the concentration of hydrogen: \([H_{2}]=\)5.60×10^-3 M

(c) Determine the concentration of ICl when rate and the relation between ICl and H2 are given

We are given the rate (\(0.0715 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}\)) and the relationship between [ICl] and [H2] (H2 = 3[ICl]). Using the rate law equation, we can determine the concentration of ICl: Rate = k[ICl][H2] We know that [H2] = 3[ICl], so substitute this relationship into the rate law equation: \(0.0715 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}= 3.80 \times 10^{-2} \mathrm{~s}^{-1}[\mathrm{ICl}](3 [\mathrm{ICl}])\) Simplify the equation: \(0.0715 = 3.80 \times 10^{-2} \cdot 3[\mathrm{ICl}]^{2}\) Next, divide both sides of the equation by 3.80 × 10^-2 × 3: \([\mathrm{ICl}]^{2}=\frac{0.0715}{3.80 \times 10^{-2}\times 3}\) Now, calculate the square root of the result to find the concentration of ICl: \([\mathrm{ICl}]=\sqrt{\frac{0.0715}{3.80 \times 10^{-2}\times 3}}=\mathrm{0.30\,M}\)

Key concepts

Rate of Reaction

Understanding the rate of a chemical reaction is crucial for comprehending how quickly a reaction occurs. The rate of a reaction can be described as the speed at which reactants are converted into products, typically measured in moles per liter per second (mol/L·s). For example, in the given exercise, the rate of the reaction involving ICl and H2 to produce I2 and HCl was given as a numerical value.

When studying the rate of reaction, we consider the concentration of reactants and the conditions such as temperature and pressure which can greatly influence the reaction speed. In practical terms, knowing the rate of a reaction can help in designing reactors and processes in the chemical industry, as well as understanding and controlling natural processes such as digestion in biology.

A key aspect of the rate of reaction is that it changes over time as reactants are consumed. Initially, the rate is generally highest and then decreases as the reactants are depleted. In the exercise solution, calculations are based on a specific moment in the reaction, capturing a snapshot of its rate at distinct reactant concentrations.

Reaction Order

The order of a reaction is a key concept in chemical kinetics that tells us how the rate of a reaction depends on the concentration of the reactants. It is not necessarily related to the stoichiometry of the reaction but rather determined experimentally. For instance, the exercise states that the reaction is first-order with respect to both ICl and H2, meaning the rate of the reaction is directly proportional to the concentration of each of these reactants individually.

To clarify, if a reaction is first-order in a reactant A, doubling the concentration of A will double the rate of the reaction. Similarly, if a reaction is second-order in A, doubling the concentration of A will quadruple the reaction rate. The overall reaction order is the sum of the orders with respect to each reactant, which in this case is two (first-order in ICl plus first-order in H2).

The concept of reaction order is essential when it comes to predicting how changes in concentration affect the rate and when calculating the rate constant, as it impacts the units of this constant. It also helps in determining the mechanism by which the reaction occurs.

Rate Constant

The rate constant, symbolized as k, is a proportionality constant that appears in the rate law of a chemical reaction. It is a measure of the intrinsic reactivity of the reaction system under the given conditions of temperature and pressure. In essence, it lets us compare the rates of different reactions. For a given temperature, the rate constant is fixed and can be used along with the concentrations of the reactants to calculate the reaction rate.

In the exercise, once the rate and concentrations of reactants are known, the rate constant is determined algebraically. It is important to note that the units of the rate constant vary depending on the overall order of the reaction. Since our example pertains to a reaction that is first-order with respect to both ICl and H2, the rate constant has units of s^{-1}, indicating it is a first-order rate constant overall.

The value of the rate constant is critical to predicting reaction behavior over time and is influenced by factors such as temperature, as described by the Arrhenius equation. Knowing the rate constant, as derived in the example, enables adjustments in concentration or conditions to achieve a desired reaction rate for practical applications in chemical engineering and research.