The rate law for the reaction of nitrogen dioxide and chlorine is found to be rate = k [NO2]2[Cl2]. — Kinetics Chemistry Question

STEPS:

1. Identify the given rate law: The rate law for the reaction is given as:
$$\text{rate} = k [\text{NO}_2]^2[\text{Cl}_2] \quad$$
2. Determine the reaction order for each reactant: The exponents of the concentration terms in the rate law represent the reaction order for each individual reactant:
* The reaction is second-order with respect to $\text{NO}_2$ (since its exponent is $2$).
* The reaction is first-order with respect to $\text{Cl}_2$ (since its exponent is $1$).
3. Analyze the effect of doubling $[\text{NO}_2]$: Because the reaction is second-order with respect to $\text{NO}_2$, doubling its concentration multiplies the rate by a factor of:
$$2^2 = 4$$
4. Analyze the effect of doubling $[\text{Cl}_2]$: Because the reaction is first-order with respect to $\text{Cl}_2$, doubling its concentration multiplies the rate by a factor of:
$$2^1 = 2$$
5. Calculate the combined multiplier for the overall rate: When the concentrations of both reactants are doubled simultaneously, their individual kinetic effects are multiplied:
$$\text{Factor increase} = 4 \times 2 = \mathbf{8}$$
*Algebraically:*
$$\text{rate}_{\text{new}} = k (2[\text{NO}_2])^2(2[\text{Cl}_2]) = k (4[\text{NO}_2]^2)(2[\text{Cl}_2]) = 8 \left(k [\text{NO}_2]^2[\text{Cl}_2]\right) = 8 \times \text{rate}_{\text{initial}}$$
This confirms that the overall rate of the reaction changes by a factor of 8, matching Option E.

*

WHY_OTHERS_WRONG:

• A is incorrect: A factor of 2 would only occur if the concentration of $\text{Cl}_2$ was doubled while $[\text{NO}_2]$ remained constant, or if the entire reaction was first-order overall.
• B is incorrect: A factor of 3 is a mathematical mismatch that has no chemical or algebraic basis under these exponential rate laws.
• C is incorrect: A factor of 4 would occur if only the concentration of $\text{NO}_2$ was doubled while $[\text{Cl}_2]$ remained constant, or if the system was second-order overall and both concentration terms were first-order (resulting in a $2 \times 2$ effect).
• D is incorrect: A factor of 6 typically results from an arithmetic error where a student mistakenly adds the individual concentration factors ($4 + 2 = 6$) instead of multiplying them ($4 \times 2 = 8$).