Question

Few conditions are given regarding concentration of \(\mathrm{R}-\mathrm{X}\) and nucleophile with rate of reaction of \(\left(\mathrm{R}-\mathrm{X}+\mathrm{Nu}^{-} \rightarrow\right.\) Product \()\) \([\mathbf{R}-\mathbf{X}]\) (P) \(0.10\) (Q) \(0.20\) (R) \(0.10\) (S) \(0.20\) \(\left[\mathrm{Nu}^{-}\right]\) \(0.10\) \(0.10\) \(0.20\) \(0.20\) Rate $$ \begin{aligned} &1.2 \times 10^{-4} \\ &2.4 \times 10^{-4} \\ &2.4 \times 10^{-4} \\ &4.8 \times 10^{-4} \end{aligned} $$ The minimum condition regarding to know the correct rate expression is/are (A) Only P (B) Only P, Q (C) \(\mathrm{P}, \mathrm{Q}, \mathrm{R}\) (D) \(\mathrm{P}, \mathrm{Q}, \mathrm{R}\) and \(\mathrm{S}\)

Short answer

The minimum number of conditions required for determining the correct rate expression is P, Q, and R. The reaction is first order with respect to both R-X and Nu⁻.

Step-by-step solution

Two pairs of conditions share this property: P and Q share the same concentration of Nu⁻ but different concentrations of R-X, while P and R share the same concentration of R-X but different concentrations of Nu⁻. #Step 2: Compare rates for conditions P and Q to find the order with respect to R-X#

Both conditions P and Q have the same concentration of Nu⁻ (\(0.10\,\mathrm{M}\)) but different concentrations of R-X (\(0.10\,\mathrm{M}\) and \(0.20\,\mathrm{M}\), respectively). Using the general rate expression (Rate = \(k[\mathrm{R}-\mathrm{X}]^{m}[\mathrm{Nu}^{-}]^{n}\)), we can write the following expressions: For P: Rate = \(k(0.10)^{m}(0.10)^{n}\) = \(1.2 \times 10^{-4}\) For Q: Rate = \(k(0.20)^{m}(0.10)^{n}\) = \(2.4 \times 10^{-4}\) Divide the equation for Q by the equation for P to eliminate the rate constant, k: \(\frac{2.4 \times 10^{-4}}{1.2 \times 10^{-4}}\) = \(\frac{(0.20)^{m}(0.10)^{n}}{(0.10)^{m}(0.10)^{n}}\) Simplifying, we get: \(2 = (0.20)^{m}\) (Since the powers of n cancel each other out) Raising both sides to the power of 1/m, we get: \((2)^{1/m} = (0.20)\) As 0.20 is equal to 2 raised to the power of -1, we find that m = 1. So, the order of reaction with respect to R-X is 1. #Step 3: Compare rates for conditions P and R to find the order with respect to Nu⁻#

Both conditions P and R have the same concentration of R-X (\(0.10\,\mathrm{M}\)) but different concentrations of Nu⁻ (\(0.10\,\mathrm{M}\) and \(0.20\,\mathrm{M}\), respectively). Using the general rate expression (Rate = \(k[\mathrm{R}-\mathrm{X}]^{m}[\mathrm{Nu}^{-}]^{n}\)), we can write the following expressions: For P: Rate = \(k(0.10)^{1}(0.10)^{n}\) = \(1.2 \times 10^{-4}\) (as we found m = 1) For R: Rate = \(k(0.10)^{1}(0.20)^{n}\) = \(2.4 \times 10^{-4}\) Divide the equation for R by the equation for P to eliminate the rate constant, k: \(\frac{2.4 \times 10^{-4}}{1.2 \times 10^{-4}}\) = \(\frac{(0.20)^{n}}{(0.10)^{n}}\) Simplifying, we get: \(2 = (0.20)^{n}\) (Since the powers of m cancel each other out) Raising both sides to the power of 1/n, we get: \((2)^{1/n} = (0.20)\) As 0.20 is equal to 2 raised to the power of -1, we find that n = 1. So, the order of reaction with respect to Nu⁻ is 1. #Step 4: Conclude the minimum number of conditions required for determining the correct rate expression#

As we have determined the correct rate expression for the reaction (m = 1 and n = 1) using only conditions P, Q and R, the correct answer is (C) P, Q, and R.

Key concepts

Rate of Reaction

Understanding the rate of a chemical reaction is crucial for predicting how quickly a reactant converts into a product. It is defined as the change in concentration of a reactant or product per unit time. Mathematically, for a reactant A degrading over time, the rate can be expressed as \[\[\begin{align*}-\frac{d[A]}{dt}\end{align*}\]\] where [A] is the concentration of A and dt is a small increment of time.

In the context of the exercise, the rate of reaction changes when the concentration of reactants changes, indicating a direct correlation between reactant concentration and the speed of the reaction. Observing the effect of varying concentrations of reactants R-X and nucleophile Nu⁻ on the rate provides insight into the relationship between these factors and how they drive the reaction forward.

Reaction Order Determination

To fully grasp the dynamics of a chemical reaction, it's important to understand reaction order. It expresses the relationship between the concentrations of reactants and the rate of reaction. The overall order of a reaction is the sum of the powers to which all reactant concentrations are raised in the rate equation:
Rate = k[A]^m[B]^n

Here, m and n are the orders with respect to reactants A and B, and k is the rate constant. To determine the reaction order, one typically varies the concentration of one reactant while keeping the others constant and observes the effect on the rate.

Using the data from the exercise, comparing scenarios with different concentrations of R-X while keeping Nu⁻ consistent allows calculation of the order with respect to R-X. Similarly, altering Nu⁻ concentrations with steady R-X levels enables finding the order with respect to nucleophile, ultimately revealing that the reaction is first order in both R-X and Nu⁻.

Nucleophilic Substitution

Nucleophilic substitution is a fundamental type of reaction in organic chemistry, where an electron-rich nucleophile (Nu) targets a positively polarized carbon in a substrate (R-X), leading to the exchange of the nucleophile with a leaving group (X). The nature of this reaction is influenced by factors such as the structure of the substrate, the strength of the nucleophile, and the solvent.

There are two main types of nucleophilic substitution reactions, classified as SN1 and SN2. The SN1 reaction involves a two-step mechanism with the formation of a carbocation intermediate, while the SN2 reaction proceeds in one step with a backside attack, leading to inversion of configuration. The rate of SN2 reactions depends on both the nucleophile and the substrate concentrations, which is the case in the exercise provided, signifying that it's likely an SN2 reaction occurring. Understanding the kinetics of these reactions is key in predicting reaction outcomes and optimizing conditions for desired product formation.