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Chapter 2: Problem 37

For each of the questions, four choices have been provided. Select the correct alternative. For a reaction \(\mathrm{A}+\mathrm{B} \rightarrow \mathrm{C}\), the rate law is written as \(\mathrm{r}=\mathrm{k}[\mathrm{A}]^{2}[\mathrm{~B}] .\) Doubling the concentration of "A" without changing concentration of "B" increases the rate of reaction by (a) 2 times (b) 4 times (c) 8 times (d) 16 times

Short Answer

Expert verified
Answer: 4 times.

Step by step solution

01

Understand the Rate Law

The rate law is given as r=k[A]^2[B]. This equation shows the relationship between the rate of reaction (r) and the concentrations of the reactants A and B, where k is the rate constant.
02

Calculate the Initial Rate

Let's first find the initial rate of the reaction, r_initial, without changing anything. We will assume the initial concentrations of A and B to be [A_initial] and [B_initial], respectively. So, the initial rate law is: r_initial = k[A_initial]^2[B_initial]
03

Calculate the New Rate after Doubling the Concentration of A

Now, we will find the new rate after doubling the concentration of A. Let [A_new] = 2[A_initial] be the new concentration of A. The concentration of B remains unchanged as [B_initial]. Now, the rate law becomes: r_new = k[A_new]^2[B_initial] Substitute [A_new] = 2[A_initial] in the equation, we get: r_new = k[ (2[A_initial])^2] [B_initial]
04

Determine the Rate Increase Factor

Now, let's determine the factor by which the rate of reaction has increased. We will divide the new rate by the initial rate to find the increase factor. Rate Increase Factor = r_new / r_initial Using the equations from Step 2 and Step 3: Rate Increase Factor = (k[(2[A_initial])^2][B_initial]) / (k[A_initial]^2[B_initial])
05

Simplify the Expression and Find the Answer

Now, cancel out the common terms and simplify the expression for the Rate Increase Factor: Rate Increase Factor = [(2[A_initial])^2] / [A_initial]^2 The square of the factor 2 inside the brackets becomes 4: Rate Increase Factor = 4 * [A_initial]^2 / [A_initial]^2 Cancel the terms [A_initial]^2 from the numerator and the denominator: Rate Increase Factor = 4 The rate of the reaction increases by 4 times when the concentration of A is doubled, keeping the concentration of B constant. Thus, the correct answer is (b) 4 times.

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