Question

Match the plots of certain chemical reactions given in List-I with their characteristics given in List-II and choose the correct options

	List-I (Plots)		List-II (Characteristics)
(P)	$(a-x)^{-2}$ vs time plot is linear with slope = 2k	(1)	Zero order reaction
(Q)	$(\frac{t_1}{2})$ vs $[A]_0$ is linear with slope = $(2k)^{-1}$	(2)	First order reaction
(R)	$ln(a-x)$ vs time plot is linear with slope = -k	(3)	Second order reaction
(S)	$(a-x)^{-1}$ vs time plot is linear with slope = k	(4)	Fractional order reaction
		(5)	Third order reaction

• A. (P) - (1)
• B. (Q) - (2)
• C. (R) - (3)
• D. (S) - (4)
• E. (P) - (5), (Q) - (1), (R) - (2), (S) - (3)

Answer 1

Answer: (E)

P-5, Q-1, R-2, S-3

Answer 2

The problem requires matching plots of chemical reactions from List-I with their corresponding characteristics (order of reaction) from List-II. We will analyze the integrated rate laws and half-life expressions for different orders of reactions.

Let $[A]_0$ (or $a$) be the initial concentration and $[A]_t$ (or $a-x$) be the concentration at time $t$.

1. Zero Order Reaction:

• Integrated Rate Law: $[A]_t = [A]_0 - kt$ or $x = kt$.
  • A plot of $[A]_t$ vs time is linear with slope $-k$.
• Half-life ($t_{1/2}$): $t_{1/2} = \frac{[A]_0}{2k}$.
  • A plot of $t_{1/2}$ vs $[A]_0$ is linear with slope $\frac{1}{2k}$.

2. First Order Reaction:

• Integrated Rate Law: $\ln[A]_t = \ln[A]_0 - kt$ or $\ln(a-x) = \ln a - kt$.
  • A plot of $\ln[A]_t$ (or $\ln(a-x)$) vs time is linear with slope $-k$.
• Half-life ($t_{1/2}$): $t_{1/2} = \frac{\ln 2}{k} = \frac{0.693}{k}$. (Independent of initial concentration)

3. Second Order Reaction:

• Integrated Rate Law: $\frac{1}{[A]_t} = \frac{1}{[A]_0} + kt$ or $\frac{1}{(a-x)} = \frac{1}{a} + kt$.
  • A plot of $\frac{1}{[A]_t}$ (or $\frac{1}{(a-x)}$) vs time is linear with slope $k$.
• Half-life ($t_{1/2}$): $t_{1/2} = \frac{1}{k[A]_0}$.

4. Third Order Reaction (for $3A \rightarrow Products$):

• Integrated Rate Law: $\frac{1}{2[A]_t^2} - \frac{1}{2[A]_0^2} = kt$ or $\frac{1}{2(a-x)^2} - \frac{1}{2a^2} = kt$.
  • Rearranging, $\frac{1}{(a-x)^2} = \frac{1}{a^2} + 2kt$.
  • A plot of $(a-x)^{-2}$ vs time is linear with slope $2k$.

Now let's match the given plots from List-I with their characteristics from List-II:

• (P) $(a-x)^{-2}$ vs time plot is linear with slope = 2k: This corresponds to the integrated rate law for a Third order reaction. Therefore, (P) matches with (5).

• (Q) $(\frac{t_1}{2})$ vs $[A]_0$ is linear with slope = $(2k)^{-1}$: This describes the half-life dependence on initial concentration for a Zero order reaction. Therefore, (Q) matches with (1).

• (R) $ln(a-x)$ vs time plot is linear with slope = -k: This is the integrated rate law for a First order reaction. Therefore, (R) matches with (2).

• (S) $(a-x)^{-1}$ vs time plot is linear with slope = k: This is the integrated rate law for a Second order reaction. Therefore, (S) matches with (3).

Final Matching:

• (P) - (5)
• (Q) - (1)
• (R) - (2)
• (S) - (3)

The question asks to choose the correct options. Since it's a matching question, the correct options are the pairs derived above.

The final answer is $\boxed{P-5, Q-1, R-2, S-3}$

Explanation of the solution: The solution involves identifying the order of reaction based on the linearity of specific plots derived from integrated rate laws and half-life expressions.

1. P: The plot of $(a-x)^{-2}$ versus time being linear with slope $2k$ is characteristic of a third-order reaction.
2. Q: For a zero-order reaction, the half-life ($t_{1/2}$) is directly proportional to the initial concentration ($[A]_0$), specifically $t_{1/2} = \frac{[A]_0}{2k}$. Thus, a plot of $t_{1/2}$ vs $[A]_0$ is linear with slope $\frac{1}{2k}$.
3. R: The integrated rate law for a first-order reaction is $\ln(a-x) = \ln a - kt$. Therefore, a plot of $\ln(a-x)$ vs time is linear with a slope of $-k$.
4. S: The integrated rate law for a second-order reaction (of type $A \rightarrow Products$) is $\frac{1}{(a-x)} = \frac{1}{a} + kt$. Thus, a plot of $(a-x)^{-1}$ vs time is linear with a slope of $k$.