Question

Consider the reaction \( aX \rightarrow bY \), for which the rate constant at 30°C is \( 1 \times 10^{-3} \, \text{mol}^{-1} \, \text{L} \, \text{s}^{-1} \). Which of the following statements are true?}

• (A) When concentration of \( X \) is increased to four times, the rate of reaction becomes 16 times.
• (B) The reaction is a second order reaction.
• (C) The half-life period is independent of the concentration of \( X \).
• (D) Decomposition of \( \text{N}_2\text{O}_5 \) is an example of the above reaction.
• (E)

  

  is valid for the reaction

Choose the correct answer from the options given below:

• A. A and B Only
• B. A, B and C Only
• C. A, B, D and E Only
• D. C and D Only

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The order of a chemical reaction can be deduced directly from the units of its rate constant (\(k\)).
Once the order is identified, all standard kinetic properties (rate law, half-life formula, and graphical relationships) are fixed.
Step 2: Key Formula or Approach:
General unit of rate constant: \(\text{M}^{1-n}\text{s}^{-1}\) or \((\text{mol L}^{-1})^{1-n}\text{s}^{-1}\), where \(n\) is the order.
For 2nd order: Rate = \(k[\text{X}]^2\) and \(t_{1/2} = \frac{1}{k[\text{X}]_0}\).
Step 3: Detailed Explanation:
Analyze the given rate constant:
\(k = 1 \times 10^{-3} \text{ mol}^{-1}\text{ L s}^{-1} = 1 \times 10^{-3} \text{ M}^{-1}\text{s}^{-1}\).
Equating units: \(1 - n = -1 \implies n = 2\). So, this is a second-order reaction.
Now evaluate the statements:
A. Rate = \(k[\text{X}]^2\). If \([\text{X}] \to 4[\text{X}]\), Rate \(\to k(4[\text{X}])^2 = 16k[\text{X}]^2\). The rate becomes 16 times. (TRUE)
B. As derived from the units, the reaction is a second-order reaction. (TRUE)
C. For a 2nd order reaction, \(t_{1/2} = \frac{1}{k[\text{X}]_0}\). It is inversely dependent on the initial concentration, not independent. (FALSE)
D. The thermal decomposition of \(\text{N}_2\text{O}_5\) is a well-known first-order reaction. (FALSE)
E. A plot of \(\ln([\text{R}]_0 / [\text{R}])\) vs time yielding a straight line represents the integrated rate law for a first-order reaction. For a 2nd order reaction, the linear plot is \(1/[\text{R}]\) vs time. (FALSE)
Therefore, only statements A and B are correct.
Step 4: Final Answer:
The correct choice is A and B Only.