Question

Given below are two statements: 

• A. Statement I is false but Statement II is true
• B. Statement I is true but Statement II is false
• C. Both Statement I and Statement II are true
• D. Both Statement I and Statement II are false

Hint:

For first order reactions, the half-life is constant and independent of the initial concentration. Additionally, plotting \( \log [R] \) vs time gives a straight line with a slope related to the rate constant \( k \).

Answer 1

The Correct Option is B

- Statement (I) is false: For a first order reaction, the half-life (\( t_{1/2} \)) is constant and does not depend on the initial concentration. The relationship between \( t_{1/2} \) and \( [R_0] \) is incorrect, as the half-life for a first order reaction is independent of the initial concentration.
- Statement (II) is true: For a first order reaction, the integrated rate law is: \[ \log [R] = \log [R_0] - \frac{k}{2.303} \cdot t. \] Thus, the plot of \( \log [R] \) vs time gives a straight line with a slope of \( -\frac{k}{2.303} \).

Final Answer: Statement I is false but Statement II is true.

Answer 2

Step 1: Analyze Statement I.
Statement I refers to a graph showing the half-life \( t_{1/2} \) versus the concentration of the reactant \( [R] \) for a first-order reaction. For first-order reactions, the half-life is independent of the initial concentration and is constant. This is reflected in the graph, which shows that as the concentration decreases, the half-life remains constant. Therefore, Statement I is correct.

Step 2: Analyze Statement II.
Statement II describes a graph of \( \log \left( \frac{[R]}{[R_0]} \right) \) versus time for a first-order reaction. The equation for a first-order reaction is: \[ \log \left( \frac{[R]}{[R_0]} \right) = -kt. \] Here, the slope of the graph should be \( -k \), not \( k/2.303 \). The factor \( 2.303 \) is used when converting from natural logarithms to base-10 logarithms. Hence, Statement II is incorrect because the slope of the graph is not \( k/2.303 \), but simply \( -k \).

Step 3: Conclusion.
Statement I is correct, but Statement II is incorrect.

Final Answer:
\[ \boxed{\text{Statement I is true but Statement II is false.}} \]