The half-life of a first order reaction:

(A) depends on the reactant concentration raised to the first power

(B) depends on rate constant

(C) is independent of the reactant concentration

(D) is independent of rate constant

Choose the correct answer from the options given below:

B, C

The correct answer is option 2. B, C.

Let us delve deeper into the concept of half-life in the context of first-order reactions and analyze each statement carefully.

For a first-order reaction, the rate of reaction depends linearly on the concentration of a single reactant. The general form of a first-order reaction is:

\(\text{A} \rightarrow \text{Products}\)

The rate law for a first-order reaction is expressed as:

\(\text{Rate} = k[\text{A}]\)

where:

\( [\text{A}] \) is the concentration of the reactant A,

\( k \) is the rate constant.

The integrated rate law for a first-order reaction is:

\([\text{A}] = [\text{A}_0] e^{-kt}\)

where:

\( [\text{A}] \) is the concentration of A at time \( t \),

\( [\text{A}_0] \) is the initial concentration of A,

\( k \) is the rate constant,

\( t \) is the time.

The half-life (\( t_{1/2} \)) is the time required for the concentration of the reactant to decrease to half of its initial value:

\([\text{A}] = \frac{[\text{A}_0]}{2}\)

Substituting this into the integrated rate law:

\(\frac{[\text{A}_0]}{2} = [\text{A}_0] e^{-kt_{1/2}}\)

Dividing both sides by \( [\text{A}_0] \) and solving for \( t_{1/2} \):

\(\frac{1}{2} = e^{-kt_{1/2}}\)

Taking the natural logarithm of both sides:

\(\ln\left(\frac{1}{2}\right) = -kt_{1/2}\)

\(-\ln(2) = -kt_{1/2}\)

\(t_{1/2} = \frac{\ln(2)}{k}\)

Since \( \ln(2) \approx 0.693 \), the half-life for a first-order reaction can be expressed as:

\(t_{1/2} = \frac{0.693}{k}\)

Analysis of the Statements

(A) Depends on the reactant concentration raised to the first power:

The half-life \( t_{1/2} \) derived above shows that it depends only on the rate constant \( k \) and not on the concentration of the reactant. In other words, for a first-order reaction, the half-life is independent of the initial concentration of the reactant. This statement is therefore incorrect.

(B) Depends on rate constant:

The formula \( t_{1/2} = \frac{0.693}{k} \) clearly shows that the half-life is inversely proportional to the rate constant \( k \). This means that the half-life directly depends on the rate constant: a higher rate constant results in a shorter half-life, and vice versa. This statement is correct.

(C) Is independent of the reactant concentration:

As seen from the half-life formula, \( t_{1/2} \) is only a function of the rate constant \( k \) and does not depend on the initial or any other concentration of the reactant. This is a unique property of first-order reactions, where the half-life remains constant regardless of the amount of reactant present. Therefore, this statement is correct.

(D) Is independent of rate constant:

The half-life \( t_{1/2} \) depends on \( k \), as shown in the formula \( t_{1/2} = \frac{0.693}{k} \). Therefore, the half-life is not independent of the rate constant. This statement is **incorrect**.

Conclusion: Based on the detailed analysis:

B is correct because the half-life depends on the rate constant.

C is correct because the half-life is independent of the reactant concentration.

A and D are incorrect.

Thus, the correct answer is: Option 2: B, C.