Which of the following expressions correctly represents the rate constant, k, for a first-order reaction with respect to its half-life, t_{1/2}?

• A. k = rac{1}{t_{1/2}}
• B. k = rac{0.693}{t_{1/2}}
• C. k = t_{1/2} imes 0.693
• D. k = rac{2.303}{t_{1/2}}

Answer: (B)

Verified step by step guidance

1. Recall the integrated rate law for a first-order reaction, which relates concentration and time: \([A] = [A]_0 e^{-kt}\).
2. Understand that the half-life \(t_{1/2}\) is the time required for the concentration to decrease to half its initial value, so set \([A] = \frac{1}{2}[A]_0\) in the integrated rate law.
3. Substitute \([A] = \frac{1}{2}[A]_0\) into the equation: \(\frac{1}{2}[A]_0 = [A]_0 e^{-kt_{1/2}}\) and simplify by dividing both sides by \([A]_0\) to get \(\frac{1}{2} = e^{-kt_{1/2}}\).
4. Take the natural logarithm of both sides to solve for \(k\): \(\ln\left(\frac{1}{2}\right) = -kt_{1/2}\), which simplifies to \(-0.693 = -kt_{1/2}\).
5. Rearrange the equation to isolate \(k\): \(k = \frac{0.693}{t_{1/2}}\), which is the correct expression relating the rate constant to the half-life for a first-order reaction.