Which one of the following sets of units is appropriate for a third-order rate constant?

• A. mol \, L^{-1} \, s^{-1}
• B. mol^{-2} \, L^{2} \, s^{-1}
• C. s^{-1}
• D. mol^{-1} \, L \, s^{-1}

Answer: (B)

Verified step by step guidance

1. Recall that the units of a rate constant depend on the overall order of the reaction. For a reaction of order n, the rate law is generally written as \(\text{rate} = k [A]^n\), where the rate has units of concentration per time, typically \(\mathrm{mol \\ L^{-1} \\ s^{-1}}\).
2. Write the general rate law for a third-order reaction: \(\text{rate} = k [A]^3\). Here, the concentration term is raised to the third power.
3. Express the units of each term: the rate has units \(\mathrm{mol \\ L^{-1} \\ s^{-1}}\), and the concentration \([A]\) has units \(\mathrm{mol \\ L^{-1}}\). So, \([A]^3\) has units \(\mathrm{(mol \\ L^{-1})^3 = mol^3 \\ L^{-3}}\).
4. To find the units of the rate constant \(k\), rearrange the rate law to solve for \(k\): \(k = \frac{\text{rate}}{[A]^3}\). Substitute the units: \([k] = \frac{\mathrm{mol \\ L^{-1} \\ s^{-1}}}{\mathrm{mol^3 \\ L^{-3}}} = \mathrm{mol^{-2} \\ L^{2} \\ s^{-1}}\).
5. Compare this derived unit with the options given. The correct units for a third-order rate constant are \(\mathrm{mol^{-2} \\ L^{2} \\ s^{-1}}\).