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If doing a problem where you use the integrated rate law for a second order reaction
($\frac{1}{[\mathrm{A}]}-\frac{1}{[\mathrm{A_0}]}=kt$) to first find $k$, then use that to find the half-life. Does it matter seconds are used, or minutes, or whatever?
The unit of $k$ is $\frac{\mathrm{concentration}}{\mathrm{unit~time}}$. It is not necessary that you use any specific unit for concentration or time, but most often $t$ is measured in seconds, and the unit of $k$ is $\mathrm{mol~L^{-1}~s^{-1}}$.
