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The rate expression for a reaction is \( \frac{-\mathrm{d} C}{\mathrm{~d} t}=\frac{\alpha \cdot C}{1+\beta C} \), where \( \alpha, \beta \) are constants and \( C \) is the concentration of reactant at time \( t \). The half-life for this reaction is
(a) \( \frac{1}{\alpha} \ln 2+\frac{\beta \cdot C_{0}}{2 \alpha} \)
(b) \( \frac{1}{\beta} \ln 2+\frac{\beta \cdot C_{0}}{2 \alpha} \)
(c) \( \frac{\beta \cdot \ln 2}{\alpha} \)
(d) \( \frac{\alpha}{\beta} \)
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