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(D) The total decay constant $\lambda_{	ext{eq}}$ is the sum of the individual decay constants for the two processes: $\lambda_{	ext{eq}} = \lambda_

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(D) The total decay constant $\lambda_{	ext{eq}}$ is the sum of the individual decay constants for the two processes: $\lambda_{	ext{eq}} = \lambda_1 + \lambda_2$.Since the decay constant $\lambda$ is related to the half-life $T_{1/2}$ by the formula $\lambda = \frac{\ln 2}{T_{1/2}}$,we can write:$\frac{\ln 2}{(T_{1/2})_{	ext{eq}}} = \frac{\ln 2}{(T_{1/2})_1} + \frac{\ln 2}{(T_{1/2})_2}$.Dividing both sides by $\ln 2$,we get the formula for the effective half-life:$(T_{1/2})_{	ext{eq}} = \frac{(T_{1/2})_1 	imes (T_{1/2})_2}{(T_{1/2})_1 + (T_{1/2})_2}$.Substituting the given values $(T_{1/2})_1 = 3.0 \, hours$ and $(T_{1/2})_2 = 4.5 \, hours$:$(T_{1/2})_{	ext{eq}} = \frac{3.0 	imes 4.5}{3.0 + 4.5} = \frac{13.5}{7.5} = 1.8 \, hours$.

$A$ radioactive nucleus can decay by two different processes. The half-life for the first process is $3.0 \, hours$ while it is $4.5 \, hours$ for the second process. The effective half-life of the nucleus will be $......... \, hours.$

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