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

A radioactive nucleus decays by two different processes with half-lives $$t_1 = 3.0$$ hours and $$t_2 = 4.5$$ hours. The decay constant for each process is given by $$\lambda_1 = \frac{\ln 2}{t_1}$$ and $$\lambda_2 = \frac{\ln 2}{t_2}$$. When a nucleus can decay by two independent processes, the effective decay constant is the sum $$\lambda_{eff} = \lambda_1 + \lambda_2$$, so that $$\frac{\ln 2}{t_{eff}} = \frac{\ln 2}{t_1} + \frac{\ln 2}{t_2}$$, which yields $$\frac{1}{t_{eff}} = \frac{1}{t_1} + \frac{1}{t_2}$$.

Substituting the given half-lives gives $$\frac{1}{t_{eff}} = \frac{1}{3.0} + \frac{1}{4.5} = \frac{3}{9} + \frac{2}{9} = \frac{5}{9}$$,
and hence $$t_{eff} = \frac{9}{5} = 1.80 \text{ hours}$$.

Hence, the correct answer is Option D.