Question

It takes 4 hr 39 min for a 2.00-mg sample of radium-230 to decay to 0.25 mg. What is the half-life of radium-230? A) 1 hr 4 min B) 1 hr 17 min C) 1 hr 33 min D) 1 hr 49 min Eli

Answer 1

the half-life of radium-230 is 1 hr 33 min.

Answer 2

Answer : The half-life of radium-230 is, 1 hr 33 min

Solution : Given,

Initial amount of radium-230 = 2.00 mg

Amount left after time, 't' = 0.25 mg

Time = 4 hr 39 min = $4\times 60+39=279min$      (1 hr = 60 min)

Rate law expression for first order kinetics :

$N=N_o\times e^{-\lambda t}$

Taking 'ln' on both the sides, we get

$\ln(\frac{N}{N_o})=-\lambda t$

where,

N = amount left after time t

$N_0$ = initial amount

$\lambda$ = rate constant

t = time

Now put all the given values in the above expression, we get

$\ln(\frac{0.25}{2})=-\lambda \times (279min)$

By rearranging the terms, we get

$\lambda=0.00745min^{-1}$

Radioactive decay follows first order kinetics.

$t_{\frac{1}{2}}=\frac{0.693}{\lambda}$

$t_{\frac{1}{2}}=\frac{0.693}{0.00745min^{-1}}=93.020min=1hr33min$

Therefore, the half-life of radium-230 is, 1 hr 33 min