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* 60+39=279min` (1 hr = 60 min)

Rate law expression for first order kinetics :

`N=N_o* e^(-\lambda t)`

Taking 'ln' on both the sides, we get

`\ln((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((0.25)/(2))=-\lambda * (279min)`

By rearranging the terms, we get

`\lambda=0.00745min^(-1)`

Radioactive decay follows first order kinetics.

`t_{(1)/(2)}=(0.693)/(\lambda)`

`t_{(1)/(2)}=(0.693)/(0.00745min^(-1))=93.020min=1hr33min`

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