7 46022 This exercise uses the radioactive decay model After

7 4.6.022. This exercise uses the radioactive decay model. After 3 days a sample of radon-222 has decayed to 58% of its original amount. (a) What is the half-life of radon-222? (Round your answer to two decimal places.) days (b) How long will it take the sample to decay to 10% of its original amount? (Round your answer to two decimal places days

Solution

a) For nuclear decay model

N(t) = N_o(t) e^-vt

where N_o(t) is the original amount,  N(t) id the amount after time t and v is decay constant

As per question

a)  N(t) = 0.58 * N_o(t)

and t =3 days

using this in equation N(t) = N_o(t) e^-vt

^=> 0.58 *  N_o(t) =  N_o(t)  e^-v3

=> 0.58 = e^-v3

taking log

=> ln(0.58) = -3v ln(e)

=> v= -ln(0.58)/3

=> v = 0.18157

half - life t_1/2= ln(2)/v

=> t_1/2= ln(2)/0.18157

=> t_{1/2 = 3.81 days}

_{Half life is 3.81 days}

_{b) As per question}

N(t) = 10%. N_o(t)

using this value in equation N(t) = N_o(t) e^-vt

^=> 10%. N_o(t) =  N_o(t) e^-0.18157t

=> 10/100= e^-0.18157t

=> 0.1= e^-0.18157t

taking log

=> ln(0.1)= -0.18157t

=> t= -2.302585093/-0.18157

=> t = 12.68 days

It will take 12.68 days to decay to 10% of original value