On your birthday, you measure the activity of the sample `""^(210)Bi` which has a half - life of 5.01 days. The initial activity that you measure is `1muCi`. (a) What is the approximate activity of the sample on your next birthday? Calculate (b) the decay constant (c) the mean life (d) initial number of atoms.

Explanation

(a) A year of 365 days is equivalent to `(365 d)/(5.01d) approx 73` half - lives. Thus , the activity will be reduced after one year to approximately `(1/2)^(73)(1.000 mu Ci) approx 10^(-22)mu Ci`.
(b) Initial measure `R_(0) = 1.000 mu Ci`
`= 10^(-6) xx 3.7 xx 10^(10)`
` = 3.7 xx 10^(4) Bq`
After 1 year, the measure R = `10^(-22)mu Ci`
` = 10^(-22) xx 10^(-6) xx 3.7 xx 10^(10)`
` = 3.7 xx 10^(-18) Bq`
decay constant `lambda = (1)/(t) In((R_(0))/(R)) = ((1)/(1 year)) In ((3.7 xx 10^(4))/(3.7 xx 10^(-18)))`
`(1)/(3.156 xx 10^(7)) In(10^(22))`
`lambda = (50.657)/(3.1567 xx 10^(7)) = 1.6 xx 10^(-6) s^(-1)`
(c) Mean life `tau = (1)/(lambda) = (1)/(1.6 xx 10^(-6))s [ 1s = (1)/(86400) days]`
`tau = 7.24` days
(d) Initial number of atoms
`R_(0) = lambda N, N = (R_(0))/(lambda)`
`= (3.7 xx 10^(4))/(1.6 xx 10^(6)), N = 2.31 xx 10^(10)`