Answer to Question #132313 in Atomic and Nuclear Physics for hass

Question

radium with an atomic mass of 226,has a half life of 800 years. for 0.5g of radium, calculate the number of decays per seconds

Answer 1

The number of decays per second is called total activity that is defined by the formula

$A(t) = -\frac{dN}{dt}= \lambda N = \frac{ln 2}{T_{1/2}}N_0 2^{-t/T_{1/2}}= \frac{ln 2}{T_{1/2}} \frac{m}{\mu} N_A 2^{-t/T_{1/2}}$

$\mu = A_r(Ra) = 226$ g/mol, $t=1$s, $T_{1/2} = 800 \cdot 365 \cdot 24 \cdot 3600 = 2.25 \cdot 10^{10}$ s, $m = 0.5$ g.

$A(t) = \frac{0.69}{2.25 \cdot 10^{10}} \frac{0.5}{226} \cdot 6.02 \cdot 10^{23} \cdot 2^{-1/10^{10}}$ = $\backslash 2^{-10^{-10}} \approx2^{0}= 1 \backslash$ = $0.004 \cdot 10^{13} = 4 \cdot 10^{10}$ decays per second.