Chapter 30, Problem 46P

To determine

Mass of a sample of pure ${}_{19}{}^{40}K$ with an initial decay rate of $2.4\times {10}^5{s}^{-1}$? The half-life of ${}_{19}{}^{40}K$ is $1.248\times {10}^{9}yr$

Answer to Problem 46P

Solution:

Initial mass of ${}_{19}{}^{40}K$$=0.91$ g

Explanation of Solution

Formula used:

The number of nuclei remaining after a time t, $N=N_0{e}^{-\lambda t}$, $N_0$ is the number of nuclei present after time t = 0 and $\lambda$ is the decay constant.

Decay constant, $\lambda =\frac{\ln 2}{T_{1/2}}$, $T_{1/2}$ is the half life

Activity $=\lambda N$, where $\lambda$ is the decay constant and N is the number of nuclei remaining after a time t

Calculation:

We have initial decay rate $=\lambda N_0=2.4\times {10}^5{s}^{-1}$

We also have Decay constant, $\lambda =\frac{\ln 2}{1.248\times {10}^9\times 365\times 24\times 60\times 60}$

Substituting we will get $N_0=\frac{2.4\times {10}^5\times 1.248\times {10}^9\times 365\times 24\times 60\times 60}{\ln 2}$

$=1.364\times {10}^{22}$ nuclei

Initial mass of ${}_{19}{}^{40}K$$=\frac{N_0(atomicweight)}{6.02\times {10}^{23}}$

$=\frac{1.364\times {10}^{22}\times 40}{6.02\times {10}^{23}}=0.91g$