Question

The average life of a $W$ $gm$ sample of ${}^{200}Ra$ is $T$ seconds and average energy of the $\beta$-particles emitted is $E$ MeV. At what rate in watts does the sample emit energy?

• A. $\frac{8W{N}_{0}E}{T}\times {10}^{-16}$
• B. $\frac{8\left(\ln 2\right)W{N}_{0}E}{T}\times {10}^{-13}$
• C. $\frac{8W{N}_{0}E}{T}\times {10}^{-13}$
• D. None is correct

Answer 1

Answer: (A)

$\frac{8W{N}_{0}E}{T}\times {10}^{-16}$

$E_T=\frac{N_o\times M_{sample}\times E_{sample}}{T}$

Number of atoms =

$N_A\times \frac{W}{200}$

$={10}^{21}\times \frac{6.023W}{200}$

$=3.01W\times {10}^{21}{N}_0$particles

$=\frac{3.01W\times {10}^{21}{N}_0}{T}$particles/sec \quad decay

Now energy released with a single $\beta$ decay is $2.667\times {10}^{-37}MeV$ and average energy released is $E\times {10}^{3}eV$

$E_T=\frac{3.01W\times {10}^{21}{N}_0\times E\times {10}^{-37}\times 2.667}{T}$

$=\frac{8W{N}_{0}E\times {10}^{-16}}{T}$