Question

Before $1900$, the activity per mass of atmosphere carbon due to the presence of ${}^{14}C$ averaged about $0.255Bq$ per gram of $C$. What fraction of carbon atoms were ${}^{14}C$?

Given: Half life is $5730$ years.

• A. One ${}^{14}\text{C}$ in every $3\times {10}^{12}$ atoms
• B. Three ${}^{14}\text{C}$ in every ${10}^{12}$ atoms
• C. Four ${}^{14}\text{C}$ in every ${10}^{10}$ atoms
• D. Two ${}^{14}\text{C}$ in every ${10}^{11}$ atoms

Answer 1

The correct option is A One ${}^{14}\text{C}$ in every $3\times {10}^{12}$ atoms

Decay constant, $\lambda =\frac{ln2}{t_{1/2}}=1.209\times {10}^{-4}y{r}^{-1}$

Activity is given by: $\frac{dN}{dt}=\lambda N$

Hence, $N=\frac{dN/dt}{\lambda }=6.66\times {10}^{10}$

Number of carbon atoms in 1 gm is $\frac{6\times {10}^{23}}{12}=5\times {10}^{22}$

Fraction of ${}^{14}C$ is given by:

$\frac{6.66\times {10}^{10}}{5\times {10}^{22}}=1.33\times {10}^{-12}$