Question

The half-life for decay of ${}^{14}C$ by $\beta$-emission is $5730$ year. The fraction of ${}^{14}C$ decays, in a sample that is $22920$ year old, would be:

• A. $\frac{1}{8}$
• B. $\frac{1}{16}$
• C. $\frac{7}{8}$
• D. $\frac{15}{16}$

Answer 1

Answer: (D)

$\frac{15}{16}$

Number of half life involved, $n=\frac{totaltime\left(t\right)}{halftime\left(t_{1/2}\right)}=\frac{22920}{5730}=4$

Left amount, $N=N_0{\left(\frac{1}{2}\right)}^n$

$N=N_0{\left(\frac{1}{2}\right)}^4=\frac{N_0}{16}$

$\therefore$ Decayed fraction $x=N_0-\frac{N_0}{16}$

$x=\frac{16N_0-N_0}{16}=\frac{15N_0}{16}$