Question

$\frac{2}{3}$ fraction of a sample disintegrates in $7$ days. How much fraction of it will decay in $21$ days.

• A. 26/27
• B. 35/45
• C. 54/63
• D. 73/67

Answer 1

The correct option is A 26/27

Decayed $2/3$ means remaining $1/3$

using equation $N=N_0{e}^{-\lambda t}$ and putting $N=N_0/3$ with $t=7$

we get $7\lambda =ln3$ or $\lambda =\left(1/7\right)ln3$

put $t=21$ with above value of $\lambda$ we get $N=N_0{e}^{-\left(21/7\right)ln3}=N_0{e}^{-3ln3}=N_0{e}^{-ln{3}^3}=N_0{e}^{-ln27}=\frac{N_0}{e^{ln27}}=N_0/27$

Decayed fraction in 21 days will be $(N_0-N)/N_0=N_0(1-1/27)/N_0=26/27$

Answer is $26/27$