Question

Nucleus A decays into nucleus B with a half-life of 10 s. At

t = 0 s, there are 1000 A nuclei and no B nuclei. At what time

will there be 750 B nuclei?

Short answer

Therefore, at time$t=13.8629\mathrm{s}$there will be 750 B nuclei.

Step-by-step solution

Given information

Nucleus A decays into nucleus B with a half-life of 10 s. At

t = 0 s, there are 1000 A nuclei and no B nuclei.

Explanation

We have the formula as:

$N=N_o{e}^{-t/\tau }$

The values are as follows:

$$\begin{gathered} \tau =10\mathrm{s} \\ \mathrm{t}=0\mathrm{s} \\ {\mathrm{N}}_{\mathrm{o}}=1000 \\ \mathrm{N}=750 \end{gathered}$$

Calculations

Rearranging the equation and solving for t

$$\begin{gathered} N=N_o{e}^{-t/\tau } \\ \ln N=N_o\ln e^{-t/\tau } \\ \ln N=N_o-t/\tau \\ -\tau \ln N=N_o·t \\ \Rightarrow t=-\tau \ln \left(\frac{N}{N_o}\right) \end{gathered}$$

Substituting the values,

$$\begin{gathered} t=-(10\mathrm{s})\ln \left(\frac{250}{1000}\right) \\ t=-(10\mathrm{s})(-1.38629) \\ t=13.8629\mathrm{s} \end{gathered}$$