Question

At t =0 all of the atoms in a collection of 10000 atoms are in a excited state whose lifetime is 25 ns. Approximately how many atoms will still be in excited state at t= 12 ns.

Short answer

The number of excited atoms at $t=12\text{ns}$ is $6188$.

Step-by-step solution

Identification of given data

The initial number of atoms in excited state is$N_0=10000$

The lifetime of excited atoms is $\tau =25\text{ns}$

The time for remaining atoms in excited state is$t=12\text{ns}$

The lifetime of the excited atom is the duration in which an excited atom reaches to ground state or an atom in ground state reaches to excited state.

Determination of approximate number of excited atoms

The approximately number of excited atoms is given as:

$N=N_0{e}^{-t/\tau }$

Substitute all the values in the above equation.

$$\begin{gathered} N=\left(10000\right)e^{-\left(\frac{12\text{ns}}{25\text{ns}}\right)} \\ N\approx 6188 \end{gathered}$$

Therefore, the number of excited atoms at $t=12\text{ns}$ is

$6188$ .