Answer
$2.69\times 10^{15}\;\text{moles}$
Work Step by Step
The naturally occurring population ratio $\frac{N_x}{N_0}$ of the two states is due to thermal agitation of the gas atoms
$\frac{N_x}{N_0}=e^{-\frac{(E_x-E_0)}{kT}}$
where, $(E_x-E_0)$ is the energy separation between the two states.
Given:
$(E_x-E_0)=\frac{ch}{\lambda}=\frac{3\times10^{8}\times 6.63\times 10^{-34}}{550\times10^{-9}}\;J=3.616\times10^{-19}\;J=2.26\;eV$
$N_x=10\;\text{atoms}$ and $T=298\;K$
Substituting the given values
$\frac{10}{N_0}=e^{-\frac{2.26}{8.62\times10^{-5}\times298}}$
or, $N_0=10\times e^{\frac{2.26}{8.62\times10^{-5}\times298}}$
$N_0=1.62\times10^{39}\;\text{atoms}$
1 mole neon gas contains avogadro's number $(6.022\times10^{23})$ atoms
Therefore, the number of moles correspond to $1.62\times10^{39}$ number of atoms is:
$n=\frac{1.62\times10^{39}}{6.022\times10^{23}}\;\text{moles}=2.69\times 10^{15}\;\text{moles}$
Therefore, the required neon is $2.69\times 10^{15}\;\text{moles}$
