Question

A certain gas laser can emit light at wavelength 550 nm, which involves population inversion between ground state and an excited state. At room temperature, how many moles of neon are needed to put 10 atoms in that excited state by thermal agitation?

Short answer

The number of moles of neon that are needed to put 10 atoms in that excited state by thermal agitation is $1.3\times {10}^{15}$.

Step-by-step solution

The given data:

The wavelength of the emitted light,$\mathrm{\lambda }=550\mathrm{nm}$

According to the sample problem-“Population inversion in a laser”, the population ratio at room temperature,$\raisebox{1ex}{$N_x$}\!\left/ \!\raisebox{-1ex}{$N_0$}\right.=1.3\times {10}^{-38}$

The number of atoms to be put in the excited state,${\mathrm{N}}_{\mathrm{x}}=10\mathrm{atoms}$

Avogadro number,${\mathrm{N}}_{\mathrm{A}}=6.02\times {10}^{23}{\mathrm{mol}}^{-1}$

Understanding the concept of a number of moles of a laser gas:

Avogadro constant is defined as the number of particles that make up each mole of an object.

The ceaseless random motion of molecules or other small component particles of a substance that is associated with heat.

The concept of moles given by the Avogadro number calculates the required number of moles of a laser gas that are needed to put the atoms in that excited state by thermal agitation.

Formula:

The number of moles for a number of atoms in excited to that of Avogadro number at room temperature,

$\mathrm{n}=\raisebox{1ex}{${\mathrm{N}}_0$}\!\left/ \!\raisebox{-1ex}{${\mathrm{N}}_{\mathrm{A}}$}\right.$ ….. (1)

Calculation of the number of moles of neon:

Using the given population ratio in equation (1), the number of moles of neon that are needed to put 10 atoms in that excited state by thermal agitation is as follows.

$\begin{array}{l}\mathrm{n}=\frac{{\displaystyle \raisebox{1ex}{${\mathrm{N}}_{\mathrm{x}}$}\!\left/ \!\raisebox{-1ex}{$1.3$}\right.}\times {10}^{-38}}{{\mathrm{N}}_{\mathrm{A}}}\\ =\frac{{\mathrm{N}}_{\mathrm{x}}}{1.3\times {10}^{-38}\times {\mathrm{N}}_{\mathrm{A}}}\end{array}$

Substitute known values in the above equation.

$$\begin{gathered} \mathrm{n}=\frac{10}{\left(1.3\times {10}^{-38}\right)\left(6.02\times {10}^{23}\right)} \\ =1.3\times {10}^{15}\mathrm{mol} \end{gathered}$$

Hence, the value of the number of moles $1.3\times {10}^{15}$.