91Ó°ÊÓ

Minimum wavelength has correspondence to the largest transition energy as energy difference is inversely proportional to wavelength.

$$\begin{gathered} \mathrm{Δ·¡}=(\mathrm{Z}−1{)}^2(13.6\mathrm{eV}) \\ ={\left(23-1\right)}^2\left(13.6eV\right) \\ =6.528×{10}^3\mathrm{eV} \\ =1.055×{10}^{−15}\mathrm{J} \end{gathered}$$

Now, the wavelength and transition energy difference is given as,

$∆E=\frac{hc}{\mathrm{λ}}$

Solving for wavelength,

$$\begin{gathered} \mathrm{λ}=\frac{\mathrm{hc}}{\mathrm{Δ·¡}} \\ \mathrm{λ}=\frac{\mathrm{hc}}{\mathrm{Δ·¡}} \\ =\frac{\left(6.626×{10}^{−34}\mathrm{J}â‹\dots \mathrm{s}\right)\left(2.998×{10}^8\mathrm{m}/\mathrm{s}\right)}{1.055×{10}^{−15}\mathrm{J}} \\ =1.88×{10}^{−10}\mathrm{m} \\ =0.188\mathrm{nm} \end{gathered}$$

Maximum wavelength has correspondence to the smallest transition energy and here the wavelength is given by the Moseley’s formula

$$\begin{gathered} f=\left(2.48×{10}^{15}\mathrm{Hz}\right)(\mathrm{Z}−1{)}^2 \\ =\left(2.48×{10}^{15}\mathrm{Hz}\right)(23−1{)}^2 \\ =1.200×{10}^{18}\mathrm{Hz} \end{gathered}$$

Solving for wavelength,

$$\begin{gathered} \mathrm{λ}=\frac{c}{f} \\ =\frac{\left(2.998×{10}^8\mathrm{m}/\mathrm{s}\right)}{1.200×{10}^{18}\mathrm{Hz}} \\ =2.50×{10}^{−10}\mathrm{m} \\ =0.250\mathrm{nm} \end{gathered}$$

Minimum wavelength has correspondence to the largest transition energy as energy difference is inversely proportional to wavelength.

$\begin{array}{r}\mathrm{Δ}E=(Z−1{)}^2(13.6\mathrm{eV})\\ =(45−1{)}^2(13.6\mathrm{eV})\\ =2.663×{10}^4\mathrm{eV}\\ =4.218×{10}^{−15}\mathrm{J}\end{array}$

Now, the wavelength and transition energy difference is given as,

$∆E=\frac{hc}{\mathrm{λ}}$

Solving for wavelength,

$\mathrm{λ}=\frac{hc}{∆E}$

$$\begin{gathered} \mathrm{λ}=\frac{\mathrm{hc}}{\mathrm{Δ·¡}} \\ =\frac{\left(6.626×{10}^{−34}\mathrm{J}â‹\dots \mathrm{s}\right)\left(2.998×{10}^8\mathrm{m}/\mathrm{s}\right)}{4.218×{10}^{−15}\mathrm{J}} \\ =4.71×{10}^{−11}\mathrm{m} \\ =0.0471\mathrm{nm} \end{gathered}$$

Maximum wavelength has correspondence to the smallest transition energy and here the wavelength is given by the Moseley’s formula

$$\begin{gathered} f=\left(2.48×{10}^{15}\mathrm{Hz}\right)(\mathrm{z}−1{)}^2 \\ =\left(2.48×{10}^{15}\mathrm{Hz}\right)(45−1{)}^2 \\ =4.801×{10}^{18}\mathrm{Hz} \end{gathered}$$

Solving for wavelength,

$\mathrm{λ}=\frac{c}{f}$

$$\begin{gathered} =\frac{\left(2.998×{10}^8\mathrm{m}/\mathrm{s}\right)}{4.801×{10}^{18}\mathrm{Hz}} \\ =6.24×{10}^{−11}\mathrm{m} \\ =0.0624\mathrm{nm} \end{gathered}$$