Chapter 41, Problem 1P

(a)

To determine

The wavelength of the first three lines.

Answer to Problem 1P

The wavelength of the first three lines is $121.5\text{nm}$, $102.5\text{nm}$ and $97.20\text{nm}$.

Explanation of Solution

Write the expression to calculate the wavelength of the first line.

  $\lambda =\frac{1}{R_H}\left(\frac{n^2}{n^2-1}\right)$

Here, $\lambda$ is the wavelength and $R_H$ is the Rydberg constant.

Substitute $1.097\times {10}^7{\text{m}}^{-1}$ for $R_H$ and 2 for n in the above equation to calculate wavelength corresponds to first line.

  $\begin{array}{c}{\lambda }_1=\frac{1}{1.097\times {10}^7{\text{m}}^{-1}}\left(\frac{2^2}{2^2-1}\right)\\ =121.5\times {10}^{-9}\text{m}\left(\frac{1\text{nm}}{{10}^{-9}\text{m}}\right)\\ =121.5\text{nm}\end{array}$

Here, ${\lambda }_1$ is the wavelength corresponds to first line.

Substitute $1.097\times {10}^7{\text{m}}^{-1}$ for $R_H$ and 3 for n in the above equation to calculate wavelength corresponds to second line.

  $\begin{array}{c}{\lambda }_2=\frac{1}{1.097\times {10}^7{\text{m}}^{-1}}\left(\frac{3^2}{3^2-1}\right)\\ =102.5\times {10}^{-9}\text{m}\left(\frac{1\text{nm}}{{10}^{-9}\text{m}}\right)\\ =102.5\text{nm}\end{array}$

Here, ${\lambda }_2$ is the wavelength corresponds to second line.

Substitute $1.097\times {10}^7{\text{m}}^{-1}$ for $R_H$ and 4 for n in the above equation to calculate wavelength corresponds to third line.

  $\begin{array}{c}{\lambda }_3=\frac{1}{1.097\times {10}^7{\text{m}}^{-1}}\left(\frac{4^2}{4^2-1}\right)\\ =97.20\times {10}^{-9}\text{m}\left(\frac{1\text{nm}}{{10}^{-9}\text{m}}\right)\\ =97.20\text{nm}\end{array}$

Here, ${\lambda }_3$ is the wavelength corresponds to third line.

Conclusion:

Therefore, the wavelength of the first three lines is $121.5\text{nm}$, $102.5\text{nm}$ and $97.20\text{nm}$.

(b)

To determine

Which region of the electromagnetic spectrum does the part (a) lines come from?

Answer to Problem 1P

The three lines are in ultraviolet region.

Explanation of Solution

The wavelength corresponds to the first three lines is $121.5\text{nm}$, $102.5\text{nm}$ and $97.20\text{nm}$.

The electromagnetic spectrum starts from radio waves to gamma waves. In between the visible light with wavelength $400\text{nm}$ to $700\text{nm}$.

The above lines would have lesser wave length compared to the visible and is greater than X-ray wave.

Thus, the above lines would falls under the region of ultraviolet.

Conclusion:

Therefore, the three lines are in ultraviolet region.