The visible lines of the Balmer series were observed first because they are most easily seen. Show that the wavelengths of spectral lines in the Lyman (n = 1) and Paschen (n = 3) series are not in the visible region. Find the wavelengths of the four visible atomic hydrogen lines. Assume the visible

Question

The visible lines of the Balmer series were observed first because they are most easily seen. Show that the wavelengths of spectral lines in the Lyman (n = 1) and Paschen (n = 3) series are not in the visible region. Find the wavelengths of the four visible atomic hydrogen lines. Assume the visible wavelength region is [latex]\lambda=400-700[/latex] nm.

Strategy We use Equation (3.13) to determine the various wavelengths for n = 1, 2, and 3. If the wavelengths are between 400 and 700 nm, we conclude they are in the visible region. Otherwise, they are not visible.

[latex]\frac{1}{\lambda}=R_{ H }\left(\frac{1}{n^{2}}-\frac{1}{k^{2}}\right)[/latex] (3.13)

Accepted Answer

We use Equation (3.13) first to examine the Lyman series (n = 1):

[latex]\begin{aligned}\frac{1}{\lambda} &=R_{ H }\left(1-\frac{1}{k^{2}} ight) \&=1.0968 imes 10^{7}\left(1-\frac{1}{k^{2}} ight) m ^{-1}\end{aligned}[/latex]

[latex]\begin{aligned}k=2: & \frac{1}{\lambda}=1.0968 imes 10^{7}\left(1-\frac{1}{4} ight) m ^{-1} \& \lambda=1.216 imes 10^{-7} m =121.6 nm ext { (Ultraviolet) }\end{aligned}[/latex]

[latex]\begin{aligned}k=3: \frac{1}{\lambda} &=1.0968 imes 10^{7}\left(1-\frac{1}{9} ight) m ^{-1} \\lambda &=1.026 imes 10^{-7} m =102.6 nm ext { (Ultraviolet) }\end{aligned}[/latex]

Because the wavelengths are decreasing for higher k values, all the wavelengths in the Lyman series are in the ultraviolet region and not visible by eye.

For the Balmer series (n = 2) we find

[latex]\begin{aligned}k=3: \frac{1}{\lambda} &=1.0968 imes 10^{7}\left(\frac{1}{4}-\frac{1}{9} ight) m ^{-1} \\lambda &=6.565 imes 10^{-7} m =656.5 nm ( ext { Red })\end{aligned}[/latex]

[latex]\begin{aligned}k=4: & \frac{1}{\lambda}=1.0968 imes 10^{7}\left(\frac{1}{4}-\frac{1}{16} ight) m ^{-1} \& \lambda=4.863 imes 10^{-7} m =486.3 nm ext { (Blue-green) }\end{aligned}[/latex]

[latex]\begin{aligned}k=5: & \frac{1}{\lambda}=1.0968 imes 10^{7}\left(\frac{1}{4}-\frac{1}{25} ight) m ^{-1} \\lambda &=4.342 imes 10^{-7} m =434.2 nm ext { (Violet) }\end{aligned}[/latex]

[latex]\begin{aligned}k=6: \frac{1}{\lambda} &=1.0968 imes 10^{7}\left(\frac{1}{4}-\frac{1}{36} ight) m ^{-1} \\lambda &=4.103 imes 10^{-7} m =410.3 nm ext { (Violet) }\end{aligned}[/latex]

[latex]\begin{aligned}k=7: & \frac{1}{\lambda}=1.0968 imes 10^{7}\left(\frac{1}{4}-\frac{1}{49} ight) m ^{-1} \\lambda &=3.971 imes 10^{-7} m =397.1 nm ext { (Ultraviolet) }\end{aligned}[/latex]

Therefore k = 7 and higher k values will be in the ultraviolet region. The four lines k = 3, 4, 5, and 6 of the Balmer series are visible, although the 410-nm (k = 6) line is difficult to see because it is barely in the visible region and is weak in intensity.

The next series, n = 3, named after Paschen, has wavelengths of

[latex]\begin{aligned}k=4: & \frac{1}{\lambda}=1.0968 imes 10^{7}\left(\frac{1}{9}-\frac{1}{16} ight) m ^{-1} \&\lambda=1.876 imes 10^{-6} m =1876 nm ( ext { Infrared })\end{aligned}[/latex]

[latex]\begin{aligned}k=5: \quad \frac{1}{\lambda} &=1.0968 imes 10^{7}\left(\frac{1}{9}-\frac{1}{25} ight) m ^{-1} \\lambda &=1.282 imes 10^{-6} m =1282 nm ( ext { Infrared })\end{aligned}[/latex]

[latex]\begin{aligned}k=\infty:& \frac{1}{\lambda}=1.0968 imes 10^{7}\left(\frac{1}{9}-\frac{1}{\infty} ight) m ^{-1} \&\lambda=8.206 imes 10^{-7} m =820.6 nm ( ext { Infrared })\end{aligned}[/latex]

Thus the Paschen series has wavelengths entirely in the infrared region. The series limit is the smallest wavelength that can occur for each series (see Problem 9). Notice that the series limit is found for [latex]k=\infty[/latex] and is equal to 820.6 nm for the Paschen series. The higher series, [latex]n \geq 4[/latex], will all have wavelengths above the visible region.

Question 3.3: The visible lines of the Balmer series were observed first b...

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