Find the largest and shorted wavelengths in the Lyman
series for hydrogen. In what region of the electromagnetic spectrum does each
series lie?

Explanation

To find the largest and shortest wavelengths in the Lyman series for hydrogen, we will use the Rydberg formula for hydrogen spectral lines. The formula is given by:

\[ \frac{1}{\lambda} = RZ^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

where:
- \(\lambda\) is the wavelength,
- \(R\) is the Rydberg constant (\(1.097 \times 10^7 \, \text{m}^{-1}\)),
- \(Z\) is the atomic number (for hydrogen, \(Z = 1\)),
- \(n_1\) is the principal quantum number of the lower energy level,
- \(n_2\) is the principal quantum number of the higher energy level.

### Step 1: Finding the Largest Wavelength
For the Lyman series, the lower energy level is \(n_1 = 1\) and the higher energy level \(n_2\) can take values starting from 2 to infinity.

To find the largest wavelength, we take \(n_2 = 2\):

\[ \frac{1}{\lambda_{\text{max}}} = R \cdot 1^2 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) \]

Calculating this gives:

\[ \frac{1}{\lambda_{\text{max}}} = 1.097 \times 10^7 \left( 1 - \frac{1}{4} \right) = 1.097 \times 10^7 \left( \frac{3}{4} \right) \]

\[ \frac{1}{\lambda_{\text{max}}} = 1.097 \times 10^7 \times \frac{3}{4} = 8.2275 \times 10^6 \, \text{m}^{-1} \]

Now, taking the reciprocal to find \(\lambda_{\text{max}}\):

\[ \lambda_{\text{max}} = \frac{1}{8.2275 \times 10^6} \approx 1.215 \times 10^{-7} \, \text{m} = 1215 \, \text{Å} \]

### Step 2: Finding the Shortest Wavelength
For the shortest wavelength, we take \(n_2\) approaching infinity:

\[ \frac{1}{\lambda_{\text{min}}} = R \cdot 1^2 \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) = R \cdot 1^2 \left( 1 - 0 \right) \]

Thus:

\[ \frac{1}{\lambda_{\text{min}}} = R = 1.097 \times 10^7 \, \text{m}^{-1} \]

Taking the reciprocal gives:

\[ \lambda_{\text{min}} = \frac{1}{1.097 \times 10^7} \approx 9.128 \times 10^{-8} \, \text{m} = 91.28 \, \text{Å} \]

### Step 3: Identifying the Region of the Electromagnetic Spectrum
The Lyman series lies in the ultraviolet region of the electromagnetic spectrum.

### Summary of Results
- Largest wavelength (\(\lambda_{\text{max}}\)): 1215 Å
- Shortest wavelength (\(\lambda_{\text{min}}\)): 91.28 Å
- Region of the electromagnetic spectrum: Ultraviolet region