Determine the minimum wavelength that hydrogen in its ground state can obsorb. What would be the next smaller wavelength that would work?

Options

A. `133 nm`
B. `13.3 nm`
C. `10.3 nm`
D. `103 nm`

Explanation

To determine the minimum wavelength that hydrogen in its ground state can absorb, we will use the Rydberg formula for hydrogen:

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

Where:
- \(\lambda\) is the wavelength,
- \(R\) is the Rydberg constant (\(R \approx 1.097 \times 10^7 \, \text{m}^{-1}\)),
- \(n_1\) is the lower energy level,
- \(n_2\) is the higher energy level.

### Step 1: Identify the ground state and the first excited state
For hydrogen in its ground state, we have:
- \(n_1 = 1\) (ground state)
- The first excited state is \(n_2 = 2\).

### Step 2: Apply the Rydberg formula for the minimum wavelength
Substituting the values into the Rydberg formula:

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

Calculating the right side:

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

\[ \frac{1}{\lambda} = 1.097 \times 10^7 \times 0.75 = 8.2275 \times 10^6 \, \text{m}^{-1} \]

### Step 3: Calculate \(\lambda\)
Now, taking the reciprocal to find \(\lambda\):

\[ \lambda = \frac{1}{8.2275 \times 10^6} \approx 1.215 \times 10^{-7} \, \text{m} = 121.5 \, \text{nm} \]

### Step 4: Determine the next smaller wavelength
For the next smaller wavelength, we will use \(n_1 = 1\) and \(n_2 = 3\):

\[ \frac{1}{\lambda} = R \left( \frac{1}{1^2} - \frac{1}{3^2} \right) \]

Calculating this:

\[ \frac{1}{\lambda} = 1.097 \times 10^7 \left( 1 - \frac{1}{9} \right) = 1.097 \times 10^7 \left( \frac{8}{9} \right) \]

\[ \frac{1}{\lambda} = 1.097 \times 10^7 \times \frac{8}{9} \approx 9.748 \times 10^6 \, \text{m}^{-1} \]

### Step 5: Calculate the next \(\lambda\)
Taking the reciprocal:

\[ \lambda = \frac{1}{9.748 \times 10^6} \approx 1.025 \times 10^{-7} \, \text{m} = 102.5 \, \text{nm} \]

### Final Answers
- The minimum wavelength that hydrogen in its ground state can absorb is approximately **121.5 nm**.
- The next smaller wavelength that would work is approximately **102.5 nm**.