Determine the wavelength of incident electromagnetic radiation required to cause an electron transition from the n = 6 to the n=8 level in a hydrogen atom.

Options

A. `1.2 xx 10^(3)` nm
B. `7.5 xx 10^(3)` nm
C. `2.2 xx 10^(3)` nm
D. `5.9 xx 10^(3)` nm

Answer

To determine the wavelength of incident electromagnetic radiation required to cause an electron transition from the n = 6 to the n = 8 level in a hydrogen atom, we can follow these steps:

### Step 1: Understand the Transition
The transition is from the n = 6 level (initial state) to the n = 8 level (final state) in a hydrogen atom. We need to calculate the energy difference between these two levels.

### Step 2: Use the Rydberg Formula
The Rydberg formula for the wavelength of emitted or absorbed light during an electron transition in hydrogen is given by:

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

Where:
- \( \lambda \) is the wavelength,
- \( R \) is the Rydberg constant (\( R \approx 1.1 \times 10^7 \, \text{m}^{-1} \)),
- \( n_1 \) is the lower energy level (6 in this case),
- \( n_2 \) is the higher energy level (8 in this case).

### Step 3: Substitute the Values
Substituting the values into the formula:

\[ \frac{1}{\lambda} = 1.1 \times 10^7 \left( \frac{1}{6^2} - \frac{1}{8^2} \right) \]

Calculating \( \frac{1}{6^2} \) and \( \frac{1}{8^2} \):

\[ \frac{1}{6^2} = \frac{1}{36} \quad \text{and} \quad \frac{1}{8^2} = \frac{1}{64} \]

### Step 4: Calculate the Difference
Now calculate the difference:

\[ \frac{1}{36} - \frac{1}{64} \]

To compute this, find a common denominator (which is 288):

\[ \frac{1}{36} = \frac{8}{288}, \quad \frac{1}{64} = \frac{4.5}{288} \]

Thus,

\[ \frac{1}{36} - \frac{1}{64} = \frac{8 - 4.5}{288} = \frac{3.5}{288} \]

### Step 5: Substitute Back into the Rydberg Formula
Now substitute this back into the Rydberg formula:

\[ \frac{1}{\lambda} = 1.1 \times 10^7 \times \frac{3.5}{288} \]

Calculating this gives:

\[ \frac{1}{\lambda} = \frac{1.1 \times 3.5 \times 10^7}{288} \]

### Step 6: Calculate \( \lambda \)
Now calculate \( \lambda \):

\[ \frac{1}{\lambda} \approx 1.3 \times 10^6 \Rightarrow \lambda \approx \frac{1}{1.3 \times 10^6} \approx 7.5 \times 10^{-7} \, \text{m} \]

### Step 7: Convert to Nanometers
To convert meters to nanometers:

\[ \lambda \approx 7.5 \times 10^{-7} \, \text{m} = 750 \, \text{nm} \]

### Final Answer
The wavelength of the incident electromagnetic radiation required for the transition from n = 6 to n = 8 in a hydrogen atom is approximately **750 nm**.

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