Which one of the following will result in an electron transition from the n = 4 level to the n= 7 level in a hydrogen atom?

Options

A. Emission of a 0.28 eV photon
B. Absorption of a 0.28 eV photon
C. Emission of a 0.57 eV photon
D. Absorption of a 0.57 eV photon

Answer

To determine the energy required for an electron transition from the n = 4 level to the n = 7 level in a hydrogen atom, we can use the Rydberg formula and the relationship between energy and wavelength. Here’s a step-by-step solution:

### Step 1: Identify the transition levels
We have:
- Initial level (n1) = 4
- Final level (n2) = 7

### Step 2: Use the Rydberg formula
The Rydberg formula for hydrogen is given by:

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

where \( R \) is the Rydberg constant, approximately \( 1.1 \times 10^7 \, \text{m}^{-1} \).

### Step 3: Substitute the values into the formula
Substituting \( n_1 = 4 \) and \( n_2 = 7 \):

\[ \frac{1}{\lambda} = 1.1 \times 10^7 \left( \frac{1}{4^2} - \frac{1}{7^2} \right) \]

Calculating \( \frac{1}{4^2} \) and \( \frac{1}{7^2} \):

\[ \frac{1}{4^2} = \frac{1}{16} = 0.0625 \]
\[ \frac{1}{7^2} = \frac{1}{49} \approx 0.02041 \]

Now, substituting these values:

\[ \frac{1}{\lambda} = 1.1 \times 10^7 \left( 0.0625 - 0.02041 \right) \]
\[ \frac{1}{\lambda} = 1.1 \times 10^7 \left( 0.04209 \right) \]

### Step 4: Calculate \( \frac{1}{\lambda} \)
Calculating:

\[ \frac{1}{\lambda} \approx 1.1 \times 10^7 \times 0.04209 \approx 0.0463 \times 10^7 \]

### Step 5: Find \( \lambda \)
Taking the reciprocal to find \( \lambda \):

\[ \lambda \approx \frac{1}{0.0463 \times 10^7} \approx 21.598 \times 10^{-7} \, \text{m} \]

### Step 6: Calculate the energy using \( E = \frac{hc}{\lambda} \)
Using Planck's constant \( h = 6.626 \times 10^{-34} \, \text{Js} \) and speed of light \( c = 3 \times 10^8 \, \text{m/s} \):

\[ E = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{21.598 \times 10^{-7}} \]

Calculating this gives:

\[ E \approx 0.9167 \times 10^{-19} \, \text{J} \]

### Step 7: Convert energy to electron volts
To convert joules to electron volts, divide by \( 1.6 \times 10^{-19} \, \text{J/eV} \):

\[ E \approx \frac{0.9167 \times 10^{-19}}{1.6 \times 10^{-19}} \approx 0.57 \, \text{eV} \]

### Final Answer
The energy required for the electron transition from n = 4 to n = 7 in a hydrogen atom is approximately **0.57 eV**.