The line at 434nm in the Balmer series of the hydrogen spectrum corresponds to a transition of an electron from the `n^(th)` to second Bohr orbit. What is the value of n?

Answer

To find the value of \( n \) for the transition corresponding to the line at 434 nm in the Balmer series of the hydrogen spectrum, we can use the Rydberg formula for hydrogen:

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

where:
- \( \lambda \) is the wavelength of the emitted light,
- \( R_H \) is the Rydberg constant for hydrogen, approximately \( 1.097 \times 10^7 \, \text{m}^{-1} \),
- \( n_1 \) is the lower energy level (for the Balmer series, \( n_1 = 2 \)),
- \( n_2 \) is the higher energy level (which we need to find).

### Step 1: Convert the wavelength from nanometers to meters
Given \( \lambda = 434 \, \text{nm} \):

\[ \lambda = 434 \times 10^{-9} \, \text{m} \]

### Step 2: Substitute the values into the Rydberg formula
We know that for the Balmer series, \( n_1 = 2 \). Thus, we can rewrite the formula as:

\[ \frac{1}{434 \times 10^{-9}} = 1.097 \times 10^7 \left( \frac{1}{2^2} - \frac{1}{n^2} \right) \]

### Step 3: Calculate \( \frac{1}{434 \times 10^{-9}} \)
Calculating the left side:

\[ \frac{1}{434 \times 10^{-9}} \approx 2.303 \times 10^6 \, \text{m}^{-1} \]

### Step 4: Set up the equation
Now we can set up the equation:

\[ 2.303 \times 10^6 = 1.097 \times 10^7 \left( \frac{1}{4} - \frac{1}{n^2} \right) \]

### Step 5: Solve for \( n^2 \)
First, divide both sides by \( 1.097 \times 10^7 \):

\[ \frac{2.303 \times 10^6}{1.097 \times 10^7} = \frac{1}{4} - \frac{1}{n^2} \]

Calculating the left side gives:

\[ 0.209 = \frac{1}{4} - \frac{1}{n^2} \]

Now, since \( \frac{1}{4} = 0.25 \):

\[ 0.209 = 0.25 - \frac{1}{n^2} \]

Rearranging gives:

\[ \frac{1}{n^2} = 0.25 - 0.209 = 0.041 \]

### Step 6: Find \( n^2 \)
Taking the reciprocal:

\[ n^2 = \frac{1}{0.041} \approx 24.39 \]

### Step 7: Calculate \( n \)
Taking the square root gives:

\[ n \approx 4.94 \]

Since \( n \) must be an integer, we round to the nearest whole number:

\[ n = 5 \]

### Final Answer
The value of \( n \) is \( 5 \).

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