The limiting line Balmer series will have a frequency of

Options

A. `32.29 xx 10^(15) s^(-1)`
B. `3.65 xx 10^(15) s^(-1)`
C. `-8.22 xx 10^(15) s^(-1)`
D. `8.22 xx 10^(15) s^(-1)`

Explanation

To find the limiting line frequency of the Balmer series, we will follow these steps:

### Step 1: Understand the Balmer series
The Balmer series corresponds to the transitions of electrons in a hydrogen atom from higher energy levels (n ≥ 3) down to the second energy level (n = 2). The limiting line occurs when the electron transitions from n = ∞ to n = 2.

### Step 2: Use the Rydberg formula
The Rydberg formula for the wavelengths of the spectral lines is given by:
\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( R_H \) is the Rydberg constant, \( n_1 \) is the lower energy level, and \( n_2 \) is the higher energy level.

### Step 3: Identify the values for the limiting line
For the limiting line in the Balmer series:
- \( n_1 = 2 \)
- \( n_2 = \infty \)

Substituting these values into the formula:
\[ \frac{1}{\lambda} = R_H \left( \frac{1}{2^2} - \frac{1}{\infty^2} \right) = R_H \left( \frac{1}{4} - 0 \right) = \frac{R_H}{4} \]

### Step 4: Substitute the value of Rydberg constant
The value of the Rydberg constant \( R_H \) is approximately \( 1.09677 \times 10^7 \, \text{m}^{-1} \). Therefore:
\[ \frac{1}{\lambda} = \frac{1.09677 \times 10^7}{4} = 2.741925 \times 10^6 \, \text{m}^{-1} \]

### Step 5: Calculate the wavelength \( \lambda \)
To find \( \lambda \):
\[ \lambda = \frac{1}{\frac{1}{\lambda}} = \frac{1}{2.741925 \times 10^6} \approx 3.65 \times 10^{-7} \, \text{m} \]

### Step 6: Calculate the frequency \( \nu \)
Using the relationship between frequency \( \nu \), wavelength \( \lambda \), and the speed of light \( c \):
\[ \nu = \frac{c}{\lambda} \]
where \( c = 3 \times 10^8 \, \text{m/s} \).

Substituting the values:
\[ \nu = \frac{3 \times 10^8}{3.65 \times 10^{-7}} \approx 8.22 \times 10^{14} \, \text{s}^{-1} \]

### Conclusion
The limiting line frequency of the Balmer series is approximately \( 8.22 \times 10^{14} \, \text{s}^{-1} \).

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