The line with smallest wavelength in the Balmer series in the hydrogen spectrum will have the frequency :-

Options

A. `(a)8.22xx10^(14)s^(-1)`
B. (b)`3.29xx10^(15)s^(-1)`
C. `(c)7.22xx10^(16)s^(-1)`
D. (d)`8.05xx10^(13)s^(-1)`

Answer

To find the frequency of the line with the smallest wavelength in the Balmer series of the hydrogen spectrum, we can follow these steps:

### Step 1: Understand the relationship between wavelength and frequency
The speed of light (c) is related to wavelength (λ) and frequency (ν) by the equation:
\[ c = \lambda \cdot \nu \]
From this, we can express frequency as:
\[ \nu = \frac{c}{\lambda} \]

### Step 2: Identify the parameters for the Balmer series
For the Balmer series in hydrogen, the transition occurs from higher energy levels (n2) to n1 = 2. The smallest wavelength corresponds to the transition from the highest energy level (n2 = ∞) to n1 = 2.

### Step 3: Calculate the wave number (ν̅)
The wave number (ν̅) is given by:
\[ \nu̅ = \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
Where R (Rydberg constant) is approximately \( 1.097 \times 10^7 \, \text{m}^{-1} \).

For the smallest wavelength (n2 = ∞):
\[ \nu̅ = R \left( \frac{1}{2^2} - \frac{1}{\infty^2} \right) \]
This simplifies to:
\[ \nu̅ = R \left( \frac{1}{4} - 0 \right) = \frac{R}{4} \]

### Step 4: Substitute the value of R
Substituting the value of R:
\[ \nu̅ = \frac{1.097 \times 10^7}{4} \]
Calculating this gives:
\[ \nu̅ = 2.7425 \times 10^6 \, \text{m}^{-1} \]

### Step 5: Calculate the frequency (ν)
Now, we can find the frequency using the relationship:
\[ \nu = \nu̅ \cdot c \]
Where \( c = 3 \times 10^8 \, \text{m/s} \):
\[ \nu = \left( 2.7425 \times 10^6 \right) \cdot \left( 3 \times 10^8 \right) \]

### Step 6: Perform the calculation
Calculating the frequency:
\[ \nu = 2.7425 \times 10^6 \times 3 \times 10^8 = 8.2275 \times 10^{14} \, \text{s}^{-1} \]

### Step 7: Final answer
The frequency of the line with the smallest wavelength in the Balmer series is approximately:
\[ \nu \approx 8.23 \times 10^{14} \, \text{s}^{-1} \]