The wavelength of radiation required to remove the electron of hydrogen atom `("Ionisation energy "21.7xx10^(-12)"erg")` from n=2 orbit to `n = oo`is

Options

A. `3.664 xx 10^(-4) cm`
B. `3.66 xx 10^(-5) cm`
C. `3.66 xx 10^(-6) cm`
D. `3.664 xx 10^(-7) cm`

Answer

To find the wavelength of radiation required to remove the electron of a hydrogen atom from the n=2 orbit to n=∞, we can use the Rydberg formula for hydrogen:

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

where:
- \( R \) is the Rydberg constant, approximately \( 1.097 \times 10^7 \, \text{m}^{-1} \) or \( 109677 \, \text{cm}^{-1} \)
- \( n_1 \) is the initial energy level (in this case, \( n_1 = 2 \))
- \( n_2 \) is the final energy level (in this case, \( n_2 = \infty \))

### Step 1: Identify \( n_1 \) and \( n_2 \)
Here, we have:
- \( n_1 = 2 \)
- \( n_2 = \infty \)

### Step 2: Substitute values into the Rydberg formula
Substituting the values into the formula:

\[ \frac{1}{\lambda} = 109677 \left( \frac{1}{2^2} - \frac{1}{\infty^2} \right) \]

Since \( \frac{1}{\infty^2} \) approaches 0, we simplify:

\[ \frac{1}{\lambda} = 109677 \left( \frac{1}{4} - 0 \right) \]

### Step 3: Calculate \( \frac{1}{\lambda} \)
Now, calculate \( \frac{1}{\lambda} \):

\[ \frac{1}{\lambda} = 109677 \times \frac{1}{4} = \frac{109677}{4} \]

### Step 4: Perform the division
Calculating \( \frac{109677}{4} \):

\[ \frac{109677}{4} = 27419.25 \, \text{cm}^{-1} \]

### Step 5: Find \( \lambda \)
Now, take the reciprocal to find \( \lambda \):

\[ \lambda = \frac{1}{27419.25} \approx 3.646 \times 10^{-5} \, \text{cm} \]

### Step 6: Convert to appropriate units (if necessary)
The wavelength in centimeters can be converted to meters if required, but in this case, we can leave it in centimeters.

### Final Answer
Thus, the wavelength required to remove the electron from the n=2 orbit to n=∞ is approximately:

\[ \lambda \approx 3.66 \times 10^{-5} \, \text{cm} \]

### Conclusion
Among the given options, the correct answer is \( 3.66 \times 10^{-5} \, \text{cm} \).

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