Heat treatment of muscular pain involves radiation of wavelength of about 900nm. Which spectral line of H-atom is suitable for this purpose? `[R_H=1xx10^5 cm^(-1), h=6.6xx10^(-34) Js, c=3xx10^8 ms^(-1) ]`

Options

A. Paschen , `oo to 3`
B. Lyman , `oo to 1`
C. Paschen , `5 to 3`
D. Balmer , `oo to 2`

Explanation

To solve the problem of identifying which spectral line of the hydrogen atom corresponds to a radiation wavelength of about 900 nm, we will use the Rydberg formula for hydrogen spectral lines. The steps are as follows:

### Step 1: Understand the Rydberg Formula
The Rydberg formula is given by:

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

Where:
- \(\lambda\) is the wavelength of the emitted or absorbed radiation.
- \(R_H\) is the Rydberg constant for hydrogen, given as \(1 \times 10^5 \, \text{cm}^{-1}\).
- \(n_1\) is the principal quantum number of the lower energy level.
- \(n_2\) is the principal quantum number of the higher energy level.

### Step 2: Convert Wavelength to Centimeters
Given that the wavelength \(\lambda\) is 900 nm, we need to convert this to centimeters:

\[ 900 \, \text{nm} = 900 \times 10^{-9} \, \text{m} = 900 \times 10^{-7} \, \text{cm} \]

### Step 3: Substitute Values into the Rydberg Formula
We will assume that the electron transitions from a very high energy level (approaching infinity) to a lower energy level \(n_1\). Thus, we can set \(n_2 = \infty\) (which means \(\frac{1}{n_2^2} = 0\)).

Substituting into the Rydberg formula gives:

\[ \frac{1}{900 \times 10^{-7}} = 1 \times 10^5 \left( \frac{1}{n_1^2} - 0 \right) \]

### Step 4: Solve for \(n_1\)
Rearranging the equation gives:

\[ \frac{1}{n_1^2} = \frac{1}{900 \times 10^{-7}} \times 1 \times 10^5 \]

Calculating the left-hand side:

\[ \frac{1}{n_1^2} = \frac{1 \times 10^5}{900 \times 10^{-7}} = \frac{1 \times 10^{12}}{900} \]

Calculating this value:

\[ \frac{1 \times 10^{12}}{900} \approx 1.111 \times 10^{9} \]

Thus,

\[ n_1^2 = \frac{900}{1 \times 10^{12}} \approx 9 \]

Taking the square root gives:

\[ n_1 = 3 \]

### Step 5: Identify the Series
Since \(n_1 = 3\), we can identify the series. The series corresponding to \(n_1 = 3\) is the **Paschen series**, which involves transitions to the \(n=3\) level.

### Conclusion
The suitable spectral line of the hydrogen atom for the heat treatment of muscular pain, which involves radiation of wavelength about 900 nm, corresponds to the Paschen series, specifically the transition from \(n_2 = \infty\) to \(n_1 = 3\).

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