In what region of the electromagnetic spectrum would you look the spectral line resulting from the electronic transition from the tenth to the fifth electronic level in the hydrogen atoms ? `(R_(H)=1.10xx10^(5) cm^(-1))`

Options

A. Microwave
B. Infrared
C. Visible
D. Ultraviolet

Explanation

To determine the region of the electromagnetic spectrum for the spectral line resulting from the electronic transition from the tenth to the fifth electronic level in hydrogen atoms, we will follow these steps:

### Step 1: Identify the Rydberg Formula
The Rydberg formula for the wave number (\( \nu' \)) is given by:
\[ \nu' = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where:
- \( R_H \) is the Rydberg constant (\( 1.10 \times 10^5 \, \text{cm}^{-1} \))
- \( n_1 \) is the lower energy level (5 in this case)
- \( n_2 \) is the higher energy level (10 in this case)

### Step 2: Substitute Values into the Formula
Substituting \( n_1 = 5 \) and \( n_2 = 10 \) into the formula:
\[ \nu' = 1.10 \times 10^5 \left( \frac{1}{5^2} - \frac{1}{10^2} \right) \]

### Step 3: Calculate the Values
Calculate \( \frac{1}{5^2} \) and \( \frac{1}{10^2} \):
\[ \frac{1}{5^2} = \frac{1}{25} = 0.04 \]
\[ \frac{1}{10^2} = \frac{1}{100} = 0.01 \]
Now, substitute these values back into the equation:
\[ \nu' = 1.10 \times 10^5 \left( 0.04 - 0.01 \right) = 1.10 \times 10^5 \times 0.03 \]

### Step 4: Perform the Multiplication
Now calculate:
\[ \nu' = 1.10 \times 10^5 \times 0.03 = 3300 \, \text{cm}^{-1} \]

### Step 5: Find the Wavelength
The wavelength (\( \lambda \)) can be found using the relationship:
\[ \lambda = \frac{1}{\nu'} \]
Substituting the value of \( \nu' \):
\[ \lambda = \frac{1}{3300} \, \text{cm} \]
Calculating this gives:
\[ \lambda \approx 0.0003030 \, \text{cm} = 3.030 \times 10^{-4} \, \text{cm} \]

### Step 6: Convert to Meters
To convert to meters, we know that \( 1 \, \text{cm} = 0.01 \, \text{m} \):
\[ \lambda \approx 3.030 \times 10^{-4} \, \text{cm} = 3.030 \times 10^{-6} \, \text{m} = 3030 \, \text{nm} \]

### Step 7: Determine the Region of the Electromagnetic Spectrum
The wavelength of \( 3030 \, \text{nm} \) falls within the infrared region of the electromagnetic spectrum.

### Final Answer
The spectral line resulting from the electronic transition from the tenth to the fifth electronic level in hydrogen atoms would be found in the **infrared region** of the electromagnetic spectrum.

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