A line in the infrared region of the hydrogen spectrum has a wavelength of 1.875 micro meters. What is the transition responsible for this wavelength?
To determine the transition responsible for a specific wavelength in the hydrogen spectrum, we need to use the Rydberg formula and the Balmer series of hydrogen.
The Rydberg formula is expressed as:
1/λ = R (1/n₁² - 1/n₂²)
where λ is the wavelength of the spectral line, R is the Rydberg constant (approximately 1.097 x 10^7 m⁻¹), and n₁ and n₂ are integers representing the energy levels of the electron.
The Balmer series of hydrogen corresponds to electron transitions ending or starting from the second energy level (n₂ = 2). The wavelength you provided (1.875 μm or 1.875 x 10⁻⁶ m) lies in the infrared region, which corresponds to longer wavelengths and lower energy transitions.
By rearranging the formula and solving for n₁, we can determine the initial energy level of the electron:
1/n₁² = 1/n₂² - λR
Plugging in the values:
1/n₁² = 1/2² - (1.875 x 10⁻⁶)(1.097 x 10^7)
Simplifying:
1/n₁² = 0.25 - 2.065 x 10⁻¹²
1/n₁² ≈ 0.25
Taking the reciprocal:
n₁² ≈ 4
Taking the square root:
n₁ ≈ 2
Therefore, the transition responsible for the given wavelength of 1.875 μm in the infrared region of the hydrogen spectrum corresponds to an electron transitioning from the second energy level (n₂ = 2) to the first energy level (n₁ = 1).