For a doubly ionized lithium ion Li^2+, calculate the wavelengths of the first three lines in the Balmer series.
To calculate the wavelengths of the first three lines in the Balmer series for a doubly ionized lithium ion (Li^2+), we need to use the formula:
1/λ = R_H * (1/n_f² - 1/n_i²)
where:
- λ is the wavelength of the spectral line
- R_H is the Rydberg constant for hydrogen (approximately 1.097 x 10^7 m⁻¹)
- n_f is the final energy level
- n_i is the initial energy level
In the Balmer series, the initial energy level (n_i) is always 2, and the final energy levels (n_f) are greater than 2.
Let's calculate the wavelengths for the first three lines.
1. For the first line, n_f = 3:
1/λ₁ = R_H * (1/3² - 1/2²)
1/λ₁ = R_H * (1/9 - 1/4)
1/λ₁ = R_H * (4/36 - 9/36)
1/λ₁ = R_H * (-5/36)
λ₁ = -36/(5 * R_H)
2. For the second line, n_f = 4:
1/λ₂ = R_H * (1/4² - 1/2²)
1/λ₂ = R_H * (1/16 - 1/4)
1/λ₂ = R_H * (1/16 - 4/16)
1/λ₂ = R_H * (-3/16)
λ₂ = -16/(3 * R_H)
3. For the third line, n_f = 5:
1/λ₃ = R_H * (1/5² - 1/2²)
1/λ₃ = R_H * (1/25 - 1/4)
1/λ₃ = R_H * (4/100 - 25/100)
1/λ₃ = R_H * (-21/100)
λ₃ = -100/(21 * R_H)
Using the value of the Rydberg constant (R_H), we can substitute its value into the formulas to obtain the wavelengths (λ₁, λ₂, λ₃) for the first three lines in the Balmer series for the doubly ionized lithium ion (Li^2+).