What is the wavelength of the second line of Paschen series? Calculate it.

The Paschen series in the hydrogen spectrum corresponds to transitions from higher energy levels to the third energy level. The formula for the wavelength of a spectral line for the hydrogen atom is given by the Rydberg formula:

1/λ = R * (1/n1² - 1/n2²)

where:

• λ is the wavelength
• R is the Rydberg constant (approximately 1.097373 x 10^7 m^-1)
• n1 and n2 are the principal quantum numbers of the two energy levels involved in the transition (for the Paschen series, n1 = 3)

The second line of the Paschen series corresponds to a transition from the fourth energy level to the third, so n2 = 4.

Substituting these values into the Rydberg formula gives:

1/λ = R * (1/3² - 1/4²) 1/λ = R * (1/9 - 1/16) 1/λ = R * (0.1111 - 0.0625) 1/λ = R * 0.0486

Solving for λ gives:

λ = 1 / (R * 0.0486)

Substituting the value of R gives:

λ = 1 / (1.097373 x 10^7 m^-1 * 0.0486) λ = 1 / (533400.178 m^-1) λ = 1.875 x 10^-6 m

So, the wavelength of the second line of the Paschen series is approximately 1.875 micrometers.