Find the longest wavelength in the Balmer series of hydrogen.

The Balmer series of hydrogen refers to the set of spectral lines corresponding to electron transitions from higher energy levels to the second energy level (n=2). The longest wavelength in this series occurs when the electron transitions from the third energy level (n=3) to the second energy level (n=2).

Calculating the Longest Wavelength

The formula to find the wavelength (λ) of the emitted light in the Balmer series is given by the Rydberg formula:

1/λ = R_H (1/n1² - 1/n2²)

Where:

• R_H is the Rydberg constant (approximately 1.097 x 10⁷ m^-1).
• n1 is the lower energy level (for Balmer series, n1 = 2).
• n2 is the higher energy level (for the longest wavelength, n2 = 3).

Plugging in the Values

Substituting the values into the formula:

1/λ = R_H (1/2² - 1/3²)

1/λ = 1.097 x 10⁷ (1/4 - 1/9)

Calculating the fractions:

1/4 - 1/9 = 9/36 - 4/36 = 5/36

Now substituting back:

1/λ = 1.097 x 10⁷ (5/36)

Finding the Wavelength

Now, calculate λ:

λ = 36/(1.097 x 10⁷ x 5)

After performing the calculations, you will find:

λ ≈ 656.3 nm

Summary

The longest wavelength in the Balmer series of hydrogen is approximately 656.3 nanometers, which corresponds to the red color in the visible spectrum.