Question

An electric current through hydrogen gas produces several distinct wavelengths of visible light. What are the wavelengths of the hydrogen spectrum, if they form first-order maxima at angles 24.2°,25.7°,29.1°, and 41.0° when projected on a diffraction grating having 10,000 lines per centimeter?

Answer 1

The wavelengths of the hydrogen spectrum that form first-order maxima at angles 24.2°, 25.7°, 29.1°, and 41.0°, through a diffraction grating with 10,000 lines per centimeter, are calculated as 410 nm, 434 nm, 486 nm, and 656 nm, respectively.

Step-by-step explanation:

Calculating Wavelengths Using a Diffraction Grating

To calculate the wavelengths of the hydrogen spectrum, we use the diffraction grating formula:

nλ = d sin(θ),

where n is the order of the maximum, λ is the wavelength, d is the grating spacing (inverse of the number of lines per centimeter), and θ is the angle of the maximum.

Given 10,000 lines per centimeter:

• d = 1 / (10,000 cm-1) = 1 x 10-4 cm = 1 x 10-6 m

For first-order maxima (n = 1) and the angles provided:

1. λ1 = (1 x 10-6 m) sin(24.2°) = 410 nm
2. λ2 = (1 x 10-6 m) sin(25.7°) = 434 nm
3. λ3 = (1 x 10-6 m) sin(29.1°) = 486 nm
4. λ4 = (1 x 10-6 m) sin(41.0°) = 656 nm

These values represent the distinct wavelengths of light in the hydrogen spectrum that produce first-order maxima at the given angles.