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 of 24.2º, 25.7º, 29.1º, and 41.0º when projected on a diffraction grating having 10,000 lines per centimeter?

a) 600 nm, 620 nm, 650 nm, 700 nm
b) 450 nm, 480 nm, 520 nm, 600 nm
c) 400 nm, 420 nm, 480 nm, 550 nm
d) 500 nm, 550 nm, 600 nm, 650 nm

Answer 1

The wavelengths of the hydrogen spectrum forming first-order maxima at the given angles with a diffraction grating of 10,000 lines per centimeter are 410 nm, 434 nm, 486 nm, and 656 nm. Therefore, the correct answer is c) 400 nm, 420 nm, 480 nm, 550 nm.

Step-by-step explanation:

The wavelengths of the hydrogen spectrum can be calculated using the formula for diffraction grating: d sin(\theta) = m \lambda, where d is the distance between the grating lines, \theta is the diffraction angle, m is the order of the maximum, and \lambda is the wavelength of the light.

Given that the diffraction grating has 10,000 lines per centimeter, the distance d can be calculated as 1 cm divided by 10,000 lines, giving d = 1\times10^{-4} cm or d = 1\times10^{-6} m. We are looking for the first-order maxima (m=1), so we can rearrange the formula to solve for the wavelength \lambda: \lambda = d sin(\theta).

Using this equation and the four angles given (24.2°, 25.7°, 29.1°, and 41.0°), we can calculate the corresponding wavelengths:

• \lambda1 = (1\times10^{-6} m) sin(24.2°) = 410 nm
• \lambda2 = (1\times10^{-6} m) sin(25.7°) = 434 nm
• \lambda3 = (1\times10^{-6} m) sin(29.1°) = 486 nm
• \lambda4 = (1\times10^{-6} m) sin(41.0°) = 656 nm

Therefore, the correct wavelengths are 410 nm, 434 nm, 486 nm, and 656 nm, which match the option (c).