The hydrogen spectrum has a red line at 656 nm, and a blue line at 434 nm. What is the first order angular separation between the two spectral lines obtained with a diffraction grating with 5000 lines/cm?

To solve this problem, we need to use the formula for the diffraction grating:

dsin(θ) = mλ

where:

• d is the distance between the slits in the grating,
• θ is the angle of diffraction,
• m is the order of diffraction, and
• λ is the wavelength of light.

First, we need to convert the number of lines per cm to the distance between the slits in meters. There are 5000 lines/cm, so the distance between the slits is 1 cm / 5000 = 0.0002 cm = 0.000002 m.

Next, we need to convert the wavelengths from nm to m. The red line is at 656 nm = 656 x 10^-9 m, and the blue line is at 434 nm = 434 x 10^-9 m.

We can now use the formula to find the angles of diffraction for the red and blue lines. We are looking for the first order (m=1) diffraction, so:

sin(θ_red) = λ_red / d = (656 x 10^-9 m) / (0.000002 m) = 0.328 sin(θ_blue) = λ_blue / d = (434 x 10^-9 m) / (0.000002 m) = 0.217

To find the angles, we take the inverse sine (arcsin) of these values:

θ_red = arcsin(0.328) = 19.47 degrees θ_blue = arcsin(0.217) = 12.68 degrees

Finally, the angular separation between the red and blue lines is the difference between these angles:

Δθ = θ_red - θ_blue = 19.47 degrees - 12.68 degrees = 6.79 degrees.