Question

Three discrete spectral lines occur at angles of 10.1⁰, 13.7⁰ , and 14.8⁰ in the first-order spectrum of a grating spectrometer.(a) If the grating has 3660 slits cm, what are the wavelengths of the light?

Answer 1

To calculate the wavelengths of the light, we can use the grating equation and the given angles and order of the spectral lines. The slit spacing of the grating can be calculated using the lines per centimeter. By plugging in the values, we can determine the wavelengths of the three spectral lines.

Step-by-step explanation:

In order to determine the wavelengths of the light, we need to use the formula for the grating equation. The grating equation is given by:

dsinθ = mλ

Where d is the slit spacing (which can be calculated using the given information), θ is the angle of diffraction, m is the order of the spectral line, and λ is the wavelength of the light. Using the given angles and the known order of the lines (first order), we can calculate the wavelengths.

Let's calculate the slit spacing for the grating:

d = 1 / (lines per centimeter) = 1 / (3660 lines/cm)

d = 0.000273 meters = 0.273 mm

Now let's calculate the wavelengths of the three spectral lines:

For the first spectral line at an angle of 10.1°:

λ = (d * sin(θ)) / m = (0.000273 m * sin(10.1°)) / 1 = 4.636 x 10⁻⁷ m = 463.6 nm

For the second spectral line at an angle of 13.7°:

λ = (d * sin(θ)) / m = (0.000273 m * sin(13.7°)) / 1 = 6.292 x 10⁻⁷ m = 629.2 nm

For the third spectral line at an angle of 14.8°:

λ = (d * sin(θ)) / m = (0.000273 m * sin(14.8°)) / 1 = 6.799 x 10⁻⁷ m = 679.9 nm