A diffraction grating with 150 slits per centimeter is used to measure the wavelengths emitted by hydrogen gas. At what angles in the fourth-order spectrum would you expect to find the two violet lines of wavelength 434 nm and of wavelength 410 nm? (angles in radians)
To determine the angles at which the violet lines of specific wavelengths will appear in the spectrum produced by a diffraction grating, we can use the formula for the angular position of the diffraction maximum:
sin(θ) = m * λ / d
Where:
- θ is the angular position of the diffraction maximum.
- m is the order of the maximum (an integer).
- λ is the wavelength of light.
- d is the spacing between the slits of the grating.
In this case, we want to find the angles for the fourth-order spectrum (m = 4), and we have the wavelengths λ = 434 nm and λ = 410 nm.
First, we need to convert the wavelengths from nanometers to meters:
λ1 = 434 nm = 434 × 10^(-9) m
λ2 = 410 nm = 410 × 10^(-9) m
Next, we need to calculate the spacing between the slits (d) using the given information of 150 slits per centimeter:
d = 1 / (150 slits/cm * 100 cm/m)
d = 1 / 15000 m = 6.67 × 10^(-5) m
Now, we can calculate the angles using the formula:
sin(θ1) = 4 * λ1 / d
sin(θ2) = 4 * λ2 / d
Using a scientific calculator, we can find the values of sin(θ1) and sin(θ2) and then take the inverse sine (arcsin) to get the angles in radians.
Let's find the angles:
sin(θ1) = 4 * (434 × 10^(-9)) / (6.67 × 10^(-5))
sin(θ2) = 4 * (410 × 10^(-9)) / (6.67 × 10^(-5))
Now, calculate the inverse sine of both values to get the angles:
θ1 = arcsin(sin(θ1))
θ2 = arcsin(sin(θ2))
These angles, θ1 and θ2, will give you the positions of the two violet lines of wavelength 434 nm and 410 nm in the fourth-order diffraction spectrum of the given diffraction grating.