Question

A diffraction grating with 161 slits per centimeter is used to measure the wavelengths emitted by hydrogen gas. At what angles in the first-order spectrum would you expect to find the two violet lines of wavelength 434 nm and of wavelength 410 nm

Answer 1

`\theta_1 = 0.400^o`

`\theta_2 =0.378^o`

Step-by-step explanation:

From the question we are told that

The number of slits per cm is k =
`161\ slits\ per\ cm = 161 \ slits\ per\ 0.01 m`

The order of the maxima is n = 1

The wavelength are
`\lambda_1 = 434 nm = 434 *10^(-9) \ m \ \ \ , \lambda_2 = 410nm = 410 *10^(-9) \ m`

The spacing between the slit is mathematically represented as

`d = ( 0.01)/(k)`

=>
`d = ( 0.01)/(161)`

=>
`d = 6.211 *10^(-5) \ m`

Generally the condition for constructive interference is

`n\lambda = d \ sin \theta`

At
`\lambda_1`

`\theta _1 = sin^(-1) [ (1 * 434 *10^(-9))/(6.211 *10^(-5)) ]`

`\theta_1 = 0.400^o`

At
`\lambda_2`

`\theta _2 = sin^(-1) [ (1 * 410 *10^(-9))/(6.211 *10^(-5)) ]`

`\theta_2 =0.378^o`