Question

A diffraction grating with 140 slits per centimeter is used to measure the wavelengths emitted by hydrogen gas. At what angles in the third-order spectrum would you expect to find the two violet lines of wavelength 434 nm and of wavelength 410 nm? (angles in radians) The 434 nm line:

Answer 1

0.003181 radians

0.003005 radians

Step-by-step explanation:

Number of slits = 140 /cm

λ = Wavelength = 434 nm = 434×10⁻⁹ m

m = 3 Third order spectrum

Space between slits

`d=(0.01)/(140) =7.14* 10^(-5)\ m`

Now,

`dsin\theta = m\lambda\\\Rightarrow \theta=sin^(-1)\left((m\lambda)/(d)\right)\\\Rightarrow \theta=sin^(-1)\left((3* 434* 10^(-9))/(7.14* 10^(-5))\right)\\\Rightarrow \theta=sin^(-1)0.018228\\\Rightarrow \theta=0.01823^(\circ)=0.01823* (\pi)/(180)=0.003181 radians`

0.003181 radians

When λ = 410 nm = 410×10⁻⁹ m

`dsin\theta = m\lambda\\\Rightarrow \theta=sin^(-1)\left((m\lambda)/(d)\right)\\\Rightarrow \theta=sin^(-1)\left((3* 410* 10^(-9))/(7.14* 10^(-5))\right)\\\Rightarrow \theta=sin^(-1)0.01722\\\Rightarrow \theta=0.01722^(\circ)=0.01722* (\pi)/(180)=0.003005 radians`

0.003005 radians