Calculate the least width that a grating must have to resolve two components of thesodium D lines in the second order, the grating having 800 lines/cm. The wavelengthfor D1 and D2 lines of sodium are 5893 Å and 5896 Å respectively.

To solve this problem, we need to use the formula for the resolving power of a grating, which is given by:

R = mN

where R is the resolving power, m is the order of the spectrum, and N is the total number of lines in the grating.

We are asked to find the least width of the grating, so we need to find the minimum N. We are given that the grating has 800 lines/cm, and we are looking at the second order spectrum (m=2).

The resolving power needed to resolve the sodium D lines can be found using the formula:

R = λ/Δλ

where λ is the average wavelength of the two lines, and Δλ is the difference in their wavelengths.

The average wavelength λ is (5893 Å + 5896 Å)/2 = 5894.5 Å.

The difference in wavelengths Δλ is 5896 Å - 5893 Å = 3 Å.

So, the required resolving power R is 5894.5 Å / 3 Å = 1964.83.

Now we can find the minimum number of lines N using the formula for the resolving power:

N = R/m = 1964.83 / 2 = 982.415.

This is the total number of lines in the grating. To find the width of the grating, we divide by the number of lines per cm:

Width = N / (lines/cm) = 982.415 / 800 = 1.228 cm.

So, the grating must be at least 1.228 cm wide to resolve the sodium D lines in the second order.