Final answer:
To resolve a doublet of 481.1 nm and 481.7 nm in the second-order spectrum, the minimum number of lines needed in a diffraction grating is 402.
Step-by-step explanation:
To calculate the minimum number of lines needed in a grating that will resolve a doublet of 481.1 nm and 481.7 nm in the second-order spectrum, we use the formula derived from the condition for resolving power (R) in a diffraction grating:
R = λ / Δλ = nN
Where λ is the average wavelength of the doublet, Δλ is the difference between the wavelengths of the doublet, n is the spectral order, and N is the number of lines on the grating.
The average wavelength (λ) is (481.1 nm + 481.7 nm)/2 = 481.4 nm. The difference (Δλ) is 481.7 nm - 481.1 nm = 0.6 nm.
Since we're looking at the second-order spectrum (n = 2), the formula becomes:
R = 481.4 nm / 0.6 nm = 802.33
So for the second order (n = 2):
802.33 = 2 * N
N = 802.33 / 2 = 401.165
Since we cannot have a fraction of a line, we round up to the next whole number. Therefore, the minimum number of lines (N) needed on the grating is 402.