Question 25.8: Light from Sodium Atoms Goal Find the necessary resolving po...
Light from Sodium Atoms
Goal Find the necessary resolving power to distinguish spectral lines.
Problem Two bright lines in the spectrum of sodium have wavelengths of 589.00 \mathrm{~nm} and 589.59 \mathrm{~nm}, respectively. (a) What must the resolving power of a grating be in order to distinguish these wavelengths? (b) To resolve these lines in the second-order spectrum, how many lines of the grating must be illuminated?
Strategy This problem requires little more than substituting into Equations 25.11 and 25.12.
R\equiv\frac{\lambda}{\lambda_{2}-\lambda_{1}}=\frac{\lambda}{\Delta\lambda} (25.11)
R=N m (25.12)
(a) What must the resolving power of a grating be in order to distinguish the given wavelengths?
Substitute into Equation 25.11 to find R :
\begin{aligned} R & =\frac{\lambda}{\Delta \lambda}=\frac{589.00 \mathrm{~nm}}{589.59 \mathrm{~nm}-589.00 \mathrm{~nm}}=\frac{589 \mathrm{~nm}}{0.59 \mathrm{~nm}} \\ & =1.0 \times 10^{3} \end{aligned}
(b) To resolve these lines in the second-order spectrum, how many lines of the grating must be illuminated?
Solve Equation 25.12 for N and substitute:
N=\frac{R}{m}=\frac{1.0 \times 10^{3}}{2}=5.0 \times 10^{2} \text { lines }
Remarks The ability to resolve spectral lines is particularly important in experimental atomic physics.