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Chapter 32: Problem 43

What order is necessary to resolve 647.98 -nm and 648.07 -nm spectral lines using a 4500-line grating?

Short Answer

Expert verified
The minimum order for the grating to resolve the given spectral lines can be found by substituting the values into the equations from the steps above.

Step by step solution

01

Convert wavelengths to meters

First convert the given wavelengths from nanometers to meters by multiplying by \(1 x 10^{-9}\). So, \(\lambda_1 = 647.98 x 10^{-9}\) m and \(\lambda_2 = 648.07 x 10^{-9}\) m.
02

Calculate wavelength difference

Calculate the difference between the two wavelengths which is \(\Delta \lambda = \lambda_2 - \lambda_1 = (648.07 - 647.98) x 10^{-9}\) m.
03

Calculate minimum diffraction order using grating equation

Using the grating equation \( m = \frac{{\lambda_1 - \lambda_2}}{{\Delta \lambda_{min}}}\), where m is the order, \(\Delta \lambda_{min} = \frac{1}{{N \cdot \lambda_{avg}}}\), \(\lambda_{avg} = \frac{{\lambda_1 + \lambda_2}}{2}\) and N is the line density. Calculate \(\lambda_{avg}\), then \(\Delta \lambda_{min}\), substitute these values and \(\Delta \lambda\) into the equation to find m.

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Most popular questions from this chapter

You're designing a spectrometer whose specifications call for a minimum of \(5^{\circ}\) separation between the red hydrogen- \(\alpha\) line at \(656 \mathrm{nm}\) and the yellow sodium line at \(589 \mathrm{nm}\) when the two are observed in third order with a grating spectrometer. Available gratings have 2500 lines/cm, 3500 lines/cm, or 4500 lines/cm. What's the coarsest grating you can use?

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