Find the wavelength of the blue line (line spectrum)

[moenste]

Homework Statement

The line spectrum of a certain substance consists of three prominent lines; blue (B), yellow (Y) and red (R). When the spectrum is examined with a diffraction grating having d = 4 * 10^-6 m, it is found that the sequence of lines, moving from the centre, is B, Y, R, B, Y, B, R.

Further, it is found that the diffraction angles θ of the fifth and seventh lines are 17.46° and 20.49° respectively, and that the sixth line is at an angular position exactly halfway between them. Find the wavelength of the blue line.

Answer: 4.34 * 10^-7 m

2. The attempt at a solution
We find the θ₆ = (17.46 + 20.49) / 2 = 18.975°. Then plug into the formula: n λ = d sin θ to find the wavelength of the sixth line: λ₆ = (4 * 10^-6 * sin 18.975°) / 6 = 2.168 * 10^-7 m.

I also found the wavelengths for the fifth and seventh lines: 2.4 * 10^-7 m and 2 * 10^-7 m respectively.

But after this I don't know what to do. I need to find the wavelength of the first blue line, while I have the wavelengths of a yellow (5), blue (6) and a red (7) one. How to use them to find blue (1)?

 

[blue_leaf77]

No, you don't use n=6. The fringe order is for only one color. Overall taking into account all colors, the sixth line is blue, but which order does this blue appear from the center?

 

[moenste]

No, you don't use n=6. The fringe order is for only one color. Overall taking into account all colors, the sixth line is blue, but which order does this blue appear from the center?

B is the first color, so it should be the first order?

 

[blue_leaf77]

B is the first color, so it should be the first order?

What I meant is, should you remove the other colors beside blue, what will be the order of the 6th line blue?

 

[moenste]

What I meant is, should you remove the other colors beside blue, what will be the order of the 6th line blue?

If you put it like that, it should be the third order.

And indeed if we plug in 3 instead of 6 we'll get 4.33 * 10^-7.

Thank you!