The sodium doublet in the spectrum of sodium is a pair of lines with wavelengths 589.0nm and 589.06nm. Calculate the minimum number of rulings in a grating needed to resolve this doublet in the second order spectrum.

To determine the minimum number of rulings required in a diffraction grating to resolve the sodium doublet, we can use the Rayleigh criterion for resolution. This criterion states that two spectral lines can be resolved if the angular separation between them is greater than or equal to the width of the central maximum of the diffraction pattern. Let's break this down step by step.

Understanding the Problem

The sodium doublet consists of two closely spaced wavelengths: 589.0 nm and 589.06 nm. The difference in wavelength, which we will denote as Δλ, is:

• Δλ = 589.06 nm - 589.0 nm = 0.06 nm

Using the Grating Equation

The grating equation relates the angle of diffraction (θ), the wavelength (λ), and the number of rulings (N) in the grating:

d sin(θ) = mλ

where:

• d = distance between adjacent rulings (grating spacing)
• m = order of the spectrum (in this case, m = 2 for the second order)

Calculating the Angular Separation

To resolve the two wavelengths, we need to find the angular separation (Δθ) between them in the second order. The angular separation can be approximated using the formula:

Δθ ≈ (mΔλ) / d

For the second order (m = 2), we can rewrite this as:

Δθ ≈ (2Δλ) / d

Applying the Rayleigh Criterion

According to the Rayleigh criterion, the minimum angular separation that can be resolved is given by:

Δθ = λ / (N * d)

Setting the two expressions for Δθ equal gives us:

(2Δλ) / d = λ / (N * d)

We can simplify this to:

N = λ / (2Δλ)

Substituting Values

Now, we can substitute the values for λ and Δλ:

• λ = 589.0 nm = 589.0 x 10^-9 m
• Δλ = 0.06 nm = 0.06 x 10^-9 m

Plugging these into the equation gives:

N = (589.0 x 10^-9) / (2 * 0.06 x 10^-9)

Calculating this yields:

N = (589.0 / 0.12) = 4908.33

Final Result

Since the number of rulings must be a whole number, we round up to the nearest whole number. Therefore, the minimum number of rulings required in the grating to resolve the sodium doublet in the second order spectrum is:

N = 4909

In summary, a diffraction grating with at least 4909 rulings is necessary to distinguish between the two closely spaced wavelengths of the sodium doublet effectively. This calculation illustrates the importance of grating resolution in spectroscopy, particularly when analyzing closely spaced spectral lines.