Suppose you have measured the diffraction pattern of a grating with d = 0.09 mm and have found that the spots were separated by s = 1 cm. Now you want to determine d for an unknown grating. With the unknown grating, the spots are separated by s = 3.1 cm. What is d?

To determine the value of d for the unknown grating, we will use the concept of diffraction grating equation:

mλ = dsinθ

Where:
- m is the order of the diffraction maxima (1 for the central maximum),
- λ is the wavelength of the source used in the experiment (which is usually given),
- d is the spacing between adjacent slits of the grating, and
- θ is the angle between the incident light and the direction of the mth-order diffraction maxima.

From the given information, we know that for the first grating, d = 0.09 mm and the spots are separated by s = 1 cm. Let's first calculate λ for the first grating using the given values.

For the first grating:
s = mλ/d

We can rearrange this equation to solve for λ:
λ = s*d/m

Substituting the known values:
λ = (1 cm)*(0.09 mm)/(1)

Remember to convert mm to cm: 0.09 mm = 0.009 cm

λ = (1 cm) * (0.009 cm) / (1) = 0.009 cm

Now, for the unknown grating, we are given that the spots are separated by s = 3.1 cm. We can now find d for the unknown grating using the same equation:

s = mλ/d

As before, rearrange the equation to solve for d:
d = mλ/s

Substituting the known values:
d = (1)*(0.009 cm) / (3.1 cm)

Calculating:
d = 0.009 cm / 3.1 cm = 0.00290322 cm

So, the value of d for the unknown grating is approximately 0.00290322 cm.