Answer
12500 slits/cm.
Work Step by Step
See Figure 24–26. The angle of the diffracted light increases as the wavelength increases. To not see any lines in the second order, use the smallest wavelength in the visible spectrum, 400 nm. The maximum angle of diffraction is 90 degrees. If the second-order 400nm line falls at that angle, then no other second-order lines will be visible.
Solve Eq. 24–4 for the slit separation d.
$$d sin \theta = m \lambda $$
$$d = \frac{m \lambda}{ sin \theta }$$
$$d = \frac{2(400\times10^{-9}m)}{sin 90^{\circ}} = 8.00\times10^{-7}m=8.00\times10^{-5}cm $$
The reciprocal of the slit separation gives the number of slits per cm.
$$\frac{1}{d}=\frac{1}{8.00\times10^{-5}cm }=12500\frac{slits}{cm}$$
The answer is reported to 3 significant figures. It should be noted that the slit spacing is on the order of a wavelength of light, so this would require sophisticated nanofabrication techniques to fabricate. Most commercial diffraction gratings are far, far coarser than this.
