Final answer:
Second order interference is more precise when measuring the bright green mercury line with a spectrometer and diffraction grating, due to greater angular separation and sharper diffraction patterns at higher orders.
Step-by-step explanation:
Considering the precision of measurements for the bright green mercury line based on first order interference and second order interference using a spectrometer and a diffraction grating, the second order interference measurement would be more precise. This is because higher order measurements spread out the diffraction pattern more, allowing for more precise determinations of wavelength. Essentially, the angular separation between diffraction orders increases with order number, which makes it easier to measure and distinguish between closely spaced wavelengths.
Moreover, a diffraction grating with a high number of lines per centimeter will produce more sharply defined fringes, improving the resolution. When more lines are present, the maximum sharpness of the pattern is increased, which aids in making more accurate measurements. Also, using longer wavelengths can increase the interference pattern's spacing, consistent with the relationship that longer wavelengths produce larger diffraction angles for the same order.