A student shines a beam of hydrogen-lamp light through a diffraction grating whose line density is unknown. On a distant viewing screen she measures the angles (relative to the central maximum) of two first-order spectral lines: the red line at 656 nm appears at an angle of 21.5°, and the violet line at 434 nm appears at 14.4°. Assume the incident beam is perpendicular to the grating.
(1) Determine the spacing between adjacent slits (lines) in the grating.
(2) Predict the angle, relative to the central maximum, at which the second-order violet line (434 nm) should appear, if it is observable.
(3) Explain whether a second-order red line (656 nm) will be observable with this grating. Support your answer with quantitative reasoning.