Germicidal lamps are used to sterilize tools in biological and medical facilities. One type of germicidal lamp is a low-pressure mercury discharge tube, similar to a fluorescent lightbulb, that has been optimized to emit ultraviolet light with a wavelength of \(254 \mathrm{~nm}\). A 15-mm-diameter, \(25-\mathrm{cm}\)-long tube emits \(4.5 \mathrm{~W}\) of ultraviolet light. The mercury vapor pressure inside the tube is \(1.0 \mathrm{~Pa}\) at the operating temperature of \(40^{\circ} \mathrm{C}\).

a. On average, how many ultraviolet photons does each mercury atom emit per second?

b. The lifetime of the excited state in mercury is a rather long \(120 \mathrm{~ns}\). On average, what fraction of its time does a mercury atom spend in the excited state?