\( \mathrm{X^{a+}} \) and \( \mathrm{Y^{b+}} \) are hydrogen-like species. The wavelength of light absorbed during the transition between the states with principal quantum numbers \( n = 1 \) and \( n = 2 \) of \( \mathrm{X^{a+}} \) is \( \lambda \). The wavelength of light absorbed during the transition between the states with principal quantum numbers \( n = 2 \) and \( n = 4 \) of \( \mathrm{Y^{b+}} \) is \( 9\lambda \). The lowest possible value of \( (a + b) \) is __________.
Consider a helium (He) atom that absorbs a photon of wavelength \( 330 \, \text{nm} \). The change in the velocity (in \( \text{cm s}^{-1} \)) of He atom after the photon absorption is ____________ .
(Assume: Momentum is conserved when photon is absorbed.)
Use: Planck constant = \( 6.6 \times 10^{-34} \, \text{J s} \), Avogadro number = \( 6 \times 10^{23} \, \text{mol}^{-1} \), Molar mass of He = \( 4 \, \text{g mol}^{-1} \)
