X ^ a+ and 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 X ^ a+ is . The wavelength of light absorbed during the transition between the states with principal quantum numbers n = 2 and n = 4 of Y ^ b+ is 9 . The lowest possible value of (a + b) is ____.

Step-by-step solution

For a hydrogen-like species, the wavelength of the absorbed light during a transition from n₁ to n₂ is given by the Rydberg formula: 1 = R Z^2 ( 1 n₁^2 - 1 n₂^2 ) For X ^ a+ , the transition is from n = 1 to n = 2 : 1 = R Z_X^2 ( 1 1^2 - 1 2^2 ) = 3 4 R Z_X^2 = 4 3 R Z_X^2 For Y ^ b+ , the transition is from n = 2 to n = 4 : 1 9 = R Z_Y^2 ( 1 2^2 - 1 4^2 ) = R Z_Y^2 ( 1 4 - 1 16 ) = 3 16 R Z_Y^2 9 = 16 3 R Z_Y^2 Substituting the expression for from the first equation into the second: 9 ( 4 3 R Z_X^2 ) = 16 3 R Z_Y^