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 ____.
A. A
For a hydrogen-like species, the wavelength of the absorbed light during a transition from n_1 to n_2 is given by the Rydberg formula: 1 = R Z^2 ( 1 n_1^2 - 1 n_2^2 ) 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^2 12 Z_X^2 = 16 3 Z_Y^2 36 Z_Y^2 = 16 Z_X^2 9 Z_Y^2 = 4 Z_X^2 3 Z_Y = 2 Z_X Z_X Z_Y = 3 2 For the lowest possible integer values of atomic numbers, we get Z_X = 3 and Z_Y = 2. Since X ^ a+ and Y ^ b+ are hydrogen-like species, they must contain exactly 1 electron. Therefore, the charge on the ion is (Z - 1). For X ^ a+ (Z_X = 3): a = 3 - 1 = 2 For Y ^ b+ (Z_Y = 2): b = 2 - 1 = 1 The lowest possible value of (a + b) is 2 + 1 = 3. Answer: 3
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