The radial wave function for the 3s orbital of a hydrogen atom is given by the expression: R_ 3s (r) = A (27 - 18 + 2 ^2) e^ - /3 where = r a₀ and a₀ is the Bohr radius. The sum of the distances of the radial nodes from the nucleus is:

Step-by-step solution

Radial nodes occur at distances from the nucleus where the radial wave function R(r) is zero (excluding r = 0 and r ). Setting the polynomial part of the given wave function to zero: 2 ^2 - 18 + 27 = 0 This is a quadratic equation in terms of . The roots ₁ and ₂ represent the positions of the two radial nodes. Using the sum of roots formula for a quadratic equation ax^2 + bx + c = 0 (Sum = - b a ): ₁ + ₂ = - -18 2 = 9 Since = r a₀ , we have r = a₀ . The sum of the physical distances of the radial nodes from the nuc