The radial wave function for the 3 s orbital of a hydrogen atom is given by R_ 3 s = 1 9 3 ( 1 a₀ )^ 3/2 (6 - 6 + ^2) e^ - /2 where = 2r 3a₀ and a₀ is the Bohr radius. If r₁ and r₂ are the distances of the two radial nodes from the nucleus, the sum (r₁ + r₂) is

Step-by-step solution

At a radial node, the probability density is zero, which means the radial wave function R_ 3 s must be zero. Since the exponential term and the constants cannot be zero, the polynomial part must equal zero: ^2 - 6 + 6 = 0 Let the roots of this quadratic equation be ₁ and ₂ . According to the properties of quadratic equations, the sum of the roots is: ₁ + ₂ = - -6 1 = 6 Given the substitution = 2r 3a₀ , we can write the sum of the roots in terms of r₁ and r₂ : 2r₁ 3a₀ + 2r₂ 3a₀ = 6 Factoring out the common terms: 2