According to qauntum mechanical model of H-like species, and electron can be represented by a wave function `(psi)` which contain all dynamic information about the electron. The nature of wave function depends on the type of the orbital to which the electron belongs. For an orbital
`psi=[sqrt(2)/(81sqrt(3pi))]((1)/(a_(0)))^(3//2)(27-18sigma+2sigma^(2))e^((sigma)/(3))`
Where, `sigma =((Zr)/(a_(0))),r` = radial distance from nucleous, `a_(0)=52.9pm`
Which of the following represents the position of one of the radial nodes?

Options

A. `r=(1)/(2)(3+sqrt(3))a_(0)`
B. `r=(1)/(2)(3-sqrt(3))a_(0)`
C. `r=(3)/(2)(3-sqrt(6))a_(0)`
D. `r=(3)/(2)(3+sqrt(3))a_(0)`

Answer

D

Explanation

For radial node,
`psi(r )=0`
`27-18sigma+2sigma^(2)=0`
`sigma=(18pmsqrt(18^(2)-4xx2xx27))/(2xx2)=(18pmsqrt(108))/(4)=(18pmsqrt(36xx3))/(4)`
`sigma=(18pm6sqrt(3))/(4)`
`=(3)/(2)(3pmsqrt(3))`