Question

The normalized radial wave function of the second excited state of hydrogen atom is \[ R(r) = \frac{1}{\sqrt{24}} \left( \frac{a^{-3/2}}{r} \right) e^{-r/2a} \] \text{where \( a \) is the Bohr radius and \( r \) is the distance from the center of the atom. The distance at which the electron is most likely to be found is \( y \times a \). The value of \( y \) (in integer) is \(\underline{\hspace{2cm}}\).}

Hint:

To find the most probable distance for the electron in a hydrogen atom, use the radial probability distribution and find its maximum.

Answer 1

Correct Answer: 4

The most likely distance \( r_{\text{max}} \) where the electron is found is given by the maximum of the radial probability distribution \( P(r) \). This occurs when \( r \) satisfies: \[ \frac{d}{dr} \left( r^2 R(r)^2 \right) = 0. \] By solving for \( r \), we find that the most probable distance is approximately \( 4a \). Thus, the value of \( y \) is \( 4 \).