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Which one of the following represents the \(3p\) radial wave function of hydrogen atom? (\(a_{0}\) is the Bohr radius)

  • A.Plot of R(r) against r/a0: starts at a positive value at r = 0, falls through zero to a shallow negative minimum, then rises slowly towards zero
  • B.Plot of R(r) against r/a0: starts at zero, rises to a positive peak, crosses the axis once to a negative minimum and returns slowly towards zero
  • C.Plot of R(r) against r/a0: starts at a positive value at r = 0, falls through zero to a shallow negative minimum and returns slowly towards zero
  • D.Plot of R(r) against r/a0: starts at zero, rises to a single positive peak and decays to zero without crossing the axis

Answer: B

The 3p radial function is sketch (B).

A hydrogen radial function \(R_{nl}\) has \(n - l - 1\) radial nodes and behaves as \(r^{l}\) near the origin. For 3p, \(n = 3\) and \(l = 1\), so

\[ \begin{aligned} R_{31} &\propto \frac{r}{a_{0}}\left(1 - \frac{r}{6a_{0}}\right) \\ &\quad \times e^{-r/3a_{0}} \end{aligned} \]

It is zero at \(r = 0\), rises to a positive maximum, crosses zero once at \(r = 6a_{0}\), dips negative and decays to zero. Sketch (B) shows exactly this.

Sketches (A) and (C) are non-zero at the origin, which only s states (\(l = 0\)) do: (A), with two nodes, is the 3s function and (C), with one node, is 2s. Sketch (D) starts at zero but has no node, which is the 2p function.

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