Find the location of the radial node for an electron in a 3p orbital of a hydrogen atom (in meters)
The Correct Answer and Explanation is :
To determine the location of the radial node for an electron in the (3p) orbital of a hydrogen atom, we analyze the wavefunction of the orbital. For hydrogen atoms, the radial wavefunction (R_{n\ell}(r)) can be used to find radial nodes, which are the points where (R_{n\ell}(r) = 0).
1. Radial Wavefunction for the (3p) Orbital
For the hydrogen atom:
[
R_{n\ell}(r) = \text{Normalization constant} \cdot \rho^\ell \cdot e^{-\rho/2} \cdot L_{n-\ell-1}^{2\ell+1}(\rho)
]
where:
- (n = 3), (\ell = 1) (for the (p)-orbital),
- (\rho = \frac{2Zr}{na_0}),
- (a_0 = 5.29 \times 10^{-11}\ \text{m}) is the Bohr radius,
- (L_{n-\ell-1}^{2\ell+1}(\rho)) is an associated Laguerre polynomial.
For the (3p) orbital, the radial wavefunction has a single node. The location of the node can be found by solving:
[
R_{3p}(r) = 0 \quad \text{or} \quad L_1^3(\rho) = 0.
]
2. Solving for the Node
The relevant Laguerre polynomial for (3p) is:
[
L_1^3(\rho) = 6 – \rho.
]
Setting (L_1^3(\rho) = 0):
[
6 – \rho = 0 \quad \Rightarrow \quad \rho = 6.
]
Substituting (\rho = \frac{2r}{3a_0}):
[
\frac{2r}{3a_0} = 6 \quad \Rightarrow \quad r = 9a_0.
]
3. Final Answer
The location of the radial node is:
[
r = 9a_0 = 9 \cdot 5.29 \times 10^{-11}\ \text{m} = 4.76 \times 10^{-10}\ \text{m}.
]
Explanation
Radial nodes arise from the structure of the radial wavefunction, which depends on (n), (\ell), and the Laguerre polynomials. For the (3p) orbital, there is one radial node due to the relationship between the principal quantum number ((n = 3)) and the orbital angular momentum quantum number ((\ell = 1)). Solving the wavefunction yields (r = 9a_0), providing a precise location of the node.