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 find the location of the radial node for an electron in a 3p orbital of a hydrogen atom, we need to understand the nature of the wavefunction for the hydrogen atom’s orbitals.
Step 1: Understanding the Radial Wavefunction
The radial wavefunction for the hydrogen atom is given by the equation:
[
R_{nl}(r) = \left(\frac{2}{a_0}\right)^{3/2} \sqrt{\frac{(n-l-1)!}{2n[(n+l)!]}} \left(\frac{2r}{na_0}\right)^l e^{-r/na_0}
]
Where:
- ( n ) is the principal quantum number,
- ( l ) is the orbital angular momentum quantum number,
- ( r ) is the radial distance from the nucleus,
- ( a_0 ) is the Bohr radius (approximately ( 5.29 \times 10^{-11} \, \text{m} )).
For a 3p orbital, the quantum numbers are:
- ( n = 3 ) (principal quantum number),
- ( l = 1 ) (for a p-orbital).
Step 2: Finding the Radial Node
Radial nodes occur where the radial wavefunction equals zero, i.e., ( R_{nl}(r) = 0 ). For any orbital, the number of radial nodes is given by ( n – l – 1 ). For the 3p orbital:
[
\text{Number of radial nodes} = 3 – 1 – 1 = 1.
]
Thus, there is 1 radial node for a 3p orbital.
Step 3: Finding the Position of the Radial Node
The position of the radial node can be found by solving for ( r ) in the equation for the radial wavefunction where ( R_{3,1}(r) = 0 ). The radial wavefunction for a 3p orbital has a node at a distance given by:
[
r_{\text{node}} = \frac{2a_0}{3 – 1} = a_0.
]
Thus, the location of the radial node is at ( r = a_0 ), which is approximately ( 5.29 \times 10^{-11} \, \text{m} ).
Explanation
For a hydrogen atom, the radial wavefunction describes how the probability of finding an electron varies with distance from the nucleus. The nodes correspond to distances where the probability density (the square of the radial wavefunction) is zero. In the case of the 3p orbital, the node is located at one Bohr radius from the nucleus, where the probability of finding the electron is zero. This result is consistent with the nature of orbitals in quantum mechanics and reflects the underlying mathematical structure of the hydrogen atom’s wavefunctions.