Question

Which orbital in each of the following pairs is lower in energy in a many- electron atom? (a) \(2 s, 2 p\) (b) \(3 p, 3 d ;\) (c) \(3 s, 4 s ;\) (d) \(4 d, 5 f\).

Short answer

In each pair, the orbital with lower energy is (a) 2s , (b) 3p , (c) 3s and (d) 4d.

Step-by-step solution

Comparing 2s and 2p

In many-electron atoms, ordering of energy levels is based on the (n+l) rule, where n is the principal quantum number and l is the azimuthal quantum number. For the 2s orbital, the sum is 2+0=2 while for the 2p orbital, the sum is 2+1=3. Since 2 is smaller than 3, the 2s orbital has less energy than the 2p orbital.

Comparing 3p and 3d

The sum of n and l for the 3p orbital is 3+1=4 while for the 3d orbital, the sum is 3+2=5. Therefore, the 3p orbital is lower in energy than the 3d orbital.

Comparing 3s and 4s

For the 3s orbital, the sum of n and l is 3+0=3 while for the 4s orbital, the sum is 4+0=4. In this case, the 3s orbital is lower in energy than the 4s orbital.

Comparing 4d and 5f

The sum of n and l for the 4d orbital is 4+2=6 while for the 5f orbital, the sum is 5+3=8. Therefore, the 4d orbital is lower in energy than the 5f orbital.

Key concepts

Quantum Numbers

Quantum numbers are key to understanding the complex behavior of electrons in atoms. Think of them like an electron's address within an atom, specifying its location and movement.

There are four quantum numbers in total: principal (n), azimuthal (l), magnetic (m), and spin (s). However, for the context of orbital energy levels, we'll focus on the first two, which play a significant role in determining an electron's energy state and the shape of the region it occupies.

The principal quantum number (n) represents the energy level of an electron and its rough distance from the nucleus, while the azimuthal quantum number (l) shapes the orbital, which can be s, p, d, or f-type. The values for l range from 0 to (n-1). These numbers provide a framework for predicting electron configuration and the atom's chemical behavior.

Principal Quantum Number

The principal quantum number, denoted as (n), is akin to the 'floor number' in an 'electron hotel'. It specifies the possible energy levels or shells that an electron can occupy, starting from 1, 2, and so on. As n increases, the electron is further from the nucleus and possesses higher energy.

In the realm of quantum mechanics, the difference in energy between these levels decreases as n gets larger. Therefore, electrons in higher energy levels are more loosely bound to the nucleus and can be removed more easily. The principal quantum number is crucial when determining how electrons fill up orbitals, following the Aufbau principle, which states that electrons occupy the lowest available energy levels first.

Azimuthal Quantum Number

Following the principal quantum number is the azimuthal quantum number (l), which indicates the subshell or shape of the orbital where the electron can be found. If we continue our 'electron hotel' analogy, l would represent the 'suite type' within each floor. It is quantized and assumes integer values from 0 to (n-1) for each principal quantum number.

Each value of l corresponds to a specific type of orbital shape:

• 0: s-orbital (spherical)
• 1: p-orbital (dumbbell-shaped)
• 2: d-orbital (clover-shaped)
• 3: f-orbital (complex shapes)

The azimuthal quantum number is a determining factor in electron placement and ultimately influences the atom's chemical bonding. As l increases, the orbitals become more complex and have higher energy within the same principal quantum number.