Question

An electron that has the quantum number \(n=3\) and \(m=2\) must (1) have spin value \(+1 / 2\) (2) have \(l=1\) (3) have \(l=0,1\) or 2 (4) have \(l=2\)

Short answer

The electron must have l=2.

Step-by-step solution

Understand the Quantum Numbers

Quantum numbers n, l, m, and s describe the properties of electrons in atoms. For the given exercise, quantum numbers n (principal quantum number) and m (magnetic quantum number) are provided. The goal is to determine the possible value of l (angular momentum quantum number).

Identify the Range of l

For a given n, the quantum number l can take integer values from 0 to n-1. Since n=3, l can be 0, 1, or 2.

Verify the Value for m

The magnetic quantum number m can take on integer values from -l to +l. Given that m=2, we need to check which of the possible l values (0, 1, or 2) allow for m=2.

Determine the Correct l

If l=0 or l=1, the possible values for m will be constrained to the ranges -0 to 0 and -1 to 1 respectively, which do not include m=2. Only if l=2 does m=2 fall within the range -2 to 2.

Conclusion

Since l must be 2 in order for the given m value of 2 to be valid, the correct answer is (4) have l=2.

Key concepts

Principal Quantum Number

The principal quantum number, represented by the symbol n, is a fundamental concept in quantum mechanics. It describes the energy level and size of the electron's orbit around the nucleus.

For example, if an electron has a principal quantum number of n=3, it means that the electron is in the third energy level. As n increases, the energy and the average distance of the electron from the nucleus also increase.

Understanding the principal quantum number helps us predict the general behavior of an electron, such as its potential energy and how it interacts with other electrons.

Angular Momentum Quantum Number

The angular momentum quantum number, denoted as l, defines the shape of the orbital in which an electron resides. It can take any integer value from 0 to n-1, where n is the principal quantum number.

For instance, if n=3, then l can be 0, 1, or 2. Each value of l corresponds to a different orbital shape:

• l=0: s-orbital (spherical)
• l=1: p-orbital (dumbbell-shaped)
• l=2: d-orbital (cloverleaf-shaped)

These shapes determine how electrons distribute around the nucleus and how they bond with other atoms.

Magnetic Quantum Number

The magnetic quantum number, symbolized as m, specifies the orientation of the orbital in space. It can take any integer value between -l and +l, including zero.

For example, if l=2, then m can be -2, -1, 0, +1, or +2. This variety of values means that each type of orbital can align in different ways in three-dimensional space.

Understanding the magnetic quantum number is crucial because it determines the exact orbital an electron occupies. This helps us predict the magnetic properties of atoms and molecules.

Electron Properties

Electrons have several properties, described by quantum numbers n, l, m, and another quantum number called the spin quantum number (s). The spin quantum number can be either +1/2 or -1/2, indicating the two possible spin states of an electron.

In the given exercise, we have an electron with n=3 and m=2. To find the possible value of l, we use the rule that l can be any integer from 0 to n-1. Therefore, l can be 0, 1, or 2.

However, for m=2 to be valid, l must be at least 2, because m=2 is not possible if l is 0 or 1. This leads us to conclude that l must be equal to 2.