Question

An orbital is described with the help of a wave function. Since many wave functions are possible for an electron, there are many atomic orbitals. When atom is placed in a magnetic field the possible number of orientations for an orbitul of aximuthal quantum number 3 is (a) three (b) two (c) five (d) seven.

Short answer

The possible number of orientations for an orbital with an azimuthal quantum number of 3 is seven.

Step-by-step solution

Understanding Azimuthal Quantum Number

The azimuthal quantum number, also known as the angular momentum quantum number or orbital quantum number, is denoted by 'l' and determines the shape of the atomic orbital. This number can take on any integer value from 0 to 'n-1', where 'n' is the principal quantum number.

Calculate the Magnetic Quantum Numbers

For a given azimuthal quantum number 'l', the magnetic quantum number 'm_l' defines the number of orientations an orbital can have in a magnetic field. 'm_l' can range from '-l' to '+l' including zero. Therefore, the total number of possible orientations is '2l+1'.

Find the Number of Orientations for Azimuthal Quantum Number 3

Substituting the given azimuthal quantum number 'l=3' into the formula '2l+1', we get the number of possible orientations: 2(3) + 1 = 6 + 1 = 7.

Key concepts

Understanding Azimuthal Quantum Number

The azimuthal quantum number, represented by the symbol 'l', is integral to the structure of an atom. It's essential for determining the shape and, consequently, the type of atomic orbitals. This quantum number ranges from 0 to 'n-1', where 'n' is the principal quantum number. For instance, for the principal quantum number 3, 'l' can be 0, 1, or 2, corresponding to 's', 'p', or 'd' orbitals, respectively. Each type of orbital has a unique shape, 's' being spherical, 'p' dumbbell-shaped, and so on. These shapes are dictated by the solutions to the Schrödinger equation, which defines how electrons propagate as waves.

Additionally, the azimuthal quantum number also influences electron energy levels within a multi-electron atom. Electrons in orbitals with higher 'l' values typically possess higher energy. When instructing students, it's essential to use visual aids to depict orbital shapes so they can more easily comprehend why 'l' is critical for an orbital's geometry and electron configuration.

Magnetic Quantum Number: Defining Orbital Orientation

The magnetic quantum number, denoted as 'm_l', further refines our understanding of atomic orbitals by determining their orientation in a magnetic field. This value originates from quantization of angular momentum and can take on any integer from -l to +l, including zero. Consequently, for each value of the azimuthal quantum number 'l', there are '2l+1' possible values for 'm_l'. These denote different spatial orientations of the orbital. For example, with 'l=1', 'm_l' can be -1, 0, or +1, indicating three different orientations of a 'p' orbital.

It's crucial for students to grasp that the existence of multiple magnetic quantum numbers and their corresponding orientations underpin the concept of degenerate orbitals—orbitals that have the same energy but are oriented differently in space. A tip for student comprehension is to equate the 'm_l' with different directions in which orbitals can point, analogous to pivoting an object in space around an axis.

Wave Function: The Core of Quantum Mechanics

At the heart of quantum mechanics is the wave function, a mathematical function that encapsulates all information about a quantum system—in our case, an electron in an atom. It is represented by the symbol 'ψ' (psi). The square of the wave function's magnitude, |ψ|^2, translates to the probability density of finding an electron at a given point in space. Thus, while the wave function itself may not be directly observable, it predicts observable phenomena—the likelihood of where an electron might be.

Understanding the wave function is pivotal for students venturing into quantum chemistry or physics. It aids in conceptualizing how electrons, behaving both as particles and waves, are distributed around a nucleus. To foster understanding, educators sometimes use the analogy of a wave on a string: different vibrations (wave functions) lead to different patterns (orbitals), each with a specific likelihood (probability densities) of finding the string (electron) at a certain point.