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Mobile App Chapter 2: Problem 60
What are the possible values of ml for each given value of l? a. 0 b. 1 c. 2 d. 3
Short Answer
Expert verified
For l=0, ml=0; for l=1, ml= -1, 0, +1; for l=2, ml= -2, -1, 0, +1, +2; for l=3, ml= -3, -2, -1, 0, +1, +2, +3.
Step by step solution
01
Understanding Quantum Numbers
Quantum numbers describe values of conserved quantities in the dynamics of quantum systems. The azimuthal (or angular momentum) quantum number, denoted by 'l', determines the shape of the orbital, and for each value of 'l', the magnetic quantum number 'ml' can take on integer values ranging from -l to +l, including zero.
02
Determining ml for l=0
For a given azimuthal quantum number 'l' of 0, the magnetic quantum number 'ml' can only be 0, because -l to +l with l=0 gives a range of 0 to 0.
03
Determining ml for l=1
For an azimuthal quantum number 'l' of 1, the magnetic quantum number 'ml' can take on values from -1 to +1. This means 'ml' can be -1, 0, or +1.
04
Determining ml for l=2
For an azimuthal quantum number 'l' of 2, 'ml' can range from -2 to +2. Thus, 'ml' can be -2, -1, 0, +1, or +2.
05
Determining ml for l=3
For an azimuthal quantum number 'l' of 3, 'ml' can take values from -3 to +3. Therefore, 'ml' can be -3, -2, -1, 0, +1, +2, or +3.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quantum Numbers
In quantum mechanics, the quantum numbers are like the unique address for an electron within an atom. They describe characteristics like energy level, shape, and orientation of an electron's orbital. Think of them as the specifics of a room in a very large hotel, where each room has a distinct view and features.
There are four types of quantum numbers:
Together, these quantum numbers ensure that no two electrons in an atom have the same set of quantum numbers, a principle known as the Pauli exclusion principle.
There are four types of quantum numbers:
- The principal quantum number (\( n \)), which tells us the energy level and size of the orbital.
- The azimuthal quantum number (\( l \)), responsible for the shape of the orbital.
- The magnetic quantum number (\( m_l \)), which describes the orientation of the orbital in space.
- The spin quantum number (\( m_s \)), indicating the direction of the electron’s spin.
Together, these quantum numbers ensure that no two electrons in an atom have the same set of quantum numbers, a principle known as the Pauli exclusion principle.
Azimuthal Quantum Number
The azimuthal quantum number (\( l \)), also known as the angular momentum quantum number, is second in the hierarchy of quantum numbers and is integral in defining the shape of an electron's orbital. It can take on any integer value from 0 to \( n-1 \) (where \( n \) is the principal quantum number), with each number corresponding to a specific subshell:
The azimuthal quantum number not only determines the shape of the orbital but also indirectly influences electron energy levels and the overall chemistry of atoms.
- \( l = 0 \) relates to the s subshell
- \( l = 1 \) correlates with the p subshell
- \( l = 2 \) corresponds to the d subshell
- \( l = 3 \) is for the f subshell, and so on.
The azimuthal quantum number not only determines the shape of the orbital but also indirectly influences electron energy levels and the overall chemistry of atoms.
Orbital Shapes
The shape of an electron orbital is determined by the azimuthal quantum number. As we have seen, each value of \( l \) is associated with a particular shape:
The number of orbitals in a given subshell is equal to \( 2l + 1 \), which gives us one orbital for s, three for p, five for d, and seven for f subshells. These shapes are critical because they determine how electrons are distributed around an atom's nucleus, thereby influencing how atoms bond with each other.
- For \( l = 0 \) (s subshell), the orbitals are spherical.
- For \( l = 1 \) (p subshell), they take on a dumbbell shape.
- \( l = 2 \) (d subshell) orbitals are more complex, often described as clover-shaped.
- The \( l = 3 \) (f subshell) orbitals are even more intricate.
The number of orbitals in a given subshell is equal to \( 2l + 1 \), which gives us one orbital for s, three for p, five for d, and seven for f subshells. These shapes are critical because they determine how electrons are distributed around an atom's nucleus, thereby influencing how atoms bond with each other.
Quantum Mechanics
Quantum Mechanics is the branch of physics that studies the behavior of matter and energy at the atomic and subatomic levels. It deals with phenomena that classical physics cannot adequately explain, like the wave-particle duality of light and matter.
Quantum mechanics introduces the concept that particles can exist in multiple states at once, known as superposition, and their properties become definite only when measured, which is referred to as collapsing the wave function. This field of physics is fundamental to understanding how the universe operates on the smallest scales and is the backbone for emerging technologies such as quantum computing.
Quantum mechanics introduces the concept that particles can exist in multiple states at once, known as superposition, and their properties become definite only when measured, which is referred to as collapsing the wave function. This field of physics is fundamental to understanding how the universe operates on the smallest scales and is the backbone for emerging technologies such as quantum computing.
