Question

(a) For \(n=4,\) what are the possible values of \(l ?(\mathbf{b})\) For \(l=2\) what are the possible values of \(m_{l} ?(\mathbf{c})\) If \(m_{l}\) is \(2,\) what are the possible values for \(l\) ?

Short answer

For n = 4, the possible values of l are \{0, 1, 2, 3\}. For l = 2, the possible values of m_l are \{-2, -1, 0, 1, 2\}. For m_l = 2, the possible values of l are \{2, 3\}.

Step-by-step solution

Rule for l values

The possible values of l for a given value of n are 0, 1, 2, ..., n-1.

Calculate l values for n = 4

Based on the rule and given n = 4, the possible values of l are 0, 1, 2, 3. For n = 4, the possible values of l are {0, 1, 2, 3}. #b# Finding possible values of m_l for l = 2

Rule for m_l values

The possible values of m_l for a given value of l are -l, -(l-1), ..., 0, ..., (l-1), l.

Calculate m_l values for l = 2

Based on the rule and given l = 2, the possible values of m_l are -2, -1, 0, 1, 2. For l = 2, the possible values of m_l are {-2, -1, 0, 1, 2}. #c# Finding possible values of l for m_l = 2

Finding possible l values for m_l

Since m_l can have values from -l to l, we need to find possible values of l such that m_l = 2 is included in that range.

Check l values

Let's check different l values that could have m_l = 2: - If l = 0, m_l can only be {0} – 2 is not possible. - If l = 1, m_l values are {-1, 0, 1} – 2 is not possible. - If l = 2, m_l values are {-2, -1, 0, 1, 2} – 2 is possible. - If l = 3, m_l values are {-3, -2, -1, 0, 1, 2, 3} – 2 is possible. For m_l = 2, the possible values of l are {2, 3}.

Key concepts

Azimuthal Quantum Number

When we dive into the tiny world of atoms, we encounter a set of rules defined by quantum mechanics. One such rule involves the azimuthal quantum number, denoted as \( l \), which is crucial for understanding electron configurations in an atom.

The azimuthal quantum number determines the electron's angular momentum and the shape of the orbital. Orbits are the paths that electrons are likely to be found in around the atom's nucleus. Think of them as different layers of an onion, where the electrons move in specific patterns within each layer.

For an atom with principal quantum number \( n = 4 \), the azimuthal quantum number \( l \) can take on specific integer values ranging from 0 up to \( n - 1 \) which are {0, 1, 2, 3}. Each number correlates with a different orbital shape: \( l = 0 \) for s-orbitals (spherical), \( l = 1 \) for p-orbitals (dumbbell-shaped), \( l = 2 \) for d-orbitals (cloverleaf shaped), and \( l = 3 \) for f-orbitals (complex shapes). These shapes and sizes influence how electrons interact with each other and with the nucleus, affecting the atom's chemical properties.

Tying it to Electron Configurations

The knowledge of the azimuthal quantum number is fundamental when drawing electron configurations, as it guides us to place electrons into the correct orbitals. Remember, an electron configuration is like the address of electrons; it tells us where they 'live' in the atom.

Magnetic Quantum Number

Within each orbital shape defined by the azimuthal quantum number, electrons can have different orientations in space that are described by the magnetic quantum number, \( m_l \).

Consider an electron not just spinning in its orbital, but also spinning at different axes. The magnetic quantum number can range from \( -l \) to \( +l \) including zero. Therefore, for \( l = 2 \), \( m_l \) values span from -2 to +2: {-2, -1, 0, 1, 2}. This gives us five possible orientations of the d-orbitals where the electrons can be found.

Additionally, \( m_l \) determines the response of an electron to an external magnetic field, which can cause energy splitting in the orbital, a phenomenon used in techniques such as magnetic resonance imaging (MRI).

Practical Implications

This has practical applications in technology and materials science, for instance, when determining the magnetic properties of a substance or its behavior in an electromagnetic field. Knowing the possible \( m_l \) values helps predict these outcomes. It's like knowing the direction a top spins will tell you a lot about its motion in a magnetic field.

Angular Momentum Quantum Number

While the azimuthal quantum number is often used interchangeably with the angular momentum quantum number, there is a distinction to be made. The latter specifically refers to the quantized angular momentum of an electron within an orbital, which is determined by \( l \).

The concept of angular momentum in quantum mechanics can be tricky because it's not quite the same as the everyday angular momentum we're familiar with. In simple terms, think of angular momentum as the 'twirliness' of an electron around the nucleus - just like how a figure skater's spin speed changes when they pull their arms in or out.

This quantum number is so fundamental that it dictates the allowable energy levels an electron can exhibit. For a given energy level (principal quantum number \( n \) ) and orbital (azimuthal quantum number \( l \)), the quantum state’s angular momentum is restricted to defined packets of size \( \hbar \) (reduced Planck's constant) multiplied by the square root of \( l(l + 1) \), giving an idea of the momentum's intensity but not its direction.

Interconnection of Quantum Numbers

All these properties are intertwined; the spin of electrons around the nucleus (their angular momentum), their energy levels, and their orientation in space, are all pieces of a grand puzzle that allow us to understand the structure, reactivity, and properties of atoms. Together, they underpin the entire field of quantum chemistry and inform fields as diverse as astronomical spectroscopy to practical electronics.