Question

An electron is in one of the \(3 d\)-orbitals. What are the possible values of \(n, l\) and \(m\) for this electron? (a) \(n=3, l=0, m_{l}=0\) (b) \(n=3, l=1, m_{1}=-1,0,+1\) (c) \(n=3, l=2, m_{l}=-2,-1,0,+1,+2\) (d) \(n=3, l=3, m_{l}=-3,-2,-1,0,+1,+2,+3\)

Short answer

For an electron in a \(3d\)-orbital, the possible values are \(n=3\), \(l=2\), and \(m_l=-2,-1,0,+1,+2\).

Step-by-step solution

Understand Quantum Numbers

Quantum numbers describe the properties of atomic orbitals and the properties of electrons in orbitals. The principal quantum number (n) indicates the energy level and size of the orbital. The azimuthal quantum number (l) determines the shape of the orbital and can be any integer from 0 to \(n-1\). The magnetic quantum number (\(m_l\)) specifies the orientation of the orbital in space and can have values from -l through 0 to +l.

Evaluate Given Options

Assess each option by applying the restrictions set by the quantum numbers. (a) \(n=3, l=0, m_l=0\) corresponds to an s-orbital, which is spherical, not a d-orbital. (b) \(n=3, l=1, m_{l}=-1,0,+1\) corresponds to a p-orbital, which has dumbbell shapes, not a d-orbital. (c) \(n=3, l=2, m_{l}=-2,-1,0,+1,+2\) corresponds to a d-orbital, which is the correct shape for the electron's location. (d) \(n=3, l=3, m_{l}=-3,-2,-1,0,+1,+2,+3\) would correspond to an f-orbital, which is not what the electron is in according to the exercise statement.

Key concepts

Principal Quantum Number

The principal quantum number, represented by the symbol , plays a vital role in the architecture of the atom. Think of it as the 'address' of an electron, specifying the energy level in which it resides. As the value of increases, the electron orbits further from the nucleus and occupies a larger, higher energy orbital. For instance, an electron in a 3d-orbital has a value of 3, which tells us it's in the third energy level, a zone with distinct energy and size characteristics compared to levels 1 or 2. Understanding the significance of is critical for studying the periodic table and diving into electron configurations.

Azimuthal Quantum Number

The azimuthal quantum number, known as l, shapes our understanding of atomic orbitals by defining their geometry. In a sense, l is the 'shape identifier', ranging from 0 up to (-1). Each shape—a sphere for  = 0, a dumbbell for  = 1, and more complex shapes for higher values—contains electrons with distinct angular momentum characteristics. For a 3d-orbital, the  value is 2 because it's the third shape category after 's' (sphere) and 'p' (dumbbell). Think of l as a key to unlocking the visual models of where electrons like to 'hang out' around the nucleus.

Magnetic Quantum Number

The magnetic quantum number, denoted by m_l, narrows down an electron's location by describing its orbital's orientation in space relative to an external magnetic field. The value of m_l can span from -l to l, including zero. A 3d-orbital, with an l value of 2, means that m_l can be -2, -1, 0, 1, or 2. These numbers represent the various orientations a d-orbital can adopt—think of them as a suite of spatial arrangements. For students, visualizing these orientations can be as tricky as a magnetic compass in a hall of mirrors, but they are fundamental to the nature of atomic orbitals.

Atomic Orbitals

Atomic orbitals are regions in an atom where there is a high probability of finding electrons. Each orbital is defined by a unique combination of quantum numbers citing energy (), shape (l), and orientation (m_l). The various orbital types, denoted by letters such as s, p, d, and f, provide a map for predicting electron arrangements. When solving problems like determining the possible l and m_l for a 3d-orbital, one must consider the designated shape and manifold of space orientations corresponding to the given quantum numbers. The correct identification of atomic orbitals is like piecing together a cosmic puzzle—each piece, an electron with its own 'quantum ID', fits into a larger atomic picture.

Electron Configuration

Electron configuration refers to the distribution of electrons across the different orbitals in an atom. Like a game of musical chairs at the quantum level, electrons fill up orbitals following specific rules: Pauli's exclusion principle, Hund's rule, and the aufbau principle. These rules dictate how electrons are arranged in orbitals, layers, and sublayers, optimizing the atom’s stability. For a 3d-orbital, electron configuration plays a critical part, as it provides insight into the atom's chemical properties and reactivities. Mastering electron configurations means you're one step closer to predicting how atoms will interact in chemical reactions, a skill at the very heart of chemistry.