Question

Concerning the electrons in the shells, subshells, and orbitals of an atom, how many can have (a) \(n=4, \ell=2, m_{\ell}=1,\) and \(m_{s}=+\frac{1}{2} ?\) (b) \(n=4, \ell=2,\) and \(m_{\ell}=1 ?\) (c) \(n=4\) and \(\ell=2 ?\) (d) \(n=4 ?\) (e) \(n=4, \ell=2,\) and \(m_{s}=+\frac{1}{2} ?\)

Short answer

(a) 1 electron. (b) 2 electrons. (c) 10 electrons. (d) 32 electrons. (e) 5 electrons.

Step-by-step solution

(a) n=4, l=2, m_l=1, m_s=+1/2

Only one electron can have these exact quantum numbers because the Pauli Exclusion Principle states that no two electrons in an atom can have the same set of four quantum numbers.

(b) n=4, l=2, m_l=1

There can be two electrons with these quantum numbers, it can have one up-spin electron (\(m_s=+1/2\)) and one down-spin electron (\(m_s=-1/2\)).

(c) n=4, l=2

For given values of \(n=4\) and \(\ell=2\), \(m_{\ell}\) can range from \(-2\) to \(2\) so there are five possible values. In each orbital, there can be two electrons (one with spin up and one with spin down), hence a total of \(5*2 = 10\) electrons.

(d) n=4

For given value of the principal quantum number \(n=4\), there are four subshells \(\ell = 0, 1, 2, 3\). With 1 orbital for \(\ell = 0\), 3 orbitals for \(\ell = 1\), 5 orbitals for \(\ell = 2\), and 7 orbitals for \(\ell = 3\), and each orbital can accommodate 2 electrons, so the total number of electrons is \(2*(1+3+5+7) = 32\).

(e) n=4, l=2, m_s=+1/2

Here, for given \(n=4\), \(\ell=2\), and \(m_s=+1/2\), they needs the value of \(m_{\ell}\) which ranges from \(-2\) to \(2\). So there can be \(5\) electrons having \(m_s= +1/2\) in each orbital.

Key concepts

Pauli Exclusion Principle

The Pauli Exclusion Principle is a fundamental concept in quantum mechanics. It asserts that no two electrons in an atom can share the same set of four quantum numbers. This means each electron in an atom has its own unique "address," defined by the quantum numbers: the principal quantum number ( ext{n} ), the azimuthal quantum number ( ext{l} ), the magnetic quantum number ( ext{m_l} ), and the spin quantum number ( ext{m_s} ).

In practical terms, this principle is crucial for understanding the arrangement of electrons in atoms. For instance, when solving for how many electrons can have the quantum numbers (n=4, ext{l}=2, ext{m_l}=1, ext{m_s}=+1/2) , the answer is just one. This is because each set of quantum numbers represents a unique electron configuration that can accommodate only one electron.

Understanding this principle helps explain the structure and behavior of atoms. It ensures the stability of matter by dictating the electron arrangements that ultimately form the periodic table.

electron configuration

Electron configuration is a symbolic representation of the arrangement of electrons in an atom. It's hugely important in understanding how atoms form chemical bonds and engage in chemical reactions. Electron configurations indicate the default state of electrons under normal conditions.

The configuration adheres to guidelines derived from quantum mechanics and the Pauli Exclusion Principle. Each electron occupies specific orbitals, regions around the nucleus where the probability of finding an electron is high. The order of filling these orbitals follows the Aufbau principle, which states that electrons fill orbitals starting at the lowest available energy levels before moving to higher ones.

For example, in an atom with a known principal quantum number ( ext{n}=4 ), the sequence of filled sublevels is: 4s, then 3d, 4p, and so forth. It ends up showcasing how many electrons are in each energy level. The electron configurations help to predict and explain an atom's chemical properties and reactivity.

orbital structure

Orbital structure refers to the arrangement and shape of the orbitals within an atom. Each orbital can hold up to two electrons, which must have opposite spins, as dictated by the Pauli Exclusion Principle.

In quantum mechanics, orbitals are described by a set of quantum numbers:

• Principal quantum number ( ext{n} ) determines the orbital's size and energy level.
• Azimuthal quantum number ( ext{l} ) defines the orbital's shape.
• Magnetic quantum number ( ext{m_l} ) specifies the orbital's orientation in space.
• Spin quantum number ( ext{m_s} ) indicates the electron's spin direction.

For instance, when (n=4, ext{l}=2, and ext{m_l}=1) , these numbers describe specific 3-dimensional regions within the shell designated for ext{d} orbitals. Each type of orbital ( s, p, d, ext{or f} ) has a distinct shape:

• s orbitals are spherical.
• p orbitals are dumbbell-shaped.
• d orbitals have more complex cloverleaf shapes.

Understanding orbital structures is essential for learning about electron distribution in atoms and how this influences chemical bonding and molecule shapes.