Question

Answer the following questions. (a) What characteristic of an atomic orbital does the quantum number \(\mathrm{m}_{\mathrm{s}}\) describe? (b) Does a photon with a frequency of \(35.8 \mathrm{~Hz}\) have more or less energy than one with a frequency of \(125 \mathrm{~Hz}\) ? (c) How many orbitals can be associated with the following set of quantum numbers: \(\mathbf{n}=3, \ell=2\) ? (d) Is a sample of \(\mathrm{ZnCl}_{2}\) containing the \(\mathrm{Zn}^{2+}\) ion diamagnetic?

Short answer

a) The spin quantum number, represented by ms, describes the intrinsic angular momentum or "spin" of the electron within an atomic orbital. b) The photon with a frequency of 125 Hz has more energy compared to the one with a frequency of 35.8 Hz. c) There are 5 orbitals associated with the quantum numbers n=3 and l=2. d) ZnCl2 is diamagnetic, as all of its electrons are paired.

Step-by-step solution

The quantum number \(\mathrm{m}_{\mathrm{s}}\) represents the spin quantum number of an electron, which describes the intrinsic angular momentum or "spin" of the electron within an atomic orbital. #b) Compare the energy of two photons with different frequencies#

To determine whether a photon with a frequency of \(35.8 \mathrm{~Hz}\) has more or less energy than one with a frequency of \(125 \mathrm{~Hz}\), we can use the formula for the energy of a photon: \(E= h\nu\), where \(E\) is the energy of the photon, \(h\) is Planck's constant (\(6.626 \times 10^{-34} \mathrm{~Js}\)), and \(\nu\) is the frequency of the photon. Let's compute the energies for both photons: 1. For the \(35.8 \mathrm{~Hz}\) photon: \(E_1 = h\nu_1=6.626 \times 10^{-34} \mathrm{~Js} \times 35.8 \mathrm{~Hz}\) 2. For the \(125 \mathrm{~Hz}\) photon: \(E_2 = h\nu_2=6.626 \times 10^{-34} \mathrm{~Js} \times 125 \mathrm{~Hz}\) Then, let's compare the energies of the two photons: If \(E_1>E_2\), the photon with frequency \(35.8 \mathrm{~Hz}\) has more energy; otherwise, the photon with frequency \(125 \mathrm{~Hz}\) has more energy. #c) Calculate the number of orbitals associated with n=3 and l=2#

To determine the number of orbitals associated with a given set of quantum numbers, we should use the following relationship: For a given set of quantum numbers \(\mathrm{n}\) and \(\ell\), the number of orbitals (\(\mathrm{o}\)) can be calculated as \(2\ell + 1\). So, we can determine the number of orbitals associated with the quantum numbers \(\mathbf{n}=3, \ell=2\): \(\mathrm{o} = 2\ell + 1 = 2(2) + 1 = 5\). There are 5 orbitals associated with the given set of quantum numbers (n=3 and l=2). #d) Determine if ZnCl2 is diamagnetic or not#

Diamagnetic substances are those in which all electrons are paired and have no unpaired electrons. To determine if \(\mathrm{ZnCl}_{2}\) is diamagnetic, we need to examine the electronic configuration of the \(\mathrm{Zn}^{2+}\) ion. For Zn (element 30), the electronic configuration is: \(\mathrm{Zn: [Ar] 3d^{10} 4s^2}\). Losing two electrons (to form the \(\mathrm{Zn}^{2+}\) ion) gives the following configuration: \(\mathrm{Zn}^{2+}: [Ar] 3d^{10}\). Since all the electrons are paired (in this case, in the \(d\) orbitals), the \(\mathrm{Zn}^{2+}\) ion is diamagnetic, and therefore the compound \(\mathrm{ZnCl}_{2}\) is also diamagnetic.

Key concepts

Atomic Orbitals

Atomic orbitals are the regions around an atom's nucleus where there is a high probability of finding an electron. These orbitals are defined by quantum numbers, which describe their shape, orientation, and energy levels. Each orbital can hold a maximum of two electrons that have opposite spins.
To visualize atomic orbitals, one can think of them as 3D clouds around a nucleus, with different shapes such as spherical (for s orbitals) or dumbbell-shaped (for p orbitals). The principal quantum number (), azimuthal (or angular momentum) quantum number (), and magnetic quantum number () together determine the size, shape, and orientation of these orbitals.
Understanding the nature of atomic orbitals is crucial for grasping how atoms form bonds, their physical and chemical properties, and the structures of molecules.

Photon Energy

Photon energy refers to the amount of energy carried by a single photon, which is the basic unit or 'quantum' of light. This energy is directly proportional to the photon's frequency and is given by the equation (E = hu), where (E) is the energy, (h) is Planck's constant, and () is the frequency of the photon.
Higher frequency photons carry more energy than lower frequency photons. For example, ultraviolet light photons have more energy than those of visible light, which in turn have more energy than infrared light photons. Photon energy is a fundamental concept linking the quantum and classical views of light behavior and has applications in understanding atomic transitions, photoelectric effect, and many technologies such as lasers.

Electron Spin Quantum Number

The electron spin quantum number, denoted as (), describes one of the intrinsic properties of electrons called 'spin.' It can take on one of two possible values, (+1/2) or (-1/2), often referred to as 'spin up' and 'spin down.'
This quantum number is pivotal in determining the magnetic properties of an atom and explains the principles of electron pairing within an atomic orbital. Each orbital can accommodate two electrons, and according to the Pauli exclusion principle, these two electrons must have opposite spins. This concept is essential for understanding the structure of the periodic table and the behavior of atoms in magnetic fields.

Diamagnetism

Diamagnetism is a property of materials that causes them to be repelled by a magnetic field. It arises in substances where all the electrons are paired, with each pair's spins canceling out the other's magnetic effects. Diamagnetic materials, unlike ferromagnetic materials (which are strongly attracted to magnetic fields), do not retain magnetic properties once the external field is removed.
Diamagnetic behavior can be observed in elements like gold, bismuth, and in compounds where the constituent ions have completely filled electron shells or subshells, as is the case with (). Overall, diamagnetism is a subtle magnetic phenomenon compared to the more intense effects of ferromagnetism or paramagnetism.

Orbitals with Quantum Numbers

Orbitals are often identified by a set of quantum numbers: the principal quantum number (), the azimuthal (angular momentum) quantum number (), the magnetic quantum number (), and the spin quantum number (). These numbers provide a comprehensive description of an electron's state in an atom.
For example, the quantum numbers = 3, = 2, correspond to a set of five d orbitals, where () denotes the main energy level, () defines the shape of the orbital (d shape, in this case), and () would specify the orientation of these orbitals in space. The spin quantum number would then tell us the spin orientation of an electron within these orbitals. Understanding these quantum numbers is critical for predicting the electron configuration of atoms and the resulting chemical bonding and reactivity.