Question

The magnetic moment, \(\mathrm{m}\), is given by a formula in which \(\mathrm{n}\) is the number of unpaired electrons (a) \(\mu=n \times(n+2)\) (b) \(\mu=\sqrt{n \times(n+1)}\) (c) \(\mu \cdot \sqrt{n \times(n+2)}\) (d) \(\mu=n \times(n+1)\)

Short answer

The correct answer is option (b): \( \mu = \sqrt{n \times (n+2)} \).

Step-by-step solution

Recall the formula for the magnetic moment

The magnetic moment, \( \mu \), is given by a formula in terms of the number of unpaired electrons, \( n \). This formula is \( \mu = \sqrt{n \times (n+2)} \).

Compare the formula with the given options

Each option given in the exercise should be compared with the correct formula. By doing so, it is clear that the correct answer is option (b), which exactly matches the formula for the magnetic moment: \( \mu = \sqrt{n \times (n+2)} \). The other options do not correspond to the correct formula.

Key concepts

Understanding Unpaired Electrons

Unpaired electrons play a crucial role in determining the magnetic properties of an atom or molecule. In simple terms, an unpaired electron is an electron that does not have a counterpart electron with an opposite spin in the same atomic or molecular orbital. These lone electrons are responsible for certain magnetic behaviors, such as paramagnetism.
Paramagnetic materials are those that are attracted by external magnetic fields. This attraction occurs because the unpaired electrons have magnetic moments, which are tiny magnets generated due to their spin. In contrast, diamagnetic materials have no unpaired electrons and are slightly repelled by magnetic fields.
The presence of unpaired electrons is directly connected to the electron configuration of an element. Understanding the concept of unpaired electrons helps us anticipate an atom or molecule's reaction to magnetic fields. This is fundamental when studying elements in physical chemistry.

Exploring Magnetic Properties

The magnetic properties of materials arise from the motion of electrons, particularly those that are unpaired. These properties are crucial for understanding and predicting how materials will interact with magnetic fields.
Materials can be broadly categorized based on their response to a magnetic field:

• Ferromagnetic: Materials like iron that can become permanently magnetized.
• Paramagnetic: Materials such as aluminum and platinum that become magnetized only in the presence of an external magnetic field.
• Diamagnetic: Substances like copper and bismuth that are weakly repelled by a magnetic field.

Unpaired electrons in paramagnetic substances can align in response to an external magnetic field, enhancing the material's overall magnetic moment while the field is present.
In ferromagnetic substances, the electrons can align themselves in regions called domains, which results in a strong permanent magnet. These properties are essential in various applications, from electric motors to data storage devices.

Utilizing Physical Chemistry Formulas

Physical chemistry employs a variety of mathematical formulas to describe and predict the behavior of atoms and molecules. The magnetic moment formula, \( \mu = \sqrt{n \times (n+2)} \), is a prime example that helps to calculate the magnetic moment based on the number of unpaired electrons (n).
This formula is derived from quantum mechanics and reflects the magnetic momentum created by electron spins. In practice, \( \mu \) is expressed in units called Bohr magnetons. Each electron's spin contributes half a Bohr magneton to the total magnetic moment.
This equation serves as a quick method for chemists and physicists to estimate how magnetic a particular atom or molecule could be based on its unpaired electrons. The utilization of formulas like this is invaluable to scientists as they uncover new materials and further our understanding of chemical interactions. Knowing how to apply them is key to solving many theoretical and practical problems in physical chemistry.