Question

A compound of vanadium has a magnetic moment of \(1.73 \mathrm{BM}\). The electronic configuration of vanadium ion in the compound is (a) \([\mathrm{Ar}] 3 \mathrm{~d}^{2}\) (b) \([\mathrm{Ar}] 3 \mathrm{~d}^{1}\) (c) \([\mathrm{Ar}] 3 \mathrm{~d}^{3}\) (d) [Ar]4s

Short answer

The vanadium ion with electronic configuration [Ar] 3d^1 has one unpaired electron, which corresponds to the magnetic moment of 1.73 BM.

Step-by-step solution

Understanding Magnetic Moment

Magnetic moment (mu) is related to the number of unpaired electrons in the atom or ion. It can be calculated using the formula mu = sqrt{n(n + 2)} where n is the number of unpaired electrons.

Calculating the Number of Unpaired Electrons

Given that the magnetic moment is 1.73 BM (Bohr Magneton), we calculate the number of unpaired electrons by solving 1.73 = sqrt{n(n + 2)}. Squaring both sides of the equation gives 2.9929 = n(n + 2).

Estimating the Number of Unpaired Electrons

Through estimation or trial and error, we determine that n is close to 1 since 1(1 + 2) approximates to 3, which is close to 2.9929. This indicates there is likely one unpaired electron.

Matching Electronic Configuration

Given the estimated number of unpaired electrons, we look at the electronic configurations given in the options to identify which one corresponds to a single unpaired electron. The electronic configuration [Ar] 3d^1 contains one unpaired electron in the d orbital.

Selecting the Correct Option

The configuration with one unpaired electron matching the magnetic moment is [Ar] 3d^1. Therefore, the answer is option (b).

Key concepts

Unpaired Electrons

The concept of unpaired electrons plays a critical role in understanding the magnetic properties of atoms and molecules. Each electron in an atom has a magnetic moment due to its spin. Electrons pair up in orbitals with their spins in opposite directions, effectively cancelling each other out magnetically. However, when electrons are unpaired, their magnetic moments don't cancel, and they contribute to the total magnetic moment of the atom or ion.

When we talk about magnetic moments in chemistry, the presence or absence of unpaired electrons determines the magnetism of a substance. Paramagnetic substances have at least one unpaired electron, which makes them attracted to magnetic fields, while diamagnetic substances have all their electrons paired and are not attracted to magnetic fields. In our textbook problem, we solved for the number of unpaired electrons to help us match the correct electronic configuration of a vanadium ion to its observed magnetic behavior.

Electronic Configuration

The electronic configuration of an atom or ion depicts the arrangement of electrons in orbitals around the nucleus. It's essential to know how electrons are distributed across different energy levels and subshells because their arrangement impacts chemical bonding and the physical properties of the element. Typically, electrons occupy the lowest available energy levels in a process known as the Aufbau principle. However, in transition metals, like vanadium, the d subshells come into play significantly.

As revealed by the textbook exercise, the magnetic moment of an ion can give clues about its electronic configuration—particularly the number of electrons in its d subshell. Calculating the magnetic moment provides insight into the electronic structure that corresponds to a particular number of unpaired electrons. This approach allows us to understand the behavior of transition metals and predict their chemical and physical properties.

Vanadium Ion

Vanadium is a transition metal that can form various ions by losing electrons from its outermost shells. When vanadium forms an ion, the electrons are removed first from the highest energy level, which is typically the 4s orbital. After the 4s electrons are removed, the ion may lose electrons from the 3d orbital, depending on its charge.

The magnetic moment of a vanadium ion can indicate its specific electronic configuration and the resultant charge of the ion. When a vanadium atom loses electrons, it affects its magnetic properties because it may lead to unpaired electrons. In the problem we analyzed, we deduced from the magnetic moment the presence of one unpaired electron, revealing the likely electronic configuration of the vanadium ion present in the compound.

Bohr Magneton

The Bohr Magneton (BM) is a physical constant that quantifies the magnetic moment of an electron caused by its angular momentum. One Bohr Magneton is approximately equal to \(9.274 \times 10^{-24}\) Joules per Tesla. The concept is particularly useful in the field of magnetism, where it provides a standard measure for expressing the strength of the magnetic moments of particles and materials.

In chemistry problems involving magnetism, such as in our textbook exercise, the magnetic moment is often expressed in units of Bohr Magnetons. By comparing the experimental magnetic moment of a compound to the calculated value using the number of unpaired electrons, we can determine the electronic configuration of an ion like vanadium. This knowledge is crucial for understanding the magnetic character of substances and can further be applied in technologies such as magnetic resonance imaging (MRI) and other applications that rely on the magnetic properties of materials.