The molar entropy of electron spins in a magnetic field \(B\) is $$ S=R\left\\{\frac{\Delta E / k T}{\mathrm{e}^{\Delta E / k T}-1}-\ln \left(1-\mathrm{e}^{-\Delta E / k T}\right)\right\\} $$ where \(\Delta E=2 \mu_{\mathrm{B}} B\) is the separation in energy of the two spin states in a magnetic field, \(\mu_{B}\) is the Bohr magneton, \(\mu_{B}=9.274 \times 10^{-24}\) \(\mathrm{J} \mathrm{T}^{-1}\). Plot this function against temperature for the following values of B: \(0.1 \mathrm{~T}, 1 \mathrm{~T}, 10 \mathrm{~T}\), and \(100 \mathrm{~T}\). (See Box 8.1. Notice that the unit of magnetic induction, tesla, \(T\), where \(1 T=1 \mathrm{~kg} \mathrm{~s}^{-2} \mathrm{~A}^{-1}\), cancels.)

Understanding the interaction between a magnetic field and electron spin is crucial for diving into the realm of quantum physics and material science. The electron spin is a fundamental property of electrons, manifesting as a form of angular momentum. It has two possible orientations, commonly referred to as 'up' and 'down' spin states.

When placed in a magnetic field, these spin states experience a force due to their magnetic moments. This results in different energy levels for the spin orientations due to the Zeeman effect. The energy difference, \(\Delta E = 2\mu_{\mathrm{B}}B\), drives the alignment of spins along or against the direction of the magnetic field, with \(\mu_{\mathrm{B}}\) representing the Bohr magneton, a physical constant that quantifies the magnetic moment of an electron due to its spin.

The impact of the magnetic field on electron spins is not only a topic of theoretical interest but also foundational in practical applications like Magnetic Resonance Imaging (MRI) and data storage systems.

The Bohr magneton (\(\mu_{B}\)) is a physical constant that represents the magnetic moment of an electron resulting from its inherent properties of electric charge and spin. It is expressed in joules per tesla (\(\mathrm{J} \mathrm{T}^{-1}\)) and is defined as \(\mu_{B} = 9.274 \times 10^{-24} \mathrm{J} \mathrm{T}^{-1}\).

The Bohr magneton is a vital factor when calculating the separation in energy levels caused by an external magnetic field. This quantification allows scientists and engineers to predict how an electron's spin will interact with magnetic fields, which is essential in designing and understanding various quantum mechanical systems and devices, such as spintronic components and quantum computers.

In our exercise, the Bohr magneton is used to calculate the energy difference \(\Delta E\) between spin-up and spin-down states in a magnetic field, which in turn is central to determining the molar entropy associated with these spins.

Entropy, denoted as \(S\), is a key concept in thermodynamics and statistical mechanics—it measures the disorder or randomness of a system. Calculating entropy involves understanding the distribution of energy states and their associated probabilities.

In the context of electron spins in a magnetic field, entropy can tell us how the randomness of electron spin orientations changes with temperature and magnetic field strength. The equation for molar entropy in our exercise is a statistical mechanical expression that incorporates the Boltzmann constant (\(k\)), the gas constant (\(R\)), temperature (\(T\)), and the energy separation \(\Delta E\).

At higher temperatures or lower magnetic field strengths, the spins are more disordered because thermal energy allows for more random orientation of spins. Conversely, lower temperatures or higher magnetic field strengths tend to align the spins, thus reducing entropy. The ability to calculate this quantity is pivotal for scientists and engineers, particularly when exploring the magnetic properties of materials and in developing technologies that rely on electron spin dynamics.