Question

When a magnetic dipole is placed in a magnetic field, it has a natural tendency to minimize its potential energy by aligning itself with the field. If there is sufficient thermal energy present, however, the dipole may rotate so that it is no longer aligned with the field. Using $k_{\mathrm{B}}T$ as a measure of the thermal energy, where $k_{\mathrm{B}}$ is Boltzmann's constant and $T$ is the temperature in kelvins, determine the temperature at which there is sufficient thermal energy to rotate the magnetic dipole associated with a hydrogen atom from an orientation parallel to an applied magnetic field to one that is antiparallel to the applied field. Assume that the strength of the field is $0.15\mathrm{T}$

Short answer

ANSWER: By calculating the expression from the step by step solution, we find that the temperature needed for the hydrogen atom's dipole to rotate is around 1.61 K.

Step-by-step solution

Determine the potential energy difference

When a magnetic dipole with a magnetic moment $\overrightarrow{\mu }$ is in a magnetic field $\overrightarrow{B}$, its potential energy $U$ is given by $U=-\overrightarrow{\mu }\cdot \overrightarrow{B}$. When the magnetic moment is parallel to the magnetic field, the potential energy is minimum, and when it is antiparallel, the potential energy is maximum. The energy difference between these two states is $\mathrm{\Delta }U=-\overrightarrow{\mu }\cdot {\overrightarrow{B}}_{\text{antiparallel}}-(-\overrightarrow{\mu }\cdot {\overrightarrow{B}}_{\text{parallel}})=2\overrightarrow{\mu }\cdot \overrightarrow{B}$. For a hydrogen atom, the magnetic moment is $\mu =g_{\text{e}}{\mu }_{\text{B}}$, where $g_{\text{e}}$ is the electron's g-factor, and ${\mu }_{\text{B}}$ is the Bohr magneton. In this case, we can assume $g_{\text{e}}=1$.

Calculate the energy difference

Given the magnetic field strength $B=0.15\text{T}$ and the Bohr magneton ${\mu }_{\text{B}}=9.27\times {10}^{-24}{\text{J T}}^{-1}$, we can calculate the energy difference: $\mathrm{\Delta }U=2\overrightarrow{\mu }\cdot \overrightarrow{B}=2{\mu }_{\text{B}}B=2(1)(9.27\times {10}^{-24})(0.15)$

Calculate the temperature

Since we want to find the temperature at which the thermal energy $k_{\text{B}}T$ is equal to the energy difference $\mathrm{\Delta }U$, we can set them equal to each other and solve for $T$: $k_{\text{B}}T=\mathrm{\Delta }U$ $T=\frac{\mathrm{\Delta }U}{k_{\text{B}}}$ where $k_{\text{B}}=1.38\times {10}^{-23}{\text{J K}}^{-1}$ is the Boltzmann's constant. Plugging in the values for $\mathrm{\Delta }U$ and $k_{\text{B}}$, we get $T=\frac{2(1)(9.27\times {10}^{-24})(0.15)}{1.38\times {10}^{-23}}$ By calculating the expression, we obtain the temperature $T$ at which there is sufficient thermal energy to rotate the magnetic dipole of a hydrogen atom from an orientation parallel to the applied magnetic field to one that is antiparallel.

Key concepts

Magnetic Dipole Potential Energy

Magnetic dipole potential energy is a key concept when dealing with magnets and magnetic fields. This type of energy depends on the orientation of the magnetic dipole in relation to the magnetic field. For instance, when a magnetic dipole is aligned with the magnetic field, it is in a lower energy state compared to when it is in opposition to the field.

Imagine a compass needle, which behaves like a magnetic dipole, that naturally aligns itself with the Earth's magnetic field. The potential energy is at a minimum when it is pointing north and increases as it rotates away from this alignment. Mathematically, the potential energy (U) of a magnetic dipole ($\mathit{\mu }$) in a magnetic field ($\mathit{B}$) is given by the equation $U=-\mathit{\mu }\times \mathit{B}$.

The change in potential energy ($\mathbf{\Delta }\mathit{U}$) when flipping from parallel to antiparallel orientation in such a magnetic field is particularly significant, because it represents the minimal energy necessary to overcome the magnetic alignment. This change is effectively the work required to rotate the dipole in the field and is crucial for understanding magnetic phenomena at the atomic scale.

Boltzmann's Constant

Boltzmann's constant ($k_{\text{B}}$) is a fundamental constant that connects the average kinetic energy of particles in a gas with the temperature of the gas. It serves as a bridge between the microscopic world of atoms and molecules and the macroscopic world that we can measure with thermometers.

This constant is named after Ludwig Boltzmann, one of the key figures in the development of statistical mechanics. The value of Boltzmann's constant is approximately $1.38\times {10}^{-23}{\text{J K}}^{-1}$, which indicates the amount of energy, in joules, corresponding to a one kelvin increase in temperature.

In the context of our magnetic dipole example, Boltzmann's constant helps us know at what temperature the thermal energy ($k_{\text{B}}T$) would be sufficient for the dipole to overcome its alignment with an external magnetic field. This gives us a fundamental insight into how thermal fluctuations can affect magnetic properties at different temperatures.

Bohr Magneton

The Bohr magneton (${\mu }_{\text{B}}$) is a physical constant that provides a natural scale for the magnetic moment of an electron orbiting a nucleus. It derives its name from Niels Bohr and is a key quantity in atomic physics for quantifying the magnitude of the magnetic moments of electron configurations.

The value of the Bohr magneton is around $9.27\times {10}^{-24}{\text{J T}}^{-1}$, signifying the energy level split per tesla of an electron in a magnetic field, which is essential when analyzing the behavior of atoms in magnetic fields.

In practice, it helps us calculate the energy differences associated with different electronic configurations in a magnetic field. Specifically, when answering questions about the potential energy of a magnetic dipole in a field, the Bohr magneton serves as a unit in the equation $\mathrm{\Delta }U=2\mu \cdot B$, especially concerning the magnetic behavior of single atoms such as hydrogen.

Electron g-factor

The electron g-factor ($g_{\text{e}}$) represents the ratio of the magnetic moment of an electron to its angular momentum and signifies the magnetic properties of an electron due to its spin. This dimensionless quantity refines our understanding of the electron's magnetic moment by accounting for its intrinsic spin, as well as the orbital motion within atoms.

Standard value for a free electron is roughly $g_{\text{e}}=2$, but due to quantum mechanical effects, the actual g-factor may deviate slightly from this value. In the context of our magnetic dipole scenario with a hydrogen atom, we've simplified the scenario by assuming that $g_{\text{e}}=1$, which is not exactly true for real electrons but simplifies our calculations.

Knowing the g-factor is crucial when calculating the precise energy levels of electrons in a magnetic field and is therefore fundamental for applications such as magnetic resonance imaging (MRI) and other spectroscopy methods that depend on the interactions of magnetic fields with matter.