A magnetic field of magnitude 0.50 T is applied to a paramagnetic gas whose atoms have an intri

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A magnetic field of magnitude 0.50 T is applied to a...

A magnetic field of magnitude $0.50 \mathrm{~T}$ is applied to a paramagnetic gas whose atoms have an intrinsic magnetic dipole moment of magnitude $1.0 \times 10^{-23} \mathrm{~J} / \mathrm{T}$. At what temperature will the mean kinetic energy of translation of the gas atoms be equal to the energy required to reverse such a dipole end for end in this magnetic field?

Key Concepts

Equipartition Theorem

The equipartition theorem states that, at thermal equilibrium, energy is equally partitioned among all available degrees of freedom of a system. For a gas, this means that the average translational kinetic energy per atom (in three dimensions) is given by (3/2)kT, where T is the temperature and k is the Boltzmann constant. This theorem connects the microscopic kinetic energy of particles to the macroscopic measurable temperature.

Boltzmann Constant

The Boltzmann constant is a fundamental constant that links the average kinetic energy of particles in a system to its temperature. It serves as a bridge between microscopic energy scales (often in joules) and macroscopic thermal energy (temperature in kelvins), making it indispensable in statistical mechanics.

Magnetic Dipole Moment

This is a measure of the strength and orientation of a magnetic source. It quantifies how much an object, such as an atom or molecule, will interact with an external magnetic field. In problems involving magnetic interactions, the magnetic dipole moment is crucial because it determines the magnitude of the energy changes associated with different orientations relative to the field.

Magnetic Potential Energy

Magnetic potential energy is the energy associated with the orientation of a magnetic dipole in an external magnetic field. It is typically expressed as the negative dot product of the magnetic dipole moment and the magnetic field. This concept is essential when calculating the energy required to flip or reverse a dipole’s orientation relative to the magnetic field.

Paramagnetism

Paramagnetism is a form of magnetism in which certain materials are attracted to an external magnetic field due to the alignment of unpaired electrons' magnetic moments. In paramagnetic materials, these moments do not interact strongly with each other, so their overall magnetization results largely from their response to the applied field, especially at higher temperatures where thermal agitation plays a role.

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