A magnetic field of 3.6T is applied to a paramagnetic gas. The atoms of the gas have magnetic dipole moment of 4.5 × 10-23 J/T. At what temperature, will the mean translation kinetic energy kinetic energy of an atom of the gas be equal to the energy required to change the alignment of atom’s magnetic dipole form antiparallel to parallel (to the magnetic field) (Boltzmann constant = 1.38 × 10-23 J/K)
15.7 K
This problem asks us to determine the temperature at which the mean translational kinetic energy of an atom in a paramagnetic gas is equal to the energy required to change the alignment of its magnetic dipole moment from an antiparallel orientation to a parallel orientation relative to an applied magnetic field. We are given the magnetic field strength, the magnetic dipole moment of the atoms, and the Boltzmann constant.
Energy for Magnetic Dipole Alignment
First, let's understand the energy associated with a magnetic dipole in an external magnetic field. The potential energy (\(U\)) of a magnetic dipole with moment \(\mu\) in a magnetic field \(\mathbf{B}\) is given by:
\[U = -\mu \cdot \mathbf{B} = -\mu B \cos(\theta)\]where \(\theta\) is the angle between the magnetic dipole moment vector and the magnetic field vector.
- Antiparallel alignment: When the magnetic dipole is antiparallel to the magnetic field, \(\theta = 180^\circ\). \[U_{\text{antiparallel}} = -\mu B \cos(180^\circ) = -\mu B (-1) = \mu B\]
- Parallel alignment: When the magnetic dipole is parallel to the magnetic field, \(\theta = 0^\circ\). \[U_{\text{parallel}} = -\mu B \cos(0^\circ) = -\mu B (1) = -\mu B\]
The energy required to change the alignment from antiparallel to parallel is the difference in potential energy between these two states:
\[\Delta E = U_{\text{parallel}} - U_{\text{antiparallel}}\] \[\Delta E = (-\mu B) - (\mu B) = -2\mu B\]However, the question asks for the "energy required", which refers to the magnitude of the energy difference that must be supplied to flip the dipole. So, the energy required is:
\[E_{\text{flip}} = |\Delta E| = |-2\mu B| = 2\mu B\]Mean Translational Kinetic Energy
For a monatomic gas, the mean translational kinetic energy (\(E_{\text{kinetic}}\)) of an atom is given by the formula based on the equipartition theorem:
\[E_{\text{kinetic}} = \frac{3}{2} k_B T\]where \(k_B\) is the Boltzmann constant and \(T\) is the absolute temperature in Kelvin.
Temperature Calculation
The problem states that the mean translational kinetic energy of an atom of the gas is equal to the energy required to change the alignment of the atom's magnetic dipole from antiparallel to parallel. Therefore, we set the two energy expressions equal:
\[E_{\text{kinetic}} = E_{\text{flip}}\] \[\frac{3}{2} k_B T = 2\mu B\]Now, we can solve for the temperature \(T\):
\[T = \frac{2\mu B}{\frac{3}{2} k_B}\] \[T = \frac{4\mu B}{3k_B}\]Substituting Given Values
We are given the following values:
- Magnetic field (\(B\)) = \(3.6 \text{ T}\)
- Magnetic dipole moment (\(\mu\)) = \(4.5 \times 10^{-23} \text{ J/T}\)
- Boltzmann constant (\(k_B\)) = \(1.38 \times 10^{-23} \text{ J/K}\)
Substitute these values into the equation for \(T\):
\[T = \frac{4 \times (4.5 \times 10^{-23} \text{ J/T}) \times (3.6 \text{ T})}{3 \times (1.38 \times 10^{-23} \text{ J/K})}\]Notice that the \(10^{-23}\) terms cancel out, simplifying the calculation:
\[T = \frac{4 \times 4.5 \times 3.6}{3 \times 1.38}\] \[T = \frac{18 \times 3.6}{4.14}\] \[T = \frac{64.8}{4.14}\] \[T \approx 15.652 \text{ K}\]Rounding to one decimal place, we get:
\[T \approx 15.7 \text{ K}\]This temperature is the point where the average thermal energy available to the gas atoms is sufficient to cause the magnetic dipoles to flip their orientation from an energetically unfavorable antiparallel alignment to a favorable parallel alignment against the magnetic field.
Important Questions from Magnetism and Maxwell's Equations
- Which type of magnetic material is characterized by an induced magnetic moment that consistently opposes the applied external magnetic field, resulting in a small, negative volume magnetic susceptibility ($\chi_v$)?
The Curie temperature for cobalt is:
A metallic rod when placed in strong magnetic field, aligns itself at right angles to the magnetic field. The nature of material is:
Identify which of the following expressions is not Maxwell’s equation for time-varying fields.
Which of the following materials is used for the generation of ultrasonic waves by using magnetostriction effect?
