Boltzmann statistics, paramagnetism

1. Boltzmann statistics, paramagnetism E E2 = + B an arbitrary choice of zero energy 0 E1 = - B Consider a single dipole. Its partition function is The average energy of a single dipole is: Therefore, the average energy of N dipoles is: Finally, we calculate the average magnetic moment of a dipole: … and the magnetization of the system:

2. Rotation of diatomic molecules Rotational energy of a molecule is: Each state j is (2j+1)-fold degenerate. The partition function of a single molecule can be written as: At large temperatures This is just the prediction of the equipartition theorem (f=2) Most molecules are in excited rotational levels at ordinary temperatures The fraction of HCl molecules in the j-th rotational state at 300oK

3. H A gas is placed in a very tall container at the temperature T. The container is in a uniform gravitational field, the acceleration of free fall, g, is given. Find the average potential energy of the molecules. h 0 # of molecules within dh: For a very tall container (mgH≫kT): At very large T (mgH≪kT):

4. The equipartition theorem If we are treating the particle motions classically, then it doesn’t make much sense to express the partition function as a sum of discrete terms as we have above. Classically, the position and momentum of a particle can vary continuously, and the ‘energy levels’ are also continuous. As a result, the classical partition function takes the form of an integral rather than a sum: where the energy E can be a function of the particle positions xiand momenta pi. If we assume that we can write the energy as a sum of independent contributions from each degree of freedom, then the exponential functional dependence on the energy means that we can separate the integral into the product of integrals over each degree of freedom. i.e., The consequence of this is that we have separated the partition function into the product of partition functions for each degree of freedom. In general, we may write the partition function for a single degree of freedom in which the energy depends quadratically on the coordinate x (i.e., E(x) = cx2with c a constant) as The energy associated with each quadratic degree of freedom is therefore 1⁄2kT, and we have proved the equipartition theorem.

5. The Maxwell speed distribution There are six parameters, the position (x, y, z) and the velocity (vx, vy, vz), per molecule to know the position and instantaneous velocity of an ideal gas. These parameters define the six-dimensional phase space. The velocity components of the molecules are more important than positions, because the energy of a gas should depend only on the velocities. Define a velocity distribution function D(v). = the probability of finding a particle with speed between v and v+dv. We know already (from the equipartition theorem) that number of vectors v corresponding to speed v probability of a molecule having velocity v vy v vx vz