- Demonstrate that the geometric series
- Let

- A sample of mineral oil is placed in an external magnetic field
. Each
proton has spin
, and a magnetic moment
. It can, therefore, have
two possible energies,
, corresponding to the two possible
orientations of its spin. An applied radio-frequency field can induce transitions
between these two energy levels if its frequency
satisfies the Bohr
condition
. The power absorbed from this radiation
field is then proportional to the difference in the number of nuclei in these
two energy levels. Assume that the protons in the mineral oil are in thermal
equilibrium at a temperature
that is sufficiently high that
. How
does the absorbed power depend on the temperature,
, of the sample?
- Consider an assembly of
weakly-interacting magnetic atoms per unit volume, held
at temperature
. According to classical physics, each atomic magnetic moment,
- Show that the classical mean magnetization is
*Langevin function*. - Demonstrate that the corresponding quantum mechanical expression for a collection of
atoms with overall angular momentum
is
- Show, finally, that the previous two expressions are identical in the classical limit . (This is the classical limit because the spacing between adjacent magnetic energy levels, , where is an integer lying between and , is , which tends to zero as .)

- Show that the classical mean magnetization is
- Consider a spin-1/2 (i.e.,
and
) paramagnetic substance containing
non-interacting atoms.
- Show
that the overall magnetic partition function,
, is such that
- Demonstrate that the mean magnetic energy of the system is
- Demonstrate that the magnetic contribution to the specific heat of the substance is
*Schottky anomaly*. - Show that the magnetic contribution to the entropy
of the substance is

- Show
that the overall magnetic partition function,
, is such that
- The nuclei of atoms in a certain crystalline solid have spin one. According
to quantum theory, each nucleus can therefore be in any one of three quantum states
labeled by the quantum number
, where
, 0, or
. This quantum number
measures the projection of the nuclear spin along a crystal axis of the solid.
Because the electric charge distribution in the nucleus is not spherically
symmetric, but ellipsoidal, the energy of a nucleus depends on its spin
orientation with respect to the internal electric field existing at its location.
Thus a nucleus has the same energy
in the state
and the state
, compared with energy
in the state
.
- Find an expression, as a function of absolute temperature, , of the nuclear contribution to the molar internal energy of the solid.
- Find an expression, as a function of , of the nuclear contribution to the molar entropy of the solid.
- By directly counting the total number of accessible states, calculate the nuclear contribution to the molar entropy of the solid at very low temperatures. Calculate it also at high temperatures. Show that the expression in part (b) reduces properly to these values as and .
- Make a qualitative graph showing the temperature dependence of the nuclear contribution to the molar heat capacity of the solid. Calculate the temperature dependence explicitly. What is the temperature dependence for large values of ?

- A dilute solution of a macromolecule (large molecules of biological
interest) at temperature
is placed in an ultracentrifuge rotating with
angular velocity
. The centripetal acceleration
acting on
a particle of mass
may then be replaced by an equivalent centrifugal
force
in the rotating frame of reference.
- Find how the relative density, , of molecules varies with their radial distance, , from the axis of rotation.
- Show qualitatively how the molecular weight of the macromolecules can be determined if the density ratio at the radii and is measured by optical means.

- Consider a homogeneous mixture of inert monatomic ideal gases at absolute
temperature
in a container of volume
. Let there be
moles of
gas 1,
moles of gas 2, ..., and
moles of gas
.
- By considering the classical partition function of this system, derive its equation of state. In other words, find an expression for its total mean pressure, .
- How is this total pressure,
, of the gas related to the so-called
*partial pressure*, , that the th gas would produce if it alone occupied the entire volume at this temperature?

- Monatomic molecules adsorbed on a surface are free to move on this surface,
and can be treated as a classical ideal two-dimensional gas. At absolute temperature
, what is the heat capacity per mole of molecules thus adsorbed on a surface of
fixed size?
- Consider a system in thermal equilibrium with a heat bath held at absolute temperature
. The
probability of observing the system in some state
of energy
is is given by the canonical
probability distribution:
- Demonstrate that the entropy can be written
- Demonstrate that the mean Helmholtz free energy is related to the partition function according to

- Demonstrate that the entropy can be written
- Show that the logarithm of the classical partition function of an ideal gas consisting of
identical molecules of mass
, held in a container of
volume
, and in thermal equilibrium with a heat bath held at absolute temperature
, is
- Use the Debye approximation to calculate the contribution of lattice vibrations to the thermodynamic
functions of a solid.
- To be more specific, show that

Here, is the number of atoms in the solid, the absolute temperature, the partition function, the mean energy, the entropy, , , and is the Debye frequency. - Show that in the limit
,

[Hint: .] - Show that in the limit
,

- Further, show that

- To be more specific, show that
- For the quantized lattice waves (phonons) in the Debye theory of specific heats, the frequency,
,
of a propagating wave is related to its wavevector,
, by the dispersion relation
,
where
is the velocity of sound. On the other hand, in a ferromagnetic solid at low
temperatures, quantized waves of magnetization (spin waves) have their frequencies,
, related
to their wavevectors,
, according to the dispersion relation
, where
is a constant. Show that, at low temperatures, the contribution of spin waves to the heat capacity of
the ferromagnet varies as
.
- Verify directly that
- Show that the mean speed of molecules effusing through a small hole in a gas-filled container is
times larger than the
mean speed of the molecules within the container.
- A vessel is closed off by a porous partition through which gases can pass by effusion and then be pumped off to some collecting
chamber. The vessel is filled with dilute gas containing two types of molecule which differ because they contain different atomic
isotopes, and thus have the different masses,
and
. The concentrations of these molecules are
and
, respectively, and are maintained constant within the vessel by constantly replenishing the supply of gas
in it.
- Let and be the concentrations of the two types of molecule in the collecting chamber. What is the ratio ?
- By using the gas , one can attempt to separate from , the first of these isotopes being the one that undergoes nuclear fission reactions. The molecules in the vessel are then and . The concentrations of these molecules in the vessel corresponds to the natural abundances of the two isotopes: percent, and percent. What is the ratio, , of the two isotopic concentrations in the gas collected after effusion, compared to the original concentration ratio, ?

- Show that the mean force per unit area exerted on the walls of a container enclosing a Maxwellian gas is
- Consider a spin-1/2 ferromagnetic material consisting of
identical atoms with
and
. Let each atom have
nearest neighbors.
- Show that
- Use the molecular field approximation to demonstrate that

where , , and . - Show that for
slightly less than
, and in the absence of an external magnetic field,

- Demonstrate that exactly at the critical temperature,
- Finally, show that for
slightly larger than
,

- Show that