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Chapter 6: 6.24 (page 236)
For an molecule, the constant is approximately . Estimate the rotational partition function for an molecule at room temperature.
The rotational partition function is .
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The analysis of this section applies also to liner polyatomic molecules, for which no rotation about the axis of symmetry is possible. An example is , with . Estimate the rotational partition function for a molecule at room temperature. (Note that the arrangement of the atoms is, and the two oxygen atoms are identical.)
At very high temperatures (as in the very early universe), the proton and the neutron can be thought of as two different states of the same particle, called the "nucleon." (The reactions that convert a proton to a neutron or vice versa require the absorption of an electron or a positron or a neutrino, but all of these particles tend to be very abundant at sufficiently high temperatures.) Since the neutron's mass is higher than the proton's by 2.3 x 10-30 kg, its energy is higher by this amount times c2. Suppose, then, that at some very early time, the nucleons were in thermal equilibrium with the rest of the universe at 1011 K. What fraction of the nucleons at that time were protons, and what fraction were neutrons?
2. Consider a classical particle moving in a one-dimensional potential well , as shown The particle is in thermal equilibrium with a reservoir at temperature , so the probabilities of its various states are determined by Boltzmann statistics.
{a) Show that the average position of the particle is given by
where each integral is over the entire axis.
A one-dimensional potential well. The higher the temperature, the farther the particle will stray from the equilibrium point.
(b) If the temperature is reasonably low (but still high enough for classical mechanics to apply), the particle will spend most of its time near the bottom of the potential well. In that case we can expand in a Taylor series about the equilibrium point :
Show that the linear term must be zero, and that truncating the series after the quadratic term results in the trivial prediction .
(c) If we keep the cubic term in the Taylor series as well, the integrals in the formula for become difficult. To simplify them, assume that the cubic term is small, so its exponential can be expanded in a Taylor series (leaving the quadratic term in the exponent). Keeping only the smallest temperature-dependent term, show that in this limit x differs from zo by a term proportional to . Express the coefficient of this term in terms of the coefficients of the Taylor series for
(d) The interaction of noble gas atoms can be modeled using the Lennard Jones potential,
Sketch this function, and show that the minimum of the potential well is at , with depth . For argon, and . Expand the Lennard-Jones potential in a Taylor series about the equilibrium point, and use the result of part ( c) to predict the linear thermal expansion coefficient of a noble gas crystal in terms of . Evaluate the result numerically for argon, and compare to the measured value
Show explicitly from the results of this section thatfor an ideal gas.
A lithium nucleus has four independent spin orientations, conventionally labeled by the quantum number m = -3/2, -1/2, 1/2, 3/2. In a magnetic field B, the energies of these four states are E = mB, where the constant is 1.03 x 10-7 eV/T. In the Purcell-Pound experiment described in Section 3.3, the maximum magnetic field strength was 0.63 T and the temperature was 300 K. Calculate the probability of a lithium nucleus being in each of its four spin states under these conditions. Then show that, if the field is suddenly reversed, the probabilities of the four states obey the Boltzmann distribution for T =-300 K.
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