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Chapter 6: Q 6.16 (page 231)
Prove that, for any system in equilibrium with a reservoir at temperature T, the average value of the energy is
where . These formulas can be extremely useful when you have an explicit formula for the partition function.
Hence proved,
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Some advances textbooks define entropy by the formula
where the sum runs over all microstates accessible to the system and is the probability of the system being in microstate .
(a) For an isolated system, role="math" localid="1647056883940" for all accessible states . Show that in this case the preceding formula reduces to our familiar definition of entropy.
(b) For a system in thermal equilibrium with a reservoir at temperature,role="math" localid="1647057328146" . Show that in this case as well, the preceding formula agrees with what we already know about entropy.
Suppose you have 10 atoms of weberium: 4 with energy 0 eV, 3 with energy 1 eV, 2 with energy 4 eV, and 1 with energy 6 eV.
(a) Compute the average energy of all your atoms, by adding up all their energies and dividing by 10.
(b) Compute the probability that one of your atoms chosen at random would have energy E, for each of the four values of E that occur.
(c) Compute the average energy again, using the formula
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 u(z) 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 differs from by a term proportional to . Express the coefficient of this term in terms of the coefficients of the Taylor series
(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
Consider an ideal gas of highly relativistic particles ( such as photons or fast-moving electrons) whose energy-momentum relation is instead of . Assume that these particles live in a one-dimensional universe. By following the same logic as above, derive a formula for the single particle partition function,, for one particle in the gas.
Consider a hypothetical atom that has just two states: a ground state with energy zero and an excited state with energy 2 eV. Draw a graph of the partition function for this system as a function of temperature, and evaluate the partition function numerically at T = 300 K, 3000 K, 30,000 K, and 300,000 K.
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