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Chapter 6: Q. 6.52 (page 256)

Consider an ideal gas of highly relativistic particles ( such as photons or fast-moving electrons) whose energy-momentum relation is E=pcinstead of E=p22m. 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,Z1, for one particle in the gas.

Short Answer

Expert verified

The partition function is given byZ=2LkThc.

Step by step solution

01

Step 1. Given information

The energy of the particle is given by

E=pc............(1)

02

Step 2. Calculation

The allowed values of the energy for the gas molecule is given by

En=hcn2L.................(2)

The formula to calculate the single particle partition function is given by

Z1=∑ne-EnkT...............(3)

Substitute the value of the allowed energy from equation (2) into equation (3) and transferring the summation by integral solve to calculate the required partition function of the gas.

Z1=∫0∞e-hcn2LkTdn=-2LkThce-hcn2LkT0∞=2LkThc

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Most popular questions from this chapter

In this section we computed the single-particle translational partition function,Ztr, by summing over all definite-energy wave functions. An alternative approach, however, is to sum over all position and momentum vectors, as we did in Section 2.5. Because position and momentum are continuous variables, the sums are really integrals, and we need to slip a factor of 1h3to get a unitless number that actually counts the independent wavefunctions. Thus we might guess the formula

role="math" localid="1647147005946" Ztr=1h3∫d3rd3pe-EtrkT

where the single integral sign actually represents six integrals, three over the position components and three over the momentum components. The region of integration includes all momentum vectors, but only those position vectors that lie within a box of volume V. By evaluating the integrals explicitly, show that this expression yields the same result for the translational partition function as that obtained in the text.

Consider a classical particle moving in a one-dimensional potential well ux, as shown. The particle is in thermal equilibrium with a reservoir at temperature T, so the probabilities of its various states are determined by Boltzmann statistics.

{a) Show that the average position of the particle is given by

x=∫xe-βuxdx∫e-βuxdx

where each integral is over the entire xaxis.

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

ux=ux0+x-x0dudxx0+12x-x02d2udx2x0+13!x-x03d3udx3x0+........

Show that the linear term must be zero, and that truncating the series after the quadratic term results in the trivial prediction x=x0.

(c) If we keep the cubic term in the Taylor series as well, the integrals in the formula for xbecome 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 by a term proportional to kT. Express the coefficient of this term in terms of the coefficients of the Taylor series ux

(d) The interaction of noble gas atoms can be modeled using the Lennard Jones potential,

ux=u0x0x12-2x0x6

Sketch this function, and show that the minimum of the potential well is at x=x0, with depth u0. For argon, x0=3.9A∘and u0=0.010eV. 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 u0. Evaluate the result numerically for argon, and compare to the measured value α=0.0007K-1

For a COmolecule, the constant €is approximately 0.00024eV.(This number is measured using microwave spectroscopy, that is, by measuring the microwave frequencies needed to excite the molecules into higher rotational states.) Calculate the rotational partition function for a COmolecule at room temperature (300K), first using the exact formula 6.30 and then using the approximate formula 6.31

A water molecule can vibrate in various ways, but the easiest type of vibration to excite is the "flexing' mode in which the hydrogen atoms move toward and away from each other but the HO bonds do not stretch. The oscillations of this mode are approximately harmonic, with a frequency of 4.8 x 1013Hz. As for any quantum harmonic oscillator, the energy levels are 12hf,32hf,52hf, and so on. None of these levels are degenerate.

(a) state and in each of the first two excited states, assuming that it is in equilibrium with a reservoir (say the atmosphere) at 300 K. (Hint: Calculate 2 by adding up the first few Boltzmann factors, until the rest are negligible.) Calculate the probability of a water molecule being in its flexing ground

(b) Repeat the calculation for a water molecule in equilibrium with a reservoir at 700 K (perhaps in a steam turbine).

Prove that, for any system in equilibrium with a reservoir at temperature T, the average value of E2 is

E2¯=1Z∂2Z∂β2

Then use this result and the results of the previous two problems to derive a formula for σEin terms of the heat capacity, C=∂E¯/∂T

You should findσE=kTC/k
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