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Chapter 7: Q 7.37 (page 293)

Prove that the peak of the Planck spectrum is at x = 2.82.

Short Answer

Expert verified

Hence proved that plank's spectrum is at x = 2.82.

Step by step solution

01

Given information

The peak of the Planck spectrum is at e = 2.82.

02

Explanation

Set the derivative of x3/ex-1with respect to x equals to zero to get the maximum, then solve for:

∂∂xx3ex-1=03x2ex-1-x3exex-12=03x2ex-1-x3ex=03x2ex-3x2-x3ex=03ex-3-xex=0

Using matlab to solve the above equation: The code is:

Therefore,x=2.8214

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Most popular questions from this chapter

In analogy with the previous problem, consider a system of identical spin‐0bosonstrapped in a region where the energy levels are evenly spaced. Assume that Nis a large number, and again let qbe the number of energy units.

(a) Draw diagrams representing all allowed system states from q=0up to q=6.Instead of using dots as in the previous problem, use numbers to indicate the number of bosons occupying each level.

(b) Compute the occupancy of each energy level, for q=6. Draw a graph of the occupancy as a function of the energy at each level.

(c) Estimate values of μand Tthat you would have to plug into the Bose-Einstein distribution to best fit the graph of part(b).

(d) As in part (d) of the previous problem, draw a graph of entropy vs energy and estimate the temperature at q=6from this graph.

Consider a gas of noninteracting spin-0 bosons at high temperatures, when T≫Tc. (Note that “high” in this sense can still mean below 1 K.)

  1. Show that, in this limit, the Bose-Einstein function can be written approximately as
    n¯BE=e−(∈−μ)/kT[1+e−∈−μ/kT+⋯].
  2. Keeping only the terms shown above, plug this result into equation 7.122 to derive the first quantum correction to the chemical potential for gas of bosons.
  3. Use the properties of the grand free energy (Problems 5.23 and 7.7) to show that the pressure of any system is given by In P=(kT/V), where Zis the grand partition function. Argue that, for gas of noninteracting particles, In Zcan be computed as the sum over all modes (or single-particle states) of In Zi, where Zi; is the grand partition function for the ithmode.
  4. Continuing with the result of part (c), write the sum over modes as an integral over energy, using the density of states. Evaluate this integral explicitly for gas of noninteracting bosons in the high-temperature limit, using the result of part (b) for the chemical potential and expanding the logarithm as appropriate. When the smoke clears, you should find
    p=NkTV(1−NvQ42V),
    again neglecting higher-order terms. Thus, quantum statistics results in a lowering of the pressure of a boson gas, as one might expect.
  5. Write the result of part (d) in the form of the virial expansion introduced in Problem 1.17, and read off the second virial coefficient, B(T). Plot the predicted B(T)for a hypothetical gas of noninteracting helium-4 atoms.
  6. Repeat this entire problem for gas of spin-1/2 fermions. (Very few modifications are necessary.) Discuss the results, and plot the predicted virial coefficient for a hypothetical gas of noninteracting helium-3 atoms.

For a system of particles at room temperature, how large must ϵ-μbe before the Fermi-Dirac, Bose-Einstein, and Boltzmann distributions agree within 1%? Is this condition ever violated for the gases in our atmosphere? Explain.

Consider a collection of 10,000 atoms of rubidium- 87 , confined inside a box of volume (10-5m)3.

(a) Calculate ε0, the energy of the ground state. (Express your answer in both joules and electron-volts.)

(b) Calculate the condensation temperature, and compare kTctoε0.

(c) Suppose that T=0.9TcHow many atoms are in the ground state? How close is the chemical potential to the ground-state energy? How many atoms are in each of the (threefold-degenerate) first excited states?

(d) Repeat parts (b) and (c) for the case of 106atoms, confined to the same volume. Discuss the conditions under which the number of atoms in the ground state will be much greater than the number in the first excited state.

For a system of fermions at room temperature, compute the probability of a single-particle state being occupied if its energy is

(a) 1eVless than μ

(b) 0.01eVless than μ

(c) equal to μ

(d) 0.01eVgreater than μ

(e) 1eVgreater thanμ
See all solutions

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