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Lecture 25 Practice problems. Final: May 11, SEC 117. 3 hours (4-7 PM), 6 problems (mostly Chapters 6,7). Boltzmann Statistics, Maxwell speed distribution Fermi-Dirac distribution, Degenerate Fermi gas Bose-Einstein distribution, BEC Blackbody radiation. Sun’s Mass Loss.

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lecture 25 practice problems
Lecture 25 Practice problems

Final: May 11, SEC 117

3 hours (4-7 PM), 6 problems

(mostly Chapters 6,7)

  • Boltzmann Statistics, Maxwell speed distribution
  • Fermi-Dirac distribution, Degenerate Fermi gas
  • Bose-Einstein distribution, BEC
  • Blackbody radiation
sun s mass loss
Sun’s Mass Loss

The spectrum of the Sun radiation is close to the black body spectrum with the maximum at a wavelength  = 0.5 m. Find the mass loss for the Sun in one second. How long it takes for the Sun to loose 1% of its mass due to radiation? Radius of the Sun: 7·108 m, mass - 2 ·1030 kg.

max = 0.5 m 

This result is consistent with the flux of the solar radiation energy received by the Earth (1370 W/m2) being multiplied by the area of a sphere with radius 1.5·1011 m (Sun-Earth distance).

the mass loss per one second

1% of Sun’s mass will be lost in

carbon monoxide poisoning
Carbon monoxide poisoning

Each Hemoglobin molecule in blood has 4 adsorption sites for carrying O2. Let’s consider one site as a system which is independent of other sites. The binding energy of O2 is  = -0.7 eV. Calculate the probability of a site being occupied by O2. The partial pressure of O2 in air is 0.2 atm and T=310 K.

The system has 2 states: empty ( =0) and occupied ( = -0.7 eV). So the grand partition function is:

The system is in diffusive equilibrium with O2 in air. Using the ideal gas approximation to calculate the chemical potential:

Plugging in numbers gives:

Therefore, the probability of occupied state is:

problem 1 partition function average energy
Problem 1 (partition function, average energy)

The neutral carbon atom has a 9-fold degenerate ground level and a 5-fold degenerate excited levelat an energy 0.82 eV above the ground level. Spectroscopic measurements of a certain star show that 10% of the neutral carbon atoms are in the excited level, and that the population of higher levels is negligible. Assuming thermal equilibrium, find the temperature.

problem 2 partition function average energy

Consider a system of N particles with only 3 possible energy levels separated by  (let the ground state energy be 0). The systemoccupies a fixed volume V and is in thermal equilibrium with a reservoir at temperature T. Ignore interactions between particles and assume that Boltzmann statistics applies.

(a) (2) What is the partition function for a single particle in the system?

(b) (5) What is the average energy per particle?

(c) (5) What is probability that the 2level is occupied in the high temperature limit, kBT >> ? Explain your answer on physical grounds.

(d) (5) What is the average energy per particle in the high temperature limit, kBT >> ?

(e) (3) At what temperature is the ground state 1.1 times as likely to be occupied as the 2 level?

(f) (25) Find the heat capacity of the system, CV, analyze the low-T (kBT<<) and high-T (kBT >> ) limits, and sketch CV as a function of T. Explain your answer on physical grounds.

Problem 2 (partition function, average energy)




all 3 levels are populated with the same probability


problem 2 partition function average energy1
Problem 2 (partition function, average energy)




Low T (>>):

high T (<<):


problem 3 boltzmann distribution
Problem 3 (Boltzmann distribution)
  • A solid is placed in an external magnetic field B = 3 T. The solid contains weakly interacting paramagnetic atoms of spin ½ so that the energy of each atom is ±  B,  =9.3·10-23 J/T.
  • Below what temperature must one cool the solid so that more than 75 percent of the atoms are polarized with their spins parallel to the external magnetic field?
  • An absorption of the radio-frequency electromagnetic waves can induce transitions between these two energy levels if the frequency f satisfies he condition h f = 2  B. The power absorbed is proportional to the difference in the number of atoms in these two energy states. Assume that the solid is in thermal equilibrium at  B << kBT. How does the absorbed power depend on the temperature?



The absorbed power is proportional to the difference in the number of atoms in these two energy states:

The absorbed power is inversely proportional to the temperature.

problem 4 maxwell boltzmann
Problem 4 (maxwell-boltzmann)

(a) Find the temperature T at which the root mean square thermal speed of a hydrogen molecule H2 exceeds its most probable speed by 400 m/s.

(b) The earth’s escape velocity (the velocity an object must have at the sea level to escape the earth’s gravitational field) is 7.9x103 m/s. Compare this velocity with the root mean square thermal velocity at 300K of (a) a nitrogen molecule N2 and (b) a hydrogen molecule H2. Explain why the earth’s atmosphere contains nitrogen but not hydrogen.

Significant percentage of hydrogen molecules in the “tail” of the Maxwell-Boltzmann distribution can escape the gravitational field of the Earth.

problem 5 degenerate fermi gas
Problem 5 (degenerate Fermi gas)

The density of mobile electrons in copper is 8.5·1028 m-3, the effective mass = the mass of a free electron.

(a) Estimate the magnitude of the thermal de Broglie wavelength for an electron at room temperature. Can you apply Boltzmann statistics to this system? Explain.

- Fermi distribution

(b) Calculate the Fermi energy for mobile electrons in Cu. Is room temperature sufficiently low to treat this system as degenerate electron gas? Explain.

- strongly degenerate

(c) If the copper is heated to 1160K, what is the average number of electrons in the state with energy F + 0.1 eV?

problem 7 bec
Problem 7 (BEC)

Consider a non-interacting gas of hydrogen atoms (bosons) with the density of 11020m-3.

a)(5) Find the temperature of Bose-Einstein condensation, TC, for this system.

b)(5) Draw aqualitative graph of the number of atoms as a function of energy

of the atoms for the cases: T >> TC and T = 0.5 TC. If the total number of atoms is 11020,

how many atoms occupy the ground state at T = 0.5 TC?

c)(5) Below TC, the pressure in a degenerate Bose gas is proportional to T5/2.

Do you expect the temperature dependence of pressure to be stronger or weaker at T > TC?

Explain and draw aqualitative graph of the temperature dependence of pressure over

the temperature range 0 <T < 2 TC.

problem 7 bec cont
Problem 7 (BEC) (cont.)
  • (c) The atoms in the ground state do not contribute to pressure. At T < TC,
  • two factors contribute to the fast increase of P with temperature:
  • an increase of the number of atoms in the excited states, and
  • an increase of the average speed of atoms with temperature.
  • Above TC, only the latter factor contributes to P(T), and the rate of the pressure increase
  • with temperature becomes smaller than that at T < TC.