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

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: 7·10 8 m, mass - 2 ·10 30 kg. max = 0.5  m  This result is consistent with the flux of the solar radiation energy received by the Earth (1370 W/m 2 ) being multiplied by the area of a sphere with radius 1.5·10 11 m (Sun-Earth distance). the mass loss per one second 1% of Sun’s mass will be lost in

Carbon monoxide poisoning Each Hemoglobin molecule in blood has 4 adsorption sites for carrying O 2. Let’s consider one site as a system which is independent of other sites. The binding energy of O 2 is  = -0.7 eV. Calculate the probability of a site being occupied by O 2. The partial pressure of O 2 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 O 2 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) The neutral carbon atom has a 9-fold degenerate ground level and a 5-fold degenerate excited level at 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 system occupies 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, k B T >>  ? Explain your answer on physical grounds. (d) (5) What is the average energy per particle in the high temperature limit, k B T >>  ? (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, C V, analyze the low-T (k B T >  ) limits, and sketch C V as a function of T. Explain your answer on physical grounds. (a) (b) (c)all 3 levels are populated with the same probability (d)

Problem 2 (partition function, average energy) (e) (f) Low T (  >>  ): high T (  <<  ): T CVCV

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· J/T. (a)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? (b)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 << k B T. How does the absorbed power depend on the temperature? (a) (b)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) (a) Find the temperature T at which the root mean square thermal speed of a hydrogen molecule H 2 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 7.9x10 3 m/s. Compare this velocity with the root mean square thermal velocity at 300K of (a) a nitrogen molecule N 2 and (b) a hydrogen molecule H 2. 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) (c) If the copper is heated to 1160K, what is the average number of electrons in the state with energy  F eV? The density of mobile electrons in copper is 8.5·10 28 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

Problem 6 (photon gas)

Problem 7 (BEC) Consider a non-interacting gas of hydrogen atoms (bosons) with the density of 1  m -3. a)(5) Find the temperature of Bose-Einstein condensation, T C, 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 >> T C and T = 0.5 T C. If the total number of atoms is 1  10 20, how many atoms occupy the ground state at T = 0.5 T C ? c)(5) Below T C, the pressure in a degenerate Bose gas is proportional to T 5/2. Do you expect the temperature dependence of pressure to be stronger or weaker at T > T C ? Explain and draw aqualitative graph of the temperature dependence of pressure over the temperature range 0 <T < 2 T C.

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