Statistical Mechanics Newton's laws in principle tell us how anything works But in a system with many particles, the actual computations can become complicated. We will therefore be happy to get some 'average' or approximate behavior of the system that will be useful for practical purposes Describing this average dynamics is the goal of thermodynamics A microscopic level fundamental understanding of thermodynamics was found later, and this field is called statistical mechanics Average energy Suppose we have probability p i to have energy E i Then the average energy is P E E = i i i h i P P i i P Continuous variables: Probability for any given value is zero, but we have a probability for a range P (v) dv v v v + dv P (v) Area under curve P (v)dv =1 Z v Sometimes we may give the number of particles per unit velocity range N(v)dv particles have momentum between v and v + dv N(v) Area under curve N(v)dv = Ntotal Z v 57. Which of the following statements is (are) true for a Maxwell-Boltzmann description of an ideal gas 57. Which of the following statements is (are) true for of atoms in equilibriumGREPracticeBook at temperature T ? a Maxwell-Boltzmann description of an ideal gas of atomsI. The inaverage equilibrium velocity at temperatureof the atoms Tis ?zero. II. The distribution of the speeds of the atoms I. The average velocity of the atoms is zero. has a maximum at u = 0. II. The distribution of the speeds of the atoms III. The probability of finding an atom with zero has a maximum at u = 0. kinetic energy is zero. III. The probability of finding an atom with zero (A) Ikinetic only energy is zero. 56. A sample of N molecules has the distribution of (B) II only (A) I only 56. Aspeeds sample shown of N in moleculesthe figure hasabove. the distributionPduu is an of (C) I and II (B) II only estimate of the number of molecules with speeds (D) I and III speeds shown in the figure above. P u du is an (C) I and II (E) II and III estimatebetween ofu theand number u + du of, and molecules this number with isspeeds (D) I and III nonzero only up to 3u , where u is constant. (E) II and III between u and u + du0 , and this number0 is 58. A monatomic ideal gas changes from an initial Whichnonzero of only the followingup to 3u ,gives where the uvalue is constant. of a ? 0 0 58. Astate monatomic (Pi , Vi , idealTi, ngasi) to changes a final fromstate an(P finitial , Vf , Which of theN following gives the value of a ? Tf , nf ), where Pi < Pf , Vi = Vf , Ti < Tf and (A) a state (Pi , Vi , Ti, ni) to a final state (Pf , Vf , 3u0 N nTi ,= nnf ),. Whichwhere ofP the < Pfollowing, V = V gives, T < the T change and (A) a N f f i f i f i f (B) a 3u 0 nin =entropy n . Which of the of gas? the following gives the change 2u i f N0 in entropy3 of theTf gas? (B) a N (A) nR ln (C) a 2 u0 2 T u 3 Tif N0 (A) nR ln (C) a 3N 23 Ti u0 (B) nR ln (D) a 2 T 2u0 3 Tf 3 N (B) nR ln i (D) a 2 u 2 Tf (E) aN 0 5 f (C) nR ln 2 TTi (E) a N 5 f (C) nR ln 25 Ti (D) nR ln 2 Tf 5 Ti (D) nR ln 2 T (E) 0 f (E) 0 GO ON TO THE NEXT PAGE. -40- 46 GO ON TO THE NEXT PAGE. -40- 57. Which of the following statements is (are) true for a Maxwell-Boltzmann description of an ideal gas 57. Which of the following statements is (are) true for of atoms in equilibriumGREPracticeBook at temperature T ? a Maxwell-Boltzmann description of an ideal gas of atomsI. The inaverage equilibrium velocity at temperatureof the atoms Tis ?zero. II. The distribution of the speeds of the atoms I. The average velocity of the atoms is zero. has a maximum at u = 0. II. The distribution of the speeds of the atoms III. The probability of finding an atom with zero has a maximum at u = 0. kinetic energy is zero. III. The probability of finding an atom with zero (A) Ikinetic only energy is zero. 56. A sample of N molecules has the distribution of (B) II only (A) I only 56. Aspeeds sample shown of N in moleculesthe figure hasabove. the distributionPduu is an of (C) I and II (B) II only estimate of the number of molecules with speeds (D) I and III speeds shown in the figure above. P u du is an (C) I and II (E) II and III estimatebetween ofu theand number u + du of, and molecules this number with isspeeds (D) I and III nonzero only up to 3u , where u is constant. (E) II and III between u and u + du0 , and this number0 is 58. A monatomic ideal gas changes from an initial Whichnonzero of only the followingup to 3u ,gives where the uvalue is constant. of a ? 0 0 58. Astate monatomic (Pi , Vi , idealTi, ngasi) to changes a final fromstate an(P finitial , Vf , Which of theN following gives the value of a ? Tf , nf ), where Pi < Pf , Vi = Vf , Ti < Tf and (A) a state (Pi , Vi , Ti, ni) to a final state (Pf , Vf , 3u0 N nTi ,= nnf ),. Whichwhere ofP the < Pfollowing, V = V gives, T < the T change and (A) a N f f i f i f i f (B) a 3u 0 nin =entropy n . Which of the of gas? the following gives the change 2u i f N0 in entropy3 of theTf gas? (B) a N (A) nR ln (C) a 2 u0 2 T u 3 Tif N0 (A) nR ln (C) a 3N 23 Ti u0 (B) nR ln (D) a 2 T 2u0 3 Tf 3 N (B) nR ln i (D) a 2 u 2 Tf (E) aN 0 5 f (C) nR ln 2 TTi (E) a N 5 f (C) nR ln 25 Ti (D) nR ln 2 Tf 5 Ti (D) nR ln 2 T (E) 0 f (E) 0 GO ON TO THE NEXT PAGE. -40- 46 GO ON TO THE NEXT PAGE. -40- Hot bath, temperature T The particle can get energy from the hot walls when it touches them Basic law of Statistical Mechanics: Probability for the particle to have energy E E P e− kT / 23 1 Here k =1.38 10− JK− is the Boltzmann constant ⇥ GRE0177 74. The Lagrangian for a mechanical system is 77. An ensemble of systems is in thermal equilibrium with a reservoir for which kT = 0.025 eV. 24 Laqbq=+ , State A has an energy that is 0.1 eV above that of state B. If it is assumed the systems obey where q is a generalized coordinate and a Maxwell-Boltzmann statistics and that the and b are constants. The equation of motion degeneracies of the two states are the same, then for this system is the ratio of the number of systems in state A to b the number in state B is (A) q = q 2 a (A) e+4 2b +0.25 (B) q = q 3 (B) e a (C) 1 2b (C) q =- q 3 (D) e-0.25 a (E) e-4 2b (D) q =+ q 3 a 78. The muon decays with a characteristic lifetime b of about 10-6 second into an electron, a muon (E) q = q 3 a neutrino, and an electron antineutrino. The muon is forbidden from decaying into an electron and Fax′ I L 12// 32 0OFax I just a single neutrino by the law of conservation of Ga′ J = M− 32// 12 0PGa J G y J M PG y J (A) charge Haz′ K NM 001QPH az K (B) mass (C) energy and momentum 75. The matrix shown above transforms the com- (D) baryon number ponents of a vector in one coordinate frame S to (E) lepton number the components of the same vector in a second coordinate frame S′. This matrix represents a 79. A particle leaving a cyclotron has a total rotation of the reference frame S by relativistic energy of 10 GeV and a relativistic momentum of 8 GeV/c. What is the rest mass (A) 30° clockwise about the x-axis of this particle? (B) 30° counterclockwise about the z-axis 2 (C) 45° clockwise about the z-axis (A) 0.25 GeV/c 2 (D) 60° clockwise about the y-axis (B) 1.20 GeV/c 2 (E) 60° counterclockwise about the z-axis (C) 2.00 GeV/c (D) 6.00 GeV/c2 76. The mean kinetic energy of the conduction (E) 16.0 GeV/c2 electrons in metals is ordinarily much higher than kT because (A) electrons have many more degrees of freedom than atoms do (B) the electrons and the lattice are not in thermal equilibrium (C) the electrons form a degenerate Fermi gas (D) electrons in metals are highly relativistic (E) electrons interact strongly with phonons Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. 50 GRE0177 74. The Lagrangian for a mechanical system is 77. An ensemble of systems is in thermal equilibrium with a reservoir for which kT = 0.025 eV.