Thermodynamics & Statistical Mechanics

Transcription

1 hysics GRE: hermodynamics & Statistical Mechanics G. J. Loges University of Rochester Dept. of hysics & Astronomy xkcd.com/66/ c Gregory Loges, 206

2 Contents Ensembles 2 Laws of hermodynamics 3 hermodynamic rocesses 3. Equations of State Adiabatic rocesses Isothermal rocesses Isobaric rocesses Ideal Gases 2 5 Equipartition heorem 3 6 Maxwell Relations 4 7 Statistical Mechanics 5 8 article Statistics 5 9 Einstein & Debye heories of Specific Heat 6 A Summary 7

3 Ensembles All possible states for a system comprise an ensemble, with each microstate may be assumed to be equally likely. he probability of a macrostate will then be the proportion of microstates in those accessible which have the given property. For example, if the system has three equivalent degrees of freedom amongst which three quanta of energy are distributed, the following arrangements comprise the ensemble: (3, 0, 0) (0, 3, 0) (0, 0, 3) (2,, 0) (2, 0, ) () (, 2, 0) (, 0, 2) (0, 2, ) (0,, 2) (,, ) (2) he probability of the system having one degree of freedom with all three quanta is thus since exactly three microstates satisfy this condition out of the total ten. It is convenient to introduce entropy, defined to be S k log Ω (3) One can show that when two systems, A and A, interact and reach thermal equilibrium, then where β k and β is defined by S + S maximal (4) β log Ω E If the two systems are not initially at the same temperature then they are in a highly unlikely configuration that will evolve to a state of higher total entropy; this is captured by the second law of thermodynamics. k E (5) 2 Laws of hermodynamics he four laws of thermodynamics are 0. If two systems are each in thermal equilibrium with a third system then they are in thermal equilibrium with each other: i.e. thermal equilibrium is transitive.. Conservation of energy: dq de + dw 2. S 0 for any thermodynamic process when considering a large enough system. 3. he entropy of a system approaches a constant value as the temperature approaches absolute zero: 0 S S 0. he zeroth law legitimizes the use of well-defined temperature scales, i.e. thermometers. he second law implies that cooler objects never heat warmer objects. 3 hermodynamic rocesses A process is said to be quasistatic if it occurs slowly enough to ensure that the system remains in equilibrium throughout. A process is said to be reversible if two states may be reached from eachother. A necessary condition for a process to be reversible is ds 0 throughout.

4 3. Equations of State Any quantity that depends only on the current state of a system is called an equation of state. Examples include: energy; entropy; enthalpy; temperature; pressure; volume. Any quantity which depends on that path taken, such as work and heat, is not an equation of state. ds dq de dq dw ds dw (6) 3.2 Adiabatic rocesses When no heat is transferred a process is called adiabatic. In this case the conservation of energy becomes de + dw 0 E W (7) From the relation ds dq, we see that the entropy is constant, and so adiabatic processes are reversible. In the case of an ideal gas one may show that the pressure, volume and temperature are related by γ const. γ const. γ γ const. (8) where γ C C is 5 3 for a monatomic ideal gas and 7 5 for a diatomic ideal gas. hus the change in energy between two states is 2 [ 2 γ E W d γ d ( ) γ ] (9) γ Isothermal rocesses Here the temperature of the system is held fixed. For an ideal gas this means that the combination is constant. he work done in moving between two configurations is 2 2 ( ) 2 W d d nr log (0) 3.4 Isobaric rocesses Here the pressure is constant, and for ideal gases this means that the temperature and volume are directly proportional. he work done through expansion is 4 Ideal Gases W () An idealized gas experiences no inter-molecular forces. Many real-world gases are approximately ideal at high temperatures and low pressures. he thermodynamic quantities associated with an ideal gas are related by the ideal gas law: nr Nk (2) n is the number of moles, while N is the number of molecules. he ideal gas constant, R, and the Boltzmann constant, k, are related through Avagadro s number by R N A k (3) Because the interactions between gas molecules are negligible, the energy of an ideal gas system only depends on the temperature: ( ) E de d (4) 2

5 A generalization of the idea gas law is an der Waals equation: ( ) + n2 a 2 ( nb) nr Nk (5) he constants a and b are determined empirically for any system and account for the intermolecular forces and nonzero volumes of the particles. he ideal gas law is the simplification a b 0. 5 Equipartition heorem Let A be a system described by the coordinates q, q 2,..., q n. Assume that the energy of the system may be written E(q, q 2,..., q n ) E (q ) + E (q 2,..., q n ) E + E (6) that is, the energy is separable in the coordinate q. In addition assume that the system is in thermal equilibrium with its surroundings at a temperature. We would like to find the average energy associated with q, E. E e βe dq dq 2 dq n E E (E ) de e βe (7) dq dq 2 dq n E e βe e βe dq dq 2 dq n E e βe dq e βe dq 2 dq n (8) e βe e βe dq dq 2 dq n e βe dq e βe dq 2 dq n β e βe dq e βe dq ( β log e βe dq ) (9) Now assume the common form E (q ) Aq r : ( β log e βaqr dq ) (20) Make a change of variables to x β /r q : (β β log /r r ) e Axr dx ( β log β /r) ( β log ) e Axr dx (2) β log β ( 0 ) rβ k (22) r Summarizing, if the energy of a system in thermal equilibrium with its surroundings has an energy which may be written as E(q, q 2,..., q n ) Aq r + E (q 2,..., q n ) (23) then we have E Aq r k (24) r For example, an idealized harmonic oscillator has energy given by E + U 2 mv2 + 2 mω2 x 2 (25) 3

6 and so has average energy E + U 2 mv2 + 2 mω2 x 2 2 k + k k (26) 2 In a gas there are three translational degrees of freedom, giving an average energy E 3 2k. At higher temperatures and a diatomic molecule the rotational modes may now be excited, giving an additional two degrees of freedom and an average energy E 5 2k. At still higher temperatures the diatomic molecules may be in excited vibrational modes. With both vibrational kinetic and potential energies, there are an additional two degrees of freedom, giving an average energy E k. Explicitly, at high energies the energy for a diatomic molecule would be of the form 7 2 E 2 M ( vx 2 + vy 2 + vz 2 ) + 2 I ( ω 2 + ω2 2 ) + 2 µ η2 + 2 kη2 (27) M is the molecule s mass, v is the center of mass velocity, I is the moment of inertia about an axis perpendicular to the symmetry axis of the molecule, ω i are the angular velocities about the two axes perpendicular to the symmetry axis of the molecule, µ is the reduced mass of the molecule, k is the effective spring constant for the bond and η is the deviation of the atomic separation distance from its natural length. he seven quadratic degrees of freedom are clearly visible. 6 Maxwell Relations he equality of mixed partial derivatives gives rise to several Maxwell relations between common thermodynamic variables: ( ) ( ) ( ) ( ) (28) S S ( ) ( ) ( ) ( ) (29) A mnemonic for remembering these is the following square, along with the phrase Good hysicists Have Studied Under ery Fine eachers. S U H F G his may be used to quickly express thermodynamic quantities in terms of conjugate variables, located in opposing corners. he two corners opposite the chosen potential are the coefficients for the differentials. For example, the Gibb s free energy, G is given by dg S d + d (30) Notice that the minus sign only appears for the coefficient S and not the differential d. Since we could have written ( ) ( ) G G dg d + d (3) we conclude that S ( ) G 4 ( ) G (32)

7 Equating the mixed partial derivatives of G results in the following: ( ) G ( ) ( ) G he other Maxwell relations arise through similar analyses. ( ) (33) 7 Statistical Mechanics he partition function for a system is defined to be Z i e βe i E Ω(E)e βe (34) In the first expression the sum is over all states, which may be degenerate in energy. In the second expression the sum is over all energies, with the degeneracy of these states denoted by Ω(E). Both methods give the same result. If a system consists of subsystems, with individual partition functions ζ i, which weakly interact then the partition function for the whole system is given by Z i ζ i (35) However, when consider a system of N indistinguishable particles we must include a combinatorial factor: Z ζn (36) N! Refined versions of this are found with a quantum mechanical foundation, giving rise to particle statistics, discussed below. he partition function allows for a statistical calculation of thermodynamic quantities, such as 8 article Statistics E log Z β (37) S k(log Z + β E ) (38) C k 2 2 log Z 2 (39) he classical treatment of multi-particle systems where particles are distinguishable leads to Maxwell- Boltzmann statistics: the expected number of particles in a given state with energy E i is n i N e βe i j e βe j N e βei Z (40) When indistiguishability is introduced in the quantum mechanical treatment of a multiparticle system two distinct behaviours arise: Bose-Einstein statistics and Fermi-Dirac statistics. Bose- Einstein statistics describe systems of bosons, which have integral spin, where the total wavefunction 5

8 is unchanged after swapping two particles. his results in the expected number of particles in a state with energy E i being n i e (Ei µ)/k (4) where µ is the chemical potential. For large this is n i k E i µ Fermi-Dirac statistics describe fermions, which have half-integral spin, where the total wavefunction changes sign upon swapping two particles. One consequence of this is that no two particles may be in the same quantum state, for then the wavefunction would vanish. he expected number of particles in a state with energy E i here is n i e (Ei µ)/k (43) + Notice that for any temperature this is never greater than one. In the case of photons there is no restriction on the number of particles and the expected number of particles in a state with energy E i is For large this is n i e E i/k 9 Einstein & Debye heories of Specific Heat (42) (44) n i k E i (45) Einstein assumed that solids may be modelled as collections of identical, isotropic harmonic oscillators with three degrees of freedom per molecule. Using the appropriate quantum statistics, the average energy of a vibrational states is he quantity hν k and the specific heat is E hν e hν/k is referred to as the Einstein temperature. he total energy per mole is given by E n 3N Ahν e hν/k (46) (47) c ( ) E hν 2 n 3N e hν/k Ak ( k e hν/k ) 2 (48) At high temperatures we have the approximate expression ( c 3N A k + hν ) 3N A k 3R (49) k his matches the classically derived Law of Dulong and etit: E 3nk N A 3nR (50) However, the low temperature behaviour does not match observation, where the specific heat is seen to go as 3. In contrast, the Debye model considers the normal modes of the solid, or phonons, and correctly matches both the high- and low-temperature limiting behaviours. 6

9 A Summary Laws of hermodynamics ( st Law) dq de + dw (2 d Law) S 0 (3 d Law) 0 S S 0 (4 th Law) hermodynamic rocesses S k log Ω ds dq β k log Ω E k E S + S maximal dq 0 γ const. γ const. γ γ const. d 0 W nr log d 0 W ( 2 ) (Entropy) (emperature) (hermal Equilibrium) (Adiabatic rocesses) (Isothermal rocesses) (Isobaric rocesses) Ideal Gases nr Nk (Ideal Gas Law) ( ) E de d (Ideal Gas Energy) ( ) + n2 a 2 ( nb) nr Nk (an der Waals Equation) Equipartition heorem E(q, q 2,..., q n ) E (q ) + E (q 2,..., q n ) E + E (Assumptions) E (q ) Aq r E k (Conclusion) r 7

10 Maxwell Relations ( ) ( ) Statistical Mechanics S ( ) ( ) H E + G H S F E S S U H F G ( ) ( ) S ( ) ( ) (Maxwell Relations) (Enthalpy) (Gibb s Free Energy) (Helmholtz Free Energy) (Mnemonic) Z i e βe i E Ω(E)e βe (artition Function) Z i ζ i (Weakly Interacting Subsystems) Z ζn N! E log Z β S k(log Z + β E ) C 2 log Z k 2 2 (Indistinguishable Subsystems) (Average Energy) (Entropy) (Specific Heat) article Statistics n i N e βe i Z n i e (Ei µ)/k n i e (Ei µ)/k + n i e Ei/k (Maxwell-Boltzmann Statistics) (Bose-Einstein Statistics) (Fermi-Dirac Statistics) (hoton Statistics) heories of Specific Heat ( ) hν 2 e hν/k c 3N A k ( k e hν/k ) 2 (Einstein heory) c α 3 c 3N A k 3R (Debye Low emperature) (Einstein & Debye High emperature) (5) 8