Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
STATISTICAL THERMODYNAMICS Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email [email protected] Tel (65) 874-2749 Fax (65) 779-5452 SECOND LAW REVISITED Lord Kelvin : A transformation whose only final result is to transform into work heat extracted from a source which is at the same temperature throughout is impossible Rudolph Clausius : A transformation whose only final result is to transfer heat from a body at a given temperature to a body at a higher temperature is impossible (this principle implies the previous one) Martian Skeptic : What temperature ? SECOND LAW REVISITED Definition : Body A has higher temperature than body B (A B) if, when we bring them into thermal contact, heat flows from A to B. Body A has the same temperature as B ( A B) if, when we bring them into thermal contact, no heat flows from A to B and no heat flows from B to A. ( [A] := {U | U A} ) Enrico Fermi : (Clausius Reformulated) If heat flows by conduction from a body A to another body B, then a transformation whose only final result is to transfer heat from B to A is impossible. SECOND LAW REVISITED Sadi Carnot : If A B then we can use a reversible engine to absorb a quantity Q A 0 of heat from A, surrender a quantity of heat Q A Q B 0 to B, and perform a quantity of work W 0. Carnot & Kelvin implies Clausius, and Q A Q B f ([A], [B]), and for B C, f ([A], [B]) f ([B], [C]) f ([A], [C]) SECOND LAW REVISITED Definition : Absolute Thermodynamic Temperature Choose a body D If A D AD DA then T( A) f ([ A], [D]) then T( A) 1 then T( A) 1 f ([ D], [A]) In a reversible process Q A T( A) Q B T( B) 0 SECOND LAW REVISITED System : Cylinder that has a movable piston and contains a fixed amount of homogeneous fluid States (Macroscopic) : Region in positive quadrant of the (V = volume, T = temperature) plane. Functions (on region) : V, T, p = pressure Paths (in region) : Oriented curves Differential Forms : can be integrated over paths WORK W( ) HEAT pdV Q( ) ? SECOND LAW REVISITED Definition : Entropy Function S Choose a state (V1, T1 ) Define S(V, T) Q( 1 ) T1 where 1 2 joins (V1, T1 ) to (V, T), 1 is isothermal, and 2 is adiabatic (by thermal equilibrium and by thermal isolation) Carnot' s cyle 1 2 is and Q( ) 2 TdS FIRST LAW REVISITED There exists an (internal energy) function U such that TdS - pdV dU Therefore U U TdS dT p dV T V FIRST & SECOND LAWS COMBINED S 1 U ; T T T S 1 U p V T V Therefore, the basic (but powerful) calculus identity S S V T T V Yields (after some tedious but straightforward algebra) p U T -p V T IDEAL GAS LAW (Chemists) Boyl, Gay-Lussac, Avogardo p(V, T) n R g / V n amount of gas in moles R ideal gas constant (8.314 joules / degree Kelvin) g ideal gas temperature in Kelvin (water freezes at 373.16 degrees) JOULE’S GAS EXPANSION EXPERIMENT We substitute the expression for p (given by the ideal gas law) to obtain U nR V V dg g T dT and observe that the outcome of Joule’s gas expansion experiment U 0 Tg V IDEAL GAS LAW (Physicists) p(V, T) N k T V N number of molecules of gas (6.0225 10 23 molecules / mole) k Boltzmann’s constant (1.38 10 23 joules / deg _Kelvin) GAS THERMODYNAMICS Experimental Result : (dilute gases) dU Nk ( 1) 1 dT Therefore TV 1 & p V constant on adiabatic paths S Nk ( 1) 1 ln(T / T1 ) Nk ln(V / V1 ) GAS KINETICS Monatomic dilute gas, m = molecular mass U N 1 2 m 2 1 2 3 m x 2 average kinetic energy / molecule N A x t F t 2m x V 2 F 1 p N( 1) m2 V 2 A 5 1 3 2 kT ( 1) m 2 GAS KINETICS Photon gases E c momentum 4 3 Maxwell Equipartition of Energy m11 m 2 2 0 (1 2 ) m1 m 2 kinetic energy 1 kT in each direction 2 2 1 degrees of freedom EQUIPARTITION Number of ways of partitioning N objects into m bins with relative frequencies (probabilities) p1, p 2 ,..., p m is C N! (p1N)!... (p m N)! Stirling’s formula ( ln N! N ln N - N ) yields ln C N H(p1,..., p m ) where H(p1,..., p m ) denotes Shannon’s information-theoretic entropy H(p1,..., p m ) p1 ln p1 p m ln p m EQUIPARTITION If the bins correspond to energies, then H(p1,..., p m ), and therefore (nearly) C, is maximized, subject to an energy constraint E N(p1E1 p m E m ), by the Gibbs distribution pi exp( Ei kT ) Z(T) S k ln C 1 T dS dE and free energy E TS NkT ln Z(T) 2 Maxwell dist. prob(x, ) exp (-m / 2kT ) Therefore THIRD LAW Nernst : The entropy of every system at absolute zero can always be taken equal to zero inherently quantum mechanical discrete microstates, a quart bottle of air has about 10 24 1022 molecules & 10 microstates 1 bit of information kT ln 2 energy Maxwell’s demon : may he rest in peace Time’s arrow : probably forward ??? REFERENCES V. Ambegaokar, Reasoning about Luck H. Baeyer, Warmth Disperses and Time Passes F. Faurote, The How and Why of the Automobile E. Fermi, Thermodynamics R. Feynman, Lectures on Physics, Volume 1 REFERENCES H. S. Green and T. Triffet, Sources of Consciousness, The Biophysical and Computational Basis of Thought K. Huang, Statistical Mechanics N. Hurt and R. Hermann, Quantum Statistical Mechanics and Lie Group Harmonic Analysis C. Shannon and W. Weaver, The Mathematical Theory of Communication