Presentation on theme: "Ch1. Statistical Basis of Thermodynamics"— Presentation transcript:
1 Ch1. Statistical Basis of Thermodynamics 1.1 The macroscopic state and the microscopic stateMacrostate: a macrostate of a physical system is specified by macroscopic variables (N,V,E).Microstate: a microstate of a system is specified by the positions, velocities, and internal coordinates of all the molecules in the system.For a quantum system, Y(r1,r2,….,rN), specifies a microstate.
2 Microstate Number W(N,V,E) For a given macrostate (N,V,E), there are a large number of possible microstates that can make the values of macroscopic variables. The actual number of all possible miscrostate is a function of macrostate variables.Consider a system of N identical particles confined to a space of volume V. N~1023. In thermodynamic limit: N¥, V¥, but n=N/V finite.Macrostate variables (N, V, E)Volume: VTotal energy:
3 Macrostate variables Volume: V Total energy: ni – the number of particles with energy eiei - energy of the individual particlesMicrostate: all independent solutions of Schrodinger equation of the system. N-particle Schrodinger equation,
4 Physical siginificance of W(N,V,E) For a given macrostate (N,V,E) of a physical system, the absolute value of entropy is given byWhere k=1.38x10-23 J/K – Boltzman constantConsider two system A1 and A2 being separately in equilibrium.When allow two systems exchanging heat by thermal contact, the whole system has E(0)=E1+E2=const. macrostate (N,V, E(0))
5 Problem 1.2Assume that the entropy S and the statistical number W of a physical system are related through an arbitrary function S=f(W). Show that the additive characters of S and the multiplicative character of W necessarily required that the function f(W) to be the form off(W) = k ln(W)Solution: Consider two spatially separated systems A and BAB
6 1.3 Future contact between statistics and thermodynamics Consider energy change between two sub-systems A1 and A2, both systems can change their volumes while keeping the total volume the constant.Energy changeVolume variableNo mass changeA1(N1,V1,E1)A2(N2,V2,E2)E(0) = E1+E2=constV(0) = V1+V2=constN(0) = N1+N2=const
8 1.3 Future contact between statistics and thermodynamics –cont. Thermal equilibrium state (N1*,V1*,E1*)m1=m2P1=P2T1=T2
9 Summary-how to derive thermodynamics from a statistical beginning? 1) Start from the macrostate (N,V,E) of the given system;2) Determine the number of all possible microstate accessible to the system, W(N,V,E).3) Calculate the entropy of the system in that macrostate4) Determine system’s parameters, T,P, m5) Determine the other parameters in thermodynamicsHelmhohz free energy: A= E-T SGibbs free energy: G = A + PV = mNEnthalpy: H = E + PV
11 1.4 Classical ideal gas Model: L N particles of nonatomic moleculesFree, nonrelativistic particlesConfined in a cubic box of side L (V=L3)LWavefunction and energy of each particle
12 1.4 Classical ideal gas-cont. Hamiltonian of each particleSeparation of variablesBoundary conditions: Y(x) vanishes on the boundary,
13 1.4 Classical ideal gas-cont. Boundary conditions: Y(x) vanishes on the boundary
14 Microstate of one particle Boundary conditions: Y(x) vanishes on the boundaryOne microstate is a combination of (nx,ny,,nz)
15 The number of microstate of one particle W(1,e,V) The number of distinct microstates for a particle with energy e is the number of independent solutions of (nx,ny,nz), satisfyingnxnynzThe number W(1,e,V) is the volume in the shell of a 3 sphere. The volume of in (nx,ny,nz) space id 1.
16 Microstates of N particles The total energy isOne microstate with a given energy E is a solution of (n1,n2,……n3N) of3N-dimension sphere with radius sqrt(E*)
17 The number of microstate of N particles W(N,E,V) The volume of 3N-sphere with radius R=sqrt(E*)(Appendix C)The number W(N,E,V) is the volume in the shell of a 3N-sphere.n1n2n3
18 Entropy and thermodynamic properties of an ideal gas Determine temperatureDetermine specific heat
19 State equation of an ideal gas Determine pressureSpecific heat ratio
20 1.5 The entropy of mixing ideal gases Consider the mixing of two ideal gases 1 and 2, which are initially at the same temperature T. The temperature of the mixing would keep as the same.mixingN1,V,TN2,V,TN1,V1,TN2,V2,TBefore mixingAfter mixing
21 P1-11Four moles of nitrogen and one mole of oxygen at P=1 atm and T=300K are mixed together to form air at the same pressure and temperature. Calculate the entropy of the mixing per mole of the air formed.