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**Ch1. Statistical Basis of Thermodynamics**

1.1 The macroscopic state and the microscopic state Macrostate: 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.

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**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: V Total energy:

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**Macrostate variables Volume: V Total energy:**

ni – the number of particles with energy ei ei - energy of the individual particles Microstate: all independent solutions of Schrodinger equation of the system. N-particle Schrodinger equation,

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**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 by Where k=1.38x10-23 J/K – Boltzman constant Consider 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))

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Problem 1.2 Assume 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 of f(W) = k ln(W) Solution: Consider two spatially separated systems A and B A B

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**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 change Volume variable No mass change A1 (N1,V1,E1) A2 (N2,V2,E2) E(0) = E1+E2=const V(0) = V1+V2=const N(0) = N1+N2=const

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**1.3 Future contact between statistics and thermodynamics –cont.**

(N1,V1,E1) A2 (N2,V2,E2) Initial states System A1: (N1,V1, E1), S1(N1,V1,E1)=k lnW1(N1,V1,E1) System A2: (N2,V2, E2), S2(N2,V2,E2)=k lnW2(N1,V1,E1) Thermal contact process E(0) = E1+E2=const, E1, E2 changeable V(0) = V1+V2=const, V1, V2 changeable N(0) = N1+N2=const, N1, N2 changeable W(0) (N1,V1,E1; N2,V2,E2)= W1(N1,V1,E1)+W2(N2,V2,E2)

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**1.3 Future contact between statistics and thermodynamics –cont.**

Thermal equilibrium state (N1*,V1*,E1*) m1=m2 P1=P2 T1=T2

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**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 macrostate 4) Determine system’s parameters, T,P, m 5) Determine the other parameters in thermodynamics Helmhohz free energy: A= E-T S Gibbs free energy: G = A + PV = mN Enthalpy: H = E + PV

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**Determine heat capacity**

6) Determine heat capacity Cv and Cp;

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**1.4 Classical ideal gas Model: L**

N particles of nonatomic molecules Free, nonrelativistic particles Confined in a cubic box of side L (V=L3) L Wavefunction and energy of each particle

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**1.4 Classical ideal gas-cont.**

Hamiltonian of each particle Separation of variables Boundary conditions: Y(x) vanishes on the boundary,

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**1.4 Classical ideal gas-cont.**

Boundary conditions: Y(x) vanishes on the boundary

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**Microstate of one particle**

Boundary conditions: Y(x) vanishes on the boundary One microstate is a combination of (nx,ny,,nz)

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**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), satisfying nx ny nz The 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.

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**Microstates of N particles**

The total energy is One microstate with a given energy E is a solution of (n1,n2,……n3N) of 3N-dimension sphere with radius sqrt(E*)

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**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. n1 n2 n3

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**Entropy and thermodynamic properties of an ideal gas**

Determine temperature Determine specific heat

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**State equation of an ideal gas**

Determine pressure Specific heat ratio

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**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. mixing N1,V,T N2,V,T N1,V1,T N2,V2,T Before mixing After mixing

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P1-11 Four 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.

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