Ch1. Statistical Basis of Thermodynamics

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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. N1023. In
  thermodynamic limit N?, V?, but nN/V finite.
• Macrostate variables (N, V, E)
• Volume V
• Total energy

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Macrostate variables
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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 k1.38x10-23 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)E1E2const.
  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 Sf(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

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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
E(0) E1E2const
V(0) V1V2const
N(0) N1N2const
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1.3 Future contact between statistics and
thermodynamics cont.

• 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) E1E2const, E1, E2 changeable
V(0) V1V2const, V1, V2 changeable
N(0) N1N2const, N1, N2 changeable
W(0) (N1,V1,E1 N2,V2,E2) W1(N1,V1,E1)W2(N2,V2,E
2)
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1.3 Future contact between statistics and
thermodynamics cont.

• Thermal equilibrium state (N1,V1,E1)

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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 systems 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
• N particles of nonatomic molecules
• Free, nonrelativistic particles
• Confined in a cubic box of side L (VL3)

• 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

• 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 Rsqrt(E)

(Appendix C)

• The number W(N,E,V) is the volume in the shell of
  a 3N-sphere.

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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
N2,V2,T
N1,V1,T

• Before mixing

• After mixing

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P1-11
Four moles of nitrogen and one mole of oxygen at
P1 atm and T300K are mixed together to form air
at the same pressure and temperature. Calculate
the entropy of the mixing per mole of the air
formed.