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Chem 402 Lecture 4: L4 1:25:17
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Washington University in St. Louis

University College - Chemistry

University College - Chemistry Chem 402

Barnes Alexander

Spring

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25 January 2017
L4: Deriving Entropy and Ideal Gas Law from Q
I. Thermodynamic Variable Derivations from Q
A. Entropy
1. From L#3 we have the Gibbs equation for S in terms of microstate probabilities
a. π = βπ β π πππ
π π π
2. For an isolated system, all microstates are degenerate in energy
a. Isolated = no energy or mass transfer between system and surroundings
24
b. For a sufficiently large system (~10 = N) that is not isolated, as the number of
molecules increases, there are fewer fluctuations in system energy over time
3. If the system is isolated, all possible states/configurations have the same total energy
and thus the same probability (Ergodic)
a. π = , which we can substitute into our above expression for S
π Ξ©
b. π = βπ β Ξ© 1ππ = βπ β Ξ© 1(ππ1 β ππΞ©) = βπ βππΞ© )β Ξ© 1 = πππΞ©
1 Ξ© Ξ© 1 Ξ© 1 Ξ©
β’ Sub in, then separate the logarithm, since (ππΞ©) is a constant, we can pull it
out in front
β’ β Ξ© 1= 1 always since the sum over all probabilities is equal to one
1Ξ©
c. Boltzman equation for S in terms of degeneracy: π = πππΞ©
ππ
β’ Also, consider Ξ© = Q = (for indistinguishable particles)
π! 24
d. Even if the system is not isolated, energy fluctuations are negligible for ~10
molecules
β’ Can treat as if all states have the same energy, with equal probability and can
still use the Boltzman equation for S
B. Translational Energy Partition Function
1. Total system energy
a. πΈ = π + π + π + π + π + π (for 1 molecule!!)
π π‘ππππ .π₯ π‘ππππ .π¦ π‘ππππ .π§ π£ππ πππ‘ ππππππ
b. Must separate out into types of energy for a system of molecules, and can use
partition functions to do so
2. Separation of Partition Functions
a. When can we write the canonical partition function as a simple product of
molecular partition functions?
π
β’ π π‘ππππ = ππ‘ππππ for distinguishable particles
β’ π = π π /π! for indistinguishable particles
π‘ππππ π‘ππππ
b. These hold when the system microstate energy Eiis a sum of independent
molecule energies π (denoted by ππ, where i represent various quantum
numbers for molecule i)
c. πΈ π βππ ππ = ππ1 + ππ2 + β―+ π ππ
d. The sum over system microstate energies is just a sum over all the possible
combinations of molecular energies1π + 2 + β―+ π π
βπΈπ βππ1+ππ2β―+π ππ
π = βπ ππ = βββ¦βπ ππ
π π1 π2 ππ
βππ1 β ππ2 βππ
= (βπ ππ )(βπ ππ)β¦βπ ππ = π1 2β¦π π
π1 π2 ππ π π
β’ = π for distinguishable particles, = π /π! for indistinguishable particles
e. So if system energy = sum over independent molecula

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