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

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. 𝑆 = βˆ’π‘˜ βˆ‘ 𝑝 𝑙𝑛𝑝 𝑖 𝑖 𝑖 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. 𝑝 = , which we can substitute into our above expression for S 𝑖 Ξ© b. 𝑆 = βˆ’π‘˜ βˆ‘ Ξ© 1𝑙𝑛 = βˆ’π‘˜ βˆ‘ Ξ© 1(𝑙𝑛1 βˆ’ 𝑙𝑛Ω) = βˆ’π‘˜ βˆ’π‘™π‘›Ξ© )βˆ‘ Ξ© 1 = π‘˜π‘™π‘›Ξ© 1 Ξ© Ξ© 1 Ξ© 1 Ξ© β€’ Sub in, then separate the logarithm, since (𝑙𝑛Ω) is a constant, we can pull it out in front β€’ βˆ‘ Ξ© 1= 1 always since the sum over all probabilities is equal to one 1Ξ© c. Boltzman equation for S in terms of degeneracy: 𝑆 = π‘˜π‘™π‘›Ξ© π‘žπ‘ β€’ Also, consider Ξ© = Q = (for indistinguishable particles) 𝑁! 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. 𝐸 = πœ€ + πœ€ + πœ€ + πœ€ + πœ€ + πœ€ (for 1 molecule!!) 𝑖 π‘‘π‘Ÿπ‘Žπ‘›π‘ .π‘₯ π‘‘π‘Ÿπ‘Žπ‘›π‘ .𝑦 π‘‘π‘Ÿπ‘Žπ‘›π‘ .𝑧 𝑣𝑖𝑏 π‘Ÿπ‘œπ‘‘ π‘π‘œπ‘›π‘“π‘–π‘” 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? 𝑁 β€’ 𝑄 π‘‘π‘Ÿπ‘Žπ‘›π‘ = π‘žπ‘‘π‘Ÿπ‘Žπ‘›π‘  for distinguishable particles β€’ 𝑄 = π‘ž 𝑁 /𝑁! for indistinguishable particles π‘‘π‘Ÿπ‘Žπ‘›π‘  π‘‘π‘Ÿπ‘Žπ‘›π‘  b. These hold when the system microstate energy Eiis a sum of independent molecule energies 𝑖 (denoted by 𝑛𝑖, where i represent various quantum numbers for molecule i) c. 𝐸 𝑖 βˆ‘π‘›π‘– 𝑛𝑖 = πœ€π‘›1 + πœ€π‘›2 + β‹―+ πœ€ 𝑛𝑁 d. The sum over system microstate energies is just a sum over all the possible combinations of molecular energies1πœ€ + 2 + β‹―+ πœ€ 𝑁 βˆ’πΈπ‘– βˆ’πœ€π‘›1+πœ€π‘›2β‹―+πœ€ 𝑛𝑁 𝑄 = βˆ‘π‘’ π‘˜π‘‡ = βˆ‘βˆ‘β€¦βˆ‘π‘’ π‘˜π‘‡ 𝑖 𝑛1 𝑛2 𝑛𝑁 βˆ’πœ€π‘›1 βˆ’ πœ€π‘›2 βˆ’πœ€π‘ = (βˆ‘π‘’ π‘˜π‘‡ )(βˆ‘π‘’ π‘˜π‘‡)β€¦βˆ‘π‘’ π‘˜π‘‡ = π‘ž1 2β€¦π‘ž 𝑁 𝑛1 𝑛2 𝑛𝑁 𝑁 𝑁 β€’ = π‘ž for distinguishable particles, = 𝑁 /𝑁! for indistinguishable particles e. So if system energy = sum over independent molecula
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