Homework Hint. Last Time

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1 Homework Hint Problem 3.3 Geometric series: ωs τ ħ e s= 0 =? a n ar = For 0< r < 1 n= 0 1 r ωs τ ħ e s= 0 1 = 1 e ħω τ Last Time Boltzmann factor Partition Function Heat Capacity The magic of the partition function 1

Last Time Reversible process slow enough to remain in equilibrium the whole time Pressure Thermodynamic Identity Thermodynamically conjugate variables You can control one or the other but not both

2 Last Time Reversible process slow enough to remain in equilibrium the whole time Pressure Thermodynamic Identity Thermodynamically conjugate variables You can control one or the other but not both Today Chapter 3 (part 2) Free energy, ideal gas Helmholtz Free Energy Differential of Free Energy Maxwell Relations Free Energy and the Partition Function Quantum Concentration Ideal Gas Equipartition of Energy Entropy of Mixing 2

3 Intensive and Extensive Variables + = Extensive Variables: Scale with the size of the system Intensive Variables: Don t scale with the size of the system Intensive and Extensive Variables + = Extensive Intensive Conjugate Pairs: One is intensive, the other is extensive. 3

4 Internal Energy Note that for the internal energy U, the variables are the extensive ones: This is the right energy to talk about if entropy and volume are held constant. But that s kind of hard it s easier to hold temperature constant. Helmholtz Free Energy Why a new energy? It s the right thing to use when V, τ are constant. The Helmholtz free energy, rather than internal energy, is minimized for thermodynamic processes at constant V and τ. Takes into account that entropy wants to maximize. This is the available work (useful energy, or free energy) when V, τ are constant. 4

5 Differential of the Helmholtz Free Energy Thermodynamic Identity: Helmholtz energy: product rule These are the right variables for F Legendre Transforms Legendre Transform from one conjugate variable to another We can do similar things with (p,v), and use the Gibbs Free Energy, for systems held at constant temperature and pressure. 5

6 Legendre Transforms Legendre Transform from one conjugate variable to another Legendre Transforms Legendre Transform from one conjugate variable to another And we see that is the energy to use when temperature and pressure are held constant. For now, concentrate on the Helmholtz free energy. We will use G in Ch. 9. We ll see other types of free energy after we learn about the chemical potential in Ch. 5. 6

7 Why Helmholtz Free Energy? It is the type of energy that stays minimized during processes holding temperature and volume constant. If then (constant volume, temperature), Extremum But is it a minimum? Helmholtz Free Energy Minimized Reservoir System Spot the temperature! The Helmholtz free energy is minimized, since the total entropy is maximized. That s why we use it! 7

8 Entropy and Pressure Helmholtz Free Energy Total Differential is like a gradient See Total Differential at mathworld.com Entropy and pressure from Helmholtz Free Energy New Pressure Relation Energy Pressure Entropy Pressure Which term is dominant in solids? Which term is dominant in gases? 8

9 Maxwell Relations Maxwell relations can be derived if you keep straight which type of energy depends on which variables. Conjugate variables go in the numerator The two control variables Not an obvious relation! But useful if you have partial information. Maxwell Relations Take the second derivative of F: You can interchange the order of partial derivatives. Each partial holds the other variable constant. Maxwell Relation (There s one Maxwell relation for each energy type.) 9

10 You Try It: Derive the Maxwell Relation Corresponding to the Gibbs Free Energy G This is the useful energy available for work when pressure and temperature are fixed. F and the Partition Function Find a differential equation for F Remember the magic of Partition function? ε = log( Z ) ( 1 τ ) 10

11 F and the Partition Function ( β F + α ) In principle can add constant: ( β F α ) log ( Z ) + = = U β F = τ Z ατ log ( ) Entropy must reduce to log (g 0 ) at low temperatures, so that only lowest energy sites are occupied ( τ log Z ατ ) τ 0 ( ) limσ = log g F + σ = = = log ( g0 ) + α τ τ V 0 Only if α=0 F and the Partition Function You can get the free energy by summing over Boltzmann factors 11

12 One atom in a box. Ideal Gas L Wavefunction Energy Partition Function Sum over all combinations of n x, n y, and n z One atom in a box. Ideal Gas L Partition Function λ τ ~ ħ / M v ~ ħ / ( M ) 1 2 Quantum concentration: concentration of an atom confined to a debroglie wavelength 12

13 One atom in a box. Partition Function Ideal Gas L n Q = quantum concentration of an atom confined to a debroglie wavelength Make the box large, and don t notice quantum mechanics. Make it small, and notice quantum mechanics. n<<n Q Classical regime n ~ n Q Quantum regime Ideal Gas is in the classical regime, i.e. dilute, with no interactions between particles. Ideal Gas Many atoms in different boxes... Can you multiply the partition function like that? 13

14 Ideal Gas Many atoms in different boxes... Many different atoms in same box. (If they all have the same mass, but are still different.) ν µ Θ σ ο Ι : τ Ideal Gas Stuff identical particles all into the same box, but not too tightly. Can we use the same approach? Z = Z 1 *Z 2 *? -no! It will overcount 1 way to choose 2 ways to choose 3 ways to choose Divide by this factor Ideal Gas Partition Function (Dilute, noninteracting classical) Fine print: If dilute, chances of two particles in same energy and state is negligible. Two particles in same energy and state makes counting harder. 14

15 Ideal Gas Internal Energy Energy of an ideal gas Ideal Gas Stirling approximation Free energy: Pressure: Ideal Gas Law 15

16 Equipartition of Energy Ideal Gas Energy 3N = Degrees of Freedom (DOF) For any quadratic Hamiltonian, Internal Energy for one particle Can also count the squared terms in the Hamiltonian = 3 squared terms X N particles = 3N In a harmonic oscillator, you have to count all squared terms: 2 squared terms X N particles = 2N DOF Equipartition of Energy 5 τ 2 7 τ 2 3 τ 2 τ 16

17 Entropy of Mixing ALLOY AABBABAAA BBAABABBB ABBBABBAA BBAABAAAB Stirling! Concentration of B Entropy is always positive Entropy of Mixing ALLOY AABBABAAA BBAABABBB ABBBABBAA BBAABAAAB Assume: same interaction energy: U AA = U BB = U AB Not mixed: ( ) Mixed: F = F0 τσ ( x) F = 1 x F + xf = F Always >0 Mixed state has lower F 17

18 Entropy of Mixing N=1000 ALLOY AABBABAAA BBAABABBB ABBBABBAA BBAABAAAB x Entropy of Mixing AAAAAAA AAAAAAA AAAAAAA AAAAAAA Atoms that do not want to mix: F0 = U0 τσ 0 BAAAAAA AAAAAAA AAAAAAA AAAAAAA replace just one atom F = F + U τσ Extra energy and entropy associated with replacement 0 AB 1 F = F0 + U AB τ log( N) The extra energy is constant, but entropy increases with N: can always make N large enough to get U τ log( N ) < 0 AB 18

19 Small Concentration of Impurities Small concentration Interaction energy A to B Taylor: Small Concentration of Impurities Repulsive energy U AB >0 Minimize Free Energy There is a natural impurity content in all crystals, driven by entropy. 19

20 Today Helmholtz Free Energy Differential of Free Energy Maxwell Relations Free Energy and the Partition Function Quantum Concentration Today Ideal Gas Law Equipartition of Energy Entropy of Mixing things want to mix; alloys are stable 20

(# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble

Recall from before: Internal energy (or Entropy): &, *, - (# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble & = /01Ω maximized Ω: fundamental statistical quantity