Specific Heats of Ideal Gases 1 Assume that a pure, ideal gas is made of tiny particles that bounce into each other and the walls of their cubic container of side `. Show the average pressure P exerted by this gas is 1 N 35 P = mv 2 3 V total SO 30 2 7 7 (J/K-mole) R = NAkB Use the ideal gas law (PV = NkB T = V 2 2 CO2 C H O CH nRT )and the conservation of energy 25 2 4 Cl2 (∆Eint = CV ∆T ) to calculate the specific 5 5 20 2 R = 2 NAkB heat of an ideal gas and show the following. H2 N2 O2 CO 3 3 15 CV = NAkB = R 3 3 R = NAkB 2 2 He Ar Ne Kr 2 2 10 Is this right? 5 3 N - number of particles V = ` 0 Molecule kB - Boltzmann constant m - atomic mass NA - Avogadro's number vtotal - atom's speed Jerry Gilfoyle Einstein Solid 1 / 36 The Results 2 1 N 2 2 N 7 P = mv = hEkini 2 NAkB 3 V total 3 V 35 30 SO2 3 (J/K-mole) V 5 CO2 C N k hEkini = NkB T H O CH 2 A B 2 25 2 4 Cl2 20 H N O CO 3 3 2 2 2 3 CV = NAkB = R 2 NAkB 2 2 15 He Ar Ne Kr 10 5 0 Molecule Jerry Gilfoyle Einstein Solid 2 / 36 Quantum mechanically 2 E qm = `(` + 1) ~ rot 2I where l is the angular momen- tum quantum number. Rotational Kinetic Energy 3 Classically L2 E = rot 2I where Z X 2 2 I = mri = r dm Jerry Gilfoyle Einstein Solid 3 / 36 Rotational Kinetic Energy 4 Classically L2 E = rot 2I where Z X 2 2 I = mri = r dm Quantum mechanically 2 E qm = `(` + 1) ~ rot 2I where l is the angular momen- tum quantum number. Jerry Gilfoyle Einstein Solid 3 / 36 The Kinetic Model of Ideal Gases 5 1 The gas consists of a large number of small, mobile particles and their average separation is large. 2 The particles obey Newton's Laws and the conservation laws, but their motion can be described statistically. 3 The particles' collisions are elastic. 4 The inter-particle forces are small until they collide. 5 The gas is pure. 6 The gas is in thermal equilibrium with the container walls. Jerry Gilfoyle Einstein Solid 4 / 36 A Hint of Quantum Mechanics 6 H2 gas Jerry Gilfoyle Einstein Solid 5 / 36 The Specific Heat of Elemental Solids - 1 7 R = 8:31 J=mol K Jerry Gilfoyle Einstein Solid 6 / 36 The Specific Heat of Elemental Solids - 2 8 Solid Molar Specific Heat Our Results (J/K-mole) (J/K-mole) Lead 26:4 22 ± 6 Zinc 25:4 36 ± 14 Aluminum 26:4 24 ± 5 Copper 24:5 23 ± 5 Tin 27:0 52 ± 15 Gold 25:4 Silver 25:4 Iron 25:0 Jerry Gilfoyle Einstein Solid 7 / 36 The Einstein Solid 9 Cell boundaries atoms Jerry Gilfoyle Einstein Solid 8 / 36 The Einstein Solid 10 Jerry Gilfoyle Einstein Solid 9 / 36 The Einstein Solid 11 Jerry Gilfoyle Einstein Solid 10 / 36 The Plan 12 Quantum Mechanics The Model Combinations Probability Peak Multiplicity and Width Entropy (S) Link S to Energy and Temperature Calculate Molar Specific Heat Compare with Data!! Jerry Gilfoyle Einstein Solid 11 / 36 Molar Specific Heat of Elemental Solids 13 An Einstein solid is made of N, three-dimensional harmonic oscillators containing q quanta of energy. 1 What is the multiplicity of Molar Specific Solid Heat Our Results a single Einstein solid? (J/K-mole) (J/K-mole) Lead 26:4 ± 0:7 22 ± 6 2 What is the multiplicity of Zinc 25:4 ± 0:6 36 ± 14 two Einstein solids in ther- Aluminum 26:4 ± 0:2 24 ± 3 mal contact? Copper 24:5 ± 0:6 23 ± 5 3 How would you determine Tin 27:0 ± 0:6 52 ± 15 the most likely microstate Gold 25:4 ± 0:6 Silver 25:4 ± 0:6 of the system? Iron 25:0 ± 0:6 4 How is entropy related to temperature? 5 How is the energy related to temperature? 6 What is the molar specific heat of an elemental solid? Jerry Gilfoyle Einstein Solid 12 / 36 Rolling Dice 14 Jerry Gilfoyle Einstein Solid 13 / 36 Rolling Dice 15 Jerry Gilfoyle Einstein Solid 14 / 36 Rolling Dice 16 Jerry Gilfoyle Einstein Solid 15 / 36 A Probability Distribution 17 Throwing Dice 6 5 Number of Events 4 3 2 1 0 0 2 4 6 8 10 12 14 16 Sum Jerry Gilfoyle2014-12-30 19:21:22 Einstein Solid 16 / 36 The Entropy of the Einstein Solid 18 An Einstein solid is made of N, three-dimensional harmonic oscillators containing q quanta of energy as shown below. 1 What is the energy of a single oscillator? of N oscillators? 2 How many microstates Ω exist for a `system' with N = 1 and q = 3? for N = 2, q = 2? 3 What is Ω for any N and q? Jerry Gilfoyle Einstein Solid 17 / 36 Microstates for NA = 2, qA = 2 19 n1 n2 n3 n4 n5 n6 n1 n2 n3 n4 n5 n6 Jerry Gilfoyle Einstein Solid 18 / 36 Microstates for NA = 2, qA = 2 20 Jerry Gilfoyle Einstein Solid 19 / 36 The Specific Heat of Elemental Solids - 1 21 R = 8:31 J=mol K Jerry Gilfoyle Einstein Solid 20 / 36 Summary So Far 22 3N X Eint = (nx + ny + nz )~! = ni ~! for N atoms i=1 multiplicity (Ω) number of microstates macrostate configuration of a solid defined by bulk proper- ties like N and E=U. microstate one of the configurations of quanta consistent with the macrostate. (qA + 3NA − 1)! Ω(NA; qA) = qA!(3NA − 1)! ΩAB - multiplicity of combined state ΩAB = ΩAΩB where ΩA;B - individual multiplicities. Jerry Gilfoyle Einstein Solid 21 / 36 Summary So Far 23 3N X Eint = (nx + ny + nz )~! = ni ~! for N atoms i=1 multiplicity (Ω) number of microstates macrostate configuration of a solid defined by bulk proper- ties like N and E=U. microstate one of the configurations of quanta consistent with the macrostate. (qA + 3NA − 1)! Ω(NA; qA) = qA!(3NA − 1)! ΩAB - multiplicity of combined state ΩAB = ΩAΩB where ΩA;B - individual multiplicities. Jerry Gilfoyle Einstein Solid 21 / 36 The Plan 24 Quantum Mechanics The Model Combinations Probability Peak Multiplicity and Width Entropy (S) Link S to Energy and Temperature Calculate Molar Specific Heat Compare with Data!! Jerry Gilfoyle Einstein Solid 22 / 36 Multiplicity and Number of Quanta 25 Einstein Solid 2.0 red-B black-A+B blue-A 1.5 x10-54 NA=30 NB=25 qAB=200 W 1.0 0.5 x10-32 x10-37 0.0 0 50 100 150 200 qa Jerry Gilfoyle Einstein Solid 23 / 36 Some Math - Logarithms 26 Jerry Gilfoyle Einstein Solid 24 / 36 Entropy of Two Einstein Solids 27 SAB SB Entropy Equilibrium Point SA qa Jerry Gilfoyle Einstein Solid 25 / 36 The Specific Heat of Elemental Solids - 1 28 R = 8:31 J=mol K Jerry Gilfoyle Einstein Solid 26 / 36 Entropy of Two Einstein Solids in Energy Terms 29 SAB SB Entropy Equilibrium Point SA E a Jerry Gilfoyle Einstein Solid 27 / 36 The Plan 30 Quantum Mechanics The Model Combinations Probability Peak Multiplicity and Width Entropy (S) Link S to Energy and Temperature Calculate Molar Specific Heat Compare with Data!! Jerry Gilfoyle Einstein Solid 28 / 36 Slopes 31 Red - dSAB/dEA dSA/dEA dE / dS dSB/dEA Ea Jerry Gilfoyle Einstein Solid 29 / 36 The Plan 32 Quantum Mechanics The Model Combinations Probability Peak Multiplicity and Width Entropy (S) Link S to Energy and Temperature Calculate Molar Specific Heat Compare with Data!! Jerry Gilfoyle Einstein Solid 30 / 36 The Specific Heat of Elemental Solids 33 Measured Our Solid Molar Specific Heat Our Results Calculation (J/K-mole) (J/K-mole) (J/K-mole) Lead 26:4 ± 0:7 22 ± 8 24:9 Zinc 25:4 ± 0:6 36 ± 14 24:9 Aluminum 26:4 ± 0:2 24 ± 3 24:9 Copper 24:5 ± 0:6 23 ± 5 24:9 Tin 27:0 ± 0:6 52 ± 15 24:9 Gold 25:4 ± 0:6 24:9 Silver 25:4 ± 0:6 24:9 Iron 25:0 ± 0:6 24:9 Jerry Gilfoyle Einstein Solid 31 / 36 The Harmonic Oscillator Approximation 34 Harmonic Oscillator Potential red Energy H L Potential True Potential green Displacement From Equilibrium Jerry Gilfoyle H L Einstein Solid 32 / 36 Quantum Oscillator 35 Jerry Gilfoyle Einstein Solid 33 / 36 Installing Capstone 1 Go to the website https://www.pasco.com/downloads/capstone and select the free trial for your platform (Windows or Mac). The installer will be downloaded to your machine. 2 Launch the installer you just downloaded. 3 Accept defaults. 4 On first launch, enter the license key listed below. 19F5C-S10o2-4o0m0-ppip3-40qr8-ece1h 5 The capstone files for each lab will be linked to the lab schedule on the course website at the following location. https://facultystaff.richmond.edu/~ggilfoyl/genphys.html Jerry Gilfoyle Einstein Solid 34 / 36 Installing statmech.exe 37 1 Go to: http://www.physics.pomona.edu/sixideas/old/sicpr.html 2 Scroll down to the section entitled \For Use With Unit T:". 3 Scroll down to the paragraph that starts with \statmech 2.7". 4 Scroll down to \Download for:" and right click on \Windows" or \Mac OSX" and save it to your Desktop. 5 On your desktop double click on the folder entitled \statmech.exe.zip".