Non-Continuum Energy Transfer: Gas Dynamics

1. Non-Continuum Energy Transfer: Gas Dynamics

2. Phonons – What We’ve Learned • Phonons are quantized lattice vibrations • store and transport thermal energy • primary energy carriers in insulators and semi-conductors (computers!) • Phonons are characterized by their • energy • wavelength (wave vector) • polarization (direction) • branch (optical/acoustic)  acoustic phonons are the primary thermal energy carriers • Phonons have a statistical occupation (Bose-Einstein), quantized (discrete) energy, and only limited numbers at each energy level • we can derive the specific heat! • We can treat phonons as particles and therefore determine the thermal conductivity based on kinetic theory

3. Electrons – What We’ve Learned • Electrons are particles with quantized energy states • store and transport thermal and electrical energy • primary energy carriers in metals • usually approximate their behavior using the Free Electron Model • energy • wavelength (wave vector) • Electrons have a statistical occupation (Fermi-Dirac), quantized (discrete) energy, and only limited numbers at each energy level (density of states) • we can derive the specific heat! • We can treat electrons as particles and therefore determine the thermal conductivity based on kinetic theory • Wiedemann Franz relates thermal conductivity to electrical conductivity • In real materials, the free electron model is limited because it does not account for interactions with the lattice • energy band is not continuous • the filling of energy bands and band gaps determine whether a material is a conductor, insulator, or semi-conductor

4. Gases – Individual Particles gas … gas • We will consider a gas as a collection of individual particles • monatomic gasses are simplest and can be analyzed from first principles fairly readily (He, Ar, Ne) • diatomic gasses are a little more difficult (H2, O2, N2)  must account for interactions between both atoms in the molecule • polyatomic gasses are even more difficult

5. Gases – How to Understand One just not possible • Understanding a gas – brute force • suppose we wanted to understand a system of N gas particles in a volume V (~1025gas molecules in 1 mm3 at STP)  position & velocity • Understanding a gas – statistically • statistical mechanics helps us understand microscopic properties and relate them to macroscopic properties • statistical mechanics obtains the equilibrium distribution of the particles • Understanding a gas – kinetically • kinetic theory considers the transport of individual particles (collisions!) under non-equilibrium conditions in order to relate microscopic properties to macroscopic transport properties  thermal conductivity!

6. Gases – Statistical Mechanics If we have a gas of N atoms, each with their own kinetic energy ε, we can organize them into “energy levels” each with Niatoms total atoms in the system: internal energy of the system: • We call each energy level εi with Niatoms a macrostate • Each macrostate consists of individual energy states called microstates • these microstates are based on quantized energy  related to the quantum mechanics  Schrödinger’s equation • Schrödinger’s equation results in discrete/quantized energy levels (macrostates) which can themselves have different quantum microstates (degeneracy, gi)  can liken it to density of states gas … gas

7. Gases – Statistical Mechanics • There can be any number of microstates in a given macrostate called that levels degeneracy gi • this number of microstates the is thermodynamic probability, Ω, of a macrostate • We describe thermodynamic equilibrium as the most probable macrostate • Three fairly important assumptions/postulates • The time-average for a thermodynamic variable is equivalent to the average over all possible microstates • All microstates are equally probable • We assume independent particles • Maxwell-Boltzmann statistics gives us the thermodynamic probability, Ω, or number of microstates per macrostate

8. Gases – Statistics and Distributions The thermodynamic probability can be determined from basic statistics but is dependant on the type of particle. Recall that we called phonons bosons and electrons fermions. Gas atoms we consider boltzons Maxwell-Boltzmann distribution Maxwell-Boltzmann statistics boltzons: distinguishable particles Bose-Einstein distribution Bose-Einstein statistics bosons: indistinguishable particles Fermi-Dirac distribution fermions: indistinguishable particles and limited occupancy (Pauli exclusion) Fermi-Dirac statistics

9. Gases – What is Entropy? Thought Experiment: consider a chamber of gas expanding into a vacuum A B A B • This process is irreversible and therefore entropy increases (additive) • The thermodynamic probability also increases because the final state is more probable than the initial state (multiplicative) How is the entropy related to the thermodynamic probability (i.e., microstates)? Only one mathematical function converts a multiplicative operation to an additive operation Boltzmann relation!

10. Gases – The Partition Function The probability of atoms in energy level iis simply the ratio of particles in ito the total number of particles in all energy levels leads directly to Maxwell-Boltzmann distribution The partition function Zis an useful statistical definition quantity that will be used to describe macroscopic thermodynamic properties from a microscopic representation

11. Gases – 1St Law from Partition Function First Law of Thermodynamics – Conservation of Energy! Heat and Work adding heat to a system affects occupancy at each energy level a system doing/receiving work does changes the energy levels

12. Gases – Equilibrium Properties Energy and entropy in terms of the partition function Z Classical definitions & Maxwell Relations then lead to the statistical definition of other properties chemical potential Gibbs free energy pressure Helmholtz free energy

13. Gases – Equilibrium Properties enthaply but classically … the Boltzmann constant is directly related to the Universal Gas Constant ideal gas law

14. Gases – Equilibrium Properties Recalling that the specific heat is the derivative of the internal energy with respect to temperature, we can rewrite intensive properties (per unit mass) statistically internal energy entropy Gibbs free energy enthaply Helmholtz free energy specific heat

15. Gases – Monatomic Gases • In diatomic/polyatomic gasses the atoms in a molecule can vibrate between each other and rotate about each other which all contributes to the internal energy of the “particle” • monatomic gasses are simpler because the internal energy of the particle is their kinetic energy and electronic energy (energy states of electrons) • an evaluation of the quantum mechanics and additional mathematics can be used to derive translational and electronic partition functions consider the translational energy only we can plug this in to our previous equations internal energy entropy specific heat

16. Gases – Monatomic Gases Where did P (pressure) come from in the entropy relation? pressure plugging in the translational partition function …. the derivative of the ln(CV)is 1/V ideal gas law

17. Gases – Monatomic Gases The electronic energy is more difficult because you have to understand the energy levels of electrons in atoms  not too bad for monatomic gases (We can look up these levels for some choice atoms) Defining derivatives as internal energy entropy specific heat

18. Gases – Monatomic Helium Consider monatomic hydrogen at 1000 K … I can look up electronic degeneracies and energies to give the following table

19. Gases – Monatomic Helium from Incropera and Dewitt

20. Gases – A Little Kinetic Theory We’ve already discussed kinetic theory in relation to thermal conductivity  individual particles carrying their energy from hot to cold G. Chen The same approach can be used to derive the flux of any property for individual particles  individual particles carrying their energy from hot to cold general flux of scalar property Φ

21. Gases – Viscosity and Mass Diffusion Consider viscosity from general kinetic theory (flux of momentum)  Newton’s Law Consider mass diffusion from general kinetic theory (flux of mass)  Fick’s Law Note that all these properties are related and depend on the average speed of the gas molecules and the mean free path between collisions

22. Gases – Average Speed The average speed can be derived from the Maxwell-Boltzmann distribution We can derive it based on assuming only translational energy, gi= 1 (good for monatomic gasses – recall that translation dominates electronic) This is a ratio is proportional to a probability density function  by definition the integral of a probability density function over all possible states must be 1 probability that a gas molecule has a given momentum p

23. Gases – Average Speed From the Maxwell-Boltzmann momentum distribution, the energy, velocity, and speed distributions easily follow

24. Gases – Mean Free Path The mean free path is the average distance traveled by a gas molecule between collisions  we can simply gas collisions using a hard-sphere, binary collision approach (billiard balls) incident particle collision with target particle rincident rincident d12 cross section defined as: rtarget General mean free path Monatomic gas

25. Gases – Transport Properties Based on this very simple approach, we can determine the transport properties for a monatomic gas to be more rigorous collision dynamics model M is molecular weight Recall, that

26. Gases – Monatomic Helium from Incropera and Dewitt only 2% difference!

27. Gases – What We’ve Learned • Gases can be treated as individual particles • store and transport thermal energy • primary energy carriers fluids  convection! • Gases have a statistical (Maxwell-Boltzmann) occupation, quantized (discrete) energy, and only limited numbers at each energy level • we can derive the specific heat, and many other gas properties using an equilibrium approach • We can use non-equilibrium kinetic theory to determine the thermal conductivity, viscosity, and diffusivity of gases • The tables in the back of the book come from somewhere!