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Lecture 1. Temperature, Ideal Gas (Ch. 1 )

Lecture 1. Temperature, Ideal Gas (Ch. 1 ). Outline : Intro, the structure of the course Temperature The Ideal Gas Model Equipartition Theorem. Professor:    Michael Gershenson Office:         Serin Physics Building Room 122W e-mail:          gersh@physics.rutgers.edu

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Lecture 1. Temperature, Ideal Gas (Ch. 1 )

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  1. Lecture 1. Temperature, Ideal Gas (Ch. 1 ) • Outline: • Intro, the structure of the course • Temperature • The Ideal Gas Model • Equipartition Theorem

  2. Professor:    Michael Gershenson Office:         Serin Physics Building Room 122W e-mail:          gersh@physics.rutgers.edu phone:    732-445-3180 (office) office hour:  Mon. 2:30 - 3:30 PM and by appointment lectures:       SEC-117,  W  10:20-11:40 AM and F 3:20-4:40 PM Textbook: An Introduction to Thermal Physics, D.V. Schroeder, Addison-Wesley-Longman, 2000 Web Site Access: http://www.physics.rutgers.edu/ugrad/351 Homework assignments, announcements and notes will be posted here.  The link must be visited regularly. Physics 351     Thermal Physics      Spring 2007

  3. Lecture Date Topic Chapter HW due 1 1/17 Introduction, Temperature, Ideal Gas 1 2 1/19 Energy and Work, the 1st Law of Thermodynamics 1 3 1/24 Transport Phenomena 1 1 4 1/26 Macrostates and Microstates 2 5 1/31 Entropy and the 2nd Law of Thermodynamics 2 2 6 2/2 Entropy and Temperature 2 7 2/7 Systems with a “limited” Energy Spectrum 3 3 8 2/9 Thermodynamic Identities 3 9 2/14 Overview Ch. 1-3 1-3 4 4 10 2/16 Midterm 11 2/21 Heat Engines 12 2/23 Toward Absolute Zero, Refrigerators 4 13 2/28 Thermodynamic Potentials 5 5 14 3/2 Equilibrium Between Phases 5 15 3/7 The van der Waals Gas 5 6 16 3/9 Phase Transformations in Binary Mixtures 5 17 3/21 Dilute Solutions 5 5 4-5 7 8 18 3/23 Chemical Equilibrium 19 3/28 Overview Ch. 4-5 20 3/30 Midterm 6 21 4/4 Boltzmann Statistics 6 22 4/6 Canonical Ensembles 6 23 4/11 Statistics of Ideal Quantum Systems 7 9 24 4/13 Degenerate Fermi Gas 7 25 4/18 Blackbody Radiation 7 10 26 4/20 Bose-Einstein Condensation 7 27 4/25 Debye Theory of Solids 7 11 28 4/27 Overview 5/8 Final

  4. Thermal Physics - conceptually, the most difficult subject of the undergraduate physics program. In the undergraduate mechanics and E&M courses, we consider a few objects (or fields) whose positions (amplitudes) can be exactly calculated from the Newton’s (Maxwell’s) equations. This approach works well only for the simplest situations (even a problem of three interacting bodies cannot be solved exactly in classical mechanics). It takes us nowhere when we consider large systems of interacting particles (one needs to solve 1023 differential equations to describe the motion of electrons in 1 cm3 of a metal). The solution is to sacrifice the exact knowledge of microscopic trajectories, and to describe a macro system using some meaningful macroscopic parameters (e.g., the temperature). Not only is the detailed dynamic evolution of the system discarded, but the entire dynamical model.

  5. Thermal Physics Thermodynamics & Statistical Mechanics • Thermodynamics provides a framework of relating the macroscopic properties of a system to one another. It is concerned only with macroscopic quantities and ignores the microscopic variables that characterize individual molecules (both strength and weakness). • Types of the questions that Thermodynamics addresses: • How does a refrigerator work? What is its maximum efficiency? • How much energy do we need to add to a kettle of water to change it to steam? • Statistical Mechanics is the bridge between the microscopic and macroscopic worlds: it links the laws of thermodynamics to the statistical behavior of molecules. • Types of the questions that Statistical Mechanics addresses: • Why are the properties of water different from those of steam, even though water and steam consist of the same type of molecules? • Why does iron lose its magnetism above a certain temperature?

  6. Macroscopic Description is Qualitatively Different! Why do we need to consider macroscopic bodies as a special class of physical objects? Because the behavior of macroscopic bodies differs from that of a single particle in a very fundamental respect: For a single particle: all equations of classical mechanics, electromagnetism, and quantum mechanics are time-reversal invariant (Newton’s second law, F = dp/dt, looks the same if the time t is replaced by –t and the momentum p by –p). For macroscopic objects: the processes are often irreversible (a time-reversed version of such a process never seems to occur). Examples: (a) living things grow old and die, but never get younger, (b) if we drop a basketball onto a floor, it will bounce several times and eventually come to rest - the arrow of time does exist. Conservation of energy does not explain why the time-reversed process does not occur - such a process also would conserve the total energy. Though the total energy is conserved, it has been transferred from one degree of freedom (motion of ball’s center of mass) to many degrees of freedom (associated with the individual molecules of the ball and the floor) – the distribution of energy changes in an irreversible manner.

  7. The Main Idea of the Course Statistical description of a large system of identical (mostly, non-interacting) particles all microstates of an isolated system occur with the same probability, the concepts of multiplicity (configuration space), entropy Equation of state for macrosystems (how macroparameters of the system and the temperature are interrelated) Irreversibility of macro processes, the 2nd Law of Thermodynamics However, most of the undergraduate courses approach this main line of attack with caution, moving in circles. Following the same logic, we’ll start with reviewing some basic facts from the general physics course.

  8. Thermodynamic Systems, Macroscopic Parameters Open systemscan exchange both matter and energy with the environment. Closed systems exchange energy but not matter with the environment. Isolated systemscan exchange neither energy nor matter with the environment. Internal and external macroscopic parameters: temperature, volume, pressure, energy, electromagnetic fields, etc. (average values, fluctuations are ignored). No matter what is the initial state of an isolated system, eventually it will reach the state ofthermodynamic equilibrium (no macroscopic processes, only microscopic motion of molecules).

  9. A very important macro-parameter: Temperature Temperature is a property associated with chaotic motion ofmanyparticles (it would be absurd to refer to the temperature of a single molecule, or to the temperature of many molecules moving with the same speed in the same direction). Introduction of the concept of temperature in thermodynamics is based on the the zeroth law of thermodynamics: A well-defined quantity called temperature exists such that two systems will be in thermal equilibrium if and only if both have the same temperature. The zeroth law implies the existence of some universal property of systems in thermal equilibrium. This law, in conjunction with the concept of entropy, will help us to mathematically define temperature in terms of statistical ideas. Also, if system C is in thermal equilibrium with systems A and B, the latter two systems, when brought in contact, also will be in thermal equilibrium. - this allows us to obtain the temperature of a system without a direct comparison to some standard.

  10. T a Temperature Measurement • Properties of a thermoscope (any device that quantifies temperature): • It should be based on an easily measured macroscopic quantity a (volume, resistance, etc.) of a common macroscopic system. • The function that relates the chosen parameter with temperature, T = f(a), should be monotonic. • The quantity should be measurable over as wide a range of T as possible. • The simplest case – linear dependence T = Aa (e.g., for the ideal gas thermometer, T = PV/NkB). the ideal gas thermometer, T = PV/NkB the resistance thermometer with a semi- conductor sensor, Thermometer athermoscope calibrated to a standard temp. scale

  11. The Absolute (Kelvin) Temp. Scale The absolute (Kelvin) temperature scale is based on fixing T of the triple point for water (a specific T=273.16K and P=611.73Pa where water can coexist in the solid, liquid, and gas phases in equilibrium). T,K 273.16 - for an ideal gas constant-volume thermoscope 0 P PTP PTP – the pressure of the gas in a constant-volume gas thermoscope at T = 273.16 K absolute zero

  12. Our first model of a many-particle system: the Ideal Gas Models of matter:gas models (random motion of particles) lattice models (positions of particles are fixed) Air at normal conditions: ~ 2.71019 molecules in 1 cm3 of air (Pr. 1.10) Size of the molecules ~ (2-3)10-10m, distance between the molecules ~ 310-9m The average speed - 500 m/s The mean free path - 10-7 m (0.1 micron) The number of collisions in 1 second - 5 109 • The ideal gas model - works well at low densities (diluted gases) • all the molecules are identical, N is huge; • the molecules are tiny compared to their average separation (point masses); • the molecules do not interact with each other; • the molecules obey Newton’s laws of motion, their motion is random; • collisions between the molecules and the container walls are elastic. The quantum version of the ideal gas model helps to understand the blackbody radiation, electrons in metals, the low-temperature behavior of crystalline solids, etc.

  13. an equilibrium state P the equation- of-state surface V T The Equation of State of Ideal Gases An equation of state - an equation that relates macroscopic variables (e.g., P, V, and T)for a given substance in thermodynamic equilibrium. In equilibrium ( no macroscopic motion), just a few macroscopic parameters are required to describe the state of a system. Geometrical representation of the equation of state: f (P,V,T) = 0 The ideal gas equation of state: P – pressure [Newtons/m2] V – volume [m3] n – number of moles of gas T – the temperature in Kelvins [K] R – a universal constant

  14. The Ideal Gas Law In terms of the total number of molecules, N = nNA Avogadro’s number the Boltzmann constant kB= R/NA  1.3810-23 J/K (introduced by Planck in 1899) NA 6.0220451023 The equations of state cannot be derived within the frame of thermodynamics: they can be either considered as experimental observations, or “borrowed” from statistical mechanics. Avogadro’s Law: equal volumes of different gases at the same P and T contain the same amount of molecules. isotherms The P-V diagram – the projection of the surface of the equation of state onto the P-V plane.

  15. The van der Waals model of real gases For real gases – both quantitative and qualitative deviations from the ideal gas model U(r) Electric interactions between electro-neutral molecules : r van der Waals attraction (~ r -7) repulsion attraction The van der Waals equation of state for real gases (the only model with interactions that we’ll consider)

  16. Connection between Ktr and T for Ideal Gases ? T of an ideal gas  the kinetic energy of molecules Pressure – the result of collisions between the molecules and walls of the container. Momentum Strategy: Pressure = Force/Area = [p / t ]/Area For each (elastic) collision: px= 2 m vx Intervals between collisions: t = 2 L/vx Piston area A vx Volume = LA L no-relativistic motion For N molecules -

  17. Connection between Ktr and T for Ideal Gases (cont.) Average kinetic energy of the translational motion of molecules: • the temperature of a gas is a direct measure of the • average translational kinetic energy of its molecules! The internal energy U of a monatomic ideal gas is independent of its volume, and depends only on T (U =0 for an isothermal process, T=const). - for an ideal gas of non-relativistic particles, kin. energy  (velocity)2.

  18. Units for Energy, Temperature - the kinetic energy is proportional to the temperature, and the Boltzmann constant kB is the coefficient of proportionality that provides one-to-one correspondence between the units of energy and temperature. Theorists usually assume that kB = 1 - the same units for energy and temperature – which makes a lot of sense. If the temperature is measured in Kelvins, and the energy – in Joules: kB= 1.3810-23 J/K In many sub-fields of physics that deal with microscopic particles (including condensed matter physics, physics of high energies, astrophysics, etc.), a convenient unit for energy is an electron-Volt (the kinetic energy acquired by an electron accelerated by the electrostatic potential difference in 1 V). K= eV 1 eV= 1.6 10-19 J At T = 300K

  19. Pressure of a photon gas - valid for non-relativistic particles - valid also for relativistic particles (m = f(v)) In particular, it applies to the gas of photons that move randomly within some cavity with mirror walls. For photons,  pre-factor 1/3 (instead 2/3) - due to a different relation between E and p for relativistic particles We will use this equation later in the course when we consider the thermal radiation.

  20. Comparison with Experiment - for a point mass with three degrees of freedom Testable prediction: if we put a known dU into a sample of gas, and measure the resulting change dT, we expect to get Conclusion: diatomic and polyatomic gases can store thermal energy in forms other than the translational kinetic energy of the molecules.

  21. Degrees of Freedom Indeed, the polyatomic molecules have more than just three degrees of freedom – they rotate and vibrate. The degrees of freedom of a system are a collection of independent variables required to characterize the system. Thus, for instance, the degrees of freedom of an ideal gas, in a thermodynamic description, are P and V (or a pair of other variables). The independent coordinates that describe the position of a mechanical system in space - e.g., for a rigid body, it is sufficient to specify the coordinates of its center of mass (x, y, z) and three angles (x, y, z). Ix – the moment of inertia for rotations around the x-axis, etc. Polyatomic molecules:6 transl.+rotat. degrees of freedom Diatomic molecules:3 + 2 = 5 transl.+rotat. degrees of freedom (QM: degrees of freedom corresponding to rotations that leave the molecule completely unchanged do not count )

  22. U(x) E4 E3 E2 E1 x Degrees of Freedom (cont.) Plus all vibrational degrees of freedom. The one-dimensional vibrational motion counts as two degrees of freedom (kin. + pot. energies): For a diatomic molecule (e.g., H2), 5 transl.+rotat. degrees of freedom plus 2 vibrational degrees of freedom = total 7 degrees of freedom Among 7 degrees of freedom, only 3 (translational) degrees correspond to a continuous energy spectrum, the other 5 – to a discrete energy spectrum. U(x) x

  23. Equipartition of Energy “Quadratic” degree of freedom – the corresponding energy = f(x2, vx2) [ translational motion, (classical) rotational and vibrational motion, etc. ] Equipartition Theorem: At temperature T, the average energy of any “quadratic”degree of freedom is 1/2kBT. - holds only for a system of particles whose kinetic energy is a quadratic form of x2, vx2 (e.g., the equipartition theorem does not work for photons, E = cp) Piston – a mechanical system with one degree of freedom. Thus, vx M – the mass of a piston, u2 the average u2, where u is the piston’s speed along the x-axis. Thus, the energy that corresponds to the one-dimensional translational motion of a macroscopic system is the same as for a molecule (in this respect, a macrosystem behaves as a giant “molecule”).

  24. U /kBT one mole of H2 “Frozen” degrees of freedom 7/2N Vibration 5/2N Rotation For an ideal gas PV = NkBT U = f/2 NkBT 3/2N Translation 100 1000 10 T, K Example of H2: An energy available to a H2 molecule colliding with a wall at T=300 K: 3/2kBT ~ 40meV. If the difference between energy levels is >> kBT, then a typical collision cannot cause transitions to the higher (excited) states and thus cannot transfer energy to this degree of freedom: it is “frozen out”. U(x) E4 E3 E2 kBT E1 x The rotational energy levels are ~15 meV apart, the difference between vibrational energy levels ~270 meV. Thus, the rotational degrees start contributing to U at T > 200 K, the vibrational degrees of freedom - at T > 3000 K.

  25. For the next lecture: Recall the ideal gas laws, look at the ideal gas problems (some problems are posted on our Web page). A useful book (covers ~ ½ of this course): T.A. Moore, Six Ideas that Shaped Physics” Unit T: Some processes are irreversible.

  26. Problem 1.16 The “exponential” atmosphere Consider a horizontal slab of air whose thickness is dz. If this slab is at rest, the pressure holding it up from below must balance both the pressure from above and the weight of the slab. Use this fact to find an expression for the variation of pressure with altitude, in terms of the density of air, . Assume that the temperature of atmosphere is independent of height, the average molecular mass m. area A P(z+dz)A z+dz z  P(z)A Mg  the density of air:

  27. The root-mean-square speed - not quite the average speed, but close... For H2 molecules (m ~21.710-27 kg ) at 300K: vrms~ 1.84 103 m/s For N2 – vrms= 493m/s (Pr. 1.18), for O2 – vrms= 461m/s This speed is close to the speed of sound in the gas – the sound wave propagates due to the thermal motion of molecules. D(v) hydrogen molecules in the “tail” escape the Earth gravitation field vrms v

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