DAY-by-DAY page

Heron's steam engine

Ph224, Spring 2024
THERMAL PHYSICS
Marcel den Nijs

red line: where we are in class

above the green line:
currently posted lecture notes

below the green line:
tentative future outline

01. INTRODUCTION

(ch1 posted lecture notes )
  • Dawn of thermodynamics
    steam engines, steam ships, steam mills
  • Power in the pre-steam era
    man power, horse power, ox power, water power, wind power
  • Dawn of statistical mechanics
    molecular made-up of nature and when it became directly accessible
  • Laws of nature at different length and time scales
    information overload and the irrelevancy of most microscopic details
  • Friction
    absence at the microscopic level
  • Time irreversibility
    Need for a statistical approach dealing with the unknown

02. MACROSCOPIC MECHANICS OF MANY PARTICLES

(ch2 posted lecture notes; Schroeder (S): chapter 1; Mazur (M): sections 18.1, 18.5 and 18.6); RHW 14.2-4; RHK:ch17)
  • Density:
    Avogadro's number
    Orders of magnitude of densities in air, liquids, and solids
  • Internal Energy:
    order of magnitude of the amount of kinetic energy stored in one cubic meter of gas
    mean free path
  • Pressure:
    tensor character of pressure, stress and strains in solids
    air pressure, Magdeburg hemispheres
  • Pascal's law:
    hydrostatic pressure
    pressure depth variation in water
    atmospheric pressure law, Pascal length
  • Equations of state:
    macroscopic conservation laws and the origin of the p(N,V,U) relation
  • Mechanical Energy Transfer and Quasi-Static Processes:
    Draw bridge and gas container with piston examples.
    Mechanical equilibrium when pressures are equal.
  • Adiabatic quasi-static expansion of a gas:
    Reversibility of the process
    The gas cools on expansion
  • Resume and Outlook:
    From random energy to useful energy
    State variables versus transaction variables
    Work versus potential energy
    Equilibrium redefinition of pressure

03. HEAT, TEMPERATURE, AND THE FIRST LAW

(ch3 posted lecture notes; S: ch1; M: 20.1-20.3 ; RHW:18.1-18.4 ; RHK: ch22)
  • Macroscopic energy exchange versus heat exchange
    Heat is not a property of an object.
    Heat it is a transaction analogous to mechanical work.
    Heat represents energy transfer escaping our macroscopic mechanical variables.
    Need a set of novel non-mechanical state variables: temperature and entropy
  • Thermal Equilibrium
    macroscopic definition of temperature:
    "two objects are in thermal equilibrium when their temperatures are equal."
    two equations of state, U(N,V,T) and p(N,V,T)
  • Linear and volume expansion coefficients of solids, liquids, and gases:
    liquids in bottles, plumbing and other examples
  • Calorimetry:
    heat, heat capacity, specific heat, examples
  • Water as cooland
    silver smiths and silver spoons specific heat of water compared to that of other liquids and solids
  • First law of thermodynamics, unification of calorimetry and mechanics:
    Mechanical equivalent of heat
    orders of magnitude of thermal versus typical macroscopic mechanical energies
    Joule's experiment.

04. KINETIC GAS THEORY AND STATISTICS

(ch4 posted lecture notes; S ch1 + ch6.4 )
  • Empirical Boyle-Gay-Lussac laws for dilute gasses
  • Kronig-Claussius-Joule derivation of U= 3/2 pV for a dilute gas
    molecular speeds in air implied by this relation
  • Primer to probability distribution functions and averages
    dice games and exam scores as examples
  • Maxwell-Boltzmann velocity distribution
    First example of the Boltzmann factor
    average molecular speeds and velocities
    speed fluctuations
    section 6.4 in Schroeder
  • Atmospheric law as a probability distribution
  • Brownian motion and the size of molecules:
    Determining the size of the molecules
    Perin's atmospheric law type colloidal pollen suspension experiment
  • Derivation of U=3/2 pV for a gas with a realistic mean free path
    Kinetic gas interpretation of temperature
  • Equipartition of energy:
    gas mixtures
    rotations and vibrations, monoatomic versus diatomic molecules.
    Law of Dulong and Petit for solids

05. THE STATISTICAL DEFINITION OF ENTROPY

(ch5 posted lecture notes)
  • Micro states
    representing micro states Γ
    information loss during coarse graining
    stochastic descriptions of nature
  • Boltzmann-Gibbs-Shannon statistical definition of entropy (information, disorder)
    information carried in probability distributions P(Γ)
    Entropy is equal to S= - kb < log P>
    minimum and maximum entropy
    entropy addition rules
  • Boltzmann's maximum entropy principle for closed systems
    Entropy increases in time in stochastic processes in closed systems
    All micro states Γ are equally likely in thermal equilibrium closed systems
    Seq= kb log (Γ)

06. ENTROPY OF N-LEVEL SYSTEMS

(ch6 posted lecture notes; S ch2+ch3+ appendices B2-B4 )
  • Stirling's formula for N!
    Gamma-function
    Integrals with extreme narrowing integrants
    maximum of the integrant approximations
  • 2-level systems
    paramagnets
    fermions
    receptors in biophysics
    stochastic card games
  • Micro states of 2-level systems at large N
    counting the number of micro states
    Combinatorics and the Binomial distribution, Pascal triangle Entropy per site
    Narrowing of the micro state distribution with increasing N
    Mixing entropy
    Entropy with and without a reservoir
  • Second Law of thermodynamics
    2-level system in contact with open and closing particle reservoir
    merging 2-level systems
    identification of partial derivatives entropy with temperature and chemical potentials
    entropic definition of thermal and chemical equilibrium
    from the first-law to the second law of thermodynamics
  • ∞-level systems
    bosons, phonons
    Bose-Einstein condensation, superfluids and superconductivity
    counting the number of micro states
    Entropy per level
  • Passive and active heat and particle reservoirs
    how large systems remain at constant temperature chemical potential
    reservoirs entropy changes linearly with heat and particles exchanged
    inverting closed system equations of state
    Fermi-Dirac and Bose-Einstein distributions

07. STATISTICAL ENTROPY OF A MONO-ATOMIC DILUTE GAS

(ch7 posted lecture notes; S ch2+ch3+ appendices B1-B4 )
  • Micro states in Newtonian mechanics
  • Entropy of an dilute classical mechanical gas
    The volume of an hyper sphere in 3N dimensional space using Gaussian integrals
    Entropy per particle of an ideal gas
    partial derivatives of the entropy and thermal and mechanical equilibrium
    form the first law to the second law of thermodynamics
  • Gibbs Paradox
    Indistinguishability in classical statistical mechanics
    Scaling with system size, mixing entropy and the Gibbs paradox
  • van der Waals gas
    entropy in the presence inter particle interactions
    van der Waals equations of state.

08. DILUTE GAS THERMODYNAMIC PROCESSES.

(ch8 posted lecture notes; S: ch1; M: 20.5-20.7; RHW: 18.5 )
  • The equations of state pVT- and UVT-surfaces :
    ideal gases versus other substances
    moving along the surface quasi-statically
    Representing specific processes along this surface in pV, pT, and VT projection
  • Isobaric versus isochoric specific heat of dilute gasses:
  • work and area in pV diagrams
  • Isothermal compression:
    who does work, and where does the energy come from?
    isothermal compressibility and bulk modulus
  • Adiabatic compression of a dilute gas revisited
    adiabatic compressibility and bulk modulus
    the velocity of sound in air
  • thermal expansion in Galileo's thermometer:
    as example of a less common path along the pVT-surface.

09. ENGINES, REFRIGERATORS, AND THE SECOND LAW.

(ch9 posted lecture notes; S: ch4 +parts ch3 )
  • Reversible and irreversible thermodynamic processes:
    Cp-Cv cycle (isobaric-isochoric heating and cooling)
    efficiency as engine
    time reversion invariance of the microscopic laws of mechanics
  • Carnot cycle
    run forward and backward
    efficiency as engine
    performance factors as refrigerator and heat pump
    reversibility
  • Kelvin-Planck and Claussius formulations of the second law:
  • Carnot formulation of the second law:
    equivalence with Kelvin-Planck and Claussus formulations
  • Entropy in macroscopic thermodynamics:
    TdS=dQ
    entropy as a thermodynamic potential function, S(V,N,U)
  • The thermodynamic temperature scale:
    the Kelvin temperature scale in terms of Carnot efficiency.
    the third law of thermodynamics
  • Stirling engines:
    efficiency as engine and refrigerator
    irreversibility and the role of the regulator
  • Other engines
    Combustion engines, Otto and Diesel cycles
    Steam engines:
    pumps for mines, pumps to keep Holland dry (Cruquius), trains, and steam ships
    electric engines
  • Examples of entropy changes in thermodynamic processes:
    isochoric an isobaric heating of coffee and gases
    isotherms and adiabatics
  • enthalpy, H=U+pV
  • Helmholtz, F=U-TS and Gibbs, G=U-TS+pV free energies
    how much energy does it take to create and place a rabbit?

10. BOLTZMANN STATISTICS

(ch10 posted lecture notes; S: ch6 )
  • Boltzmann statistics
    Derivation of the Boltzmann factor probability distribution
    Interpretation of the logarithm of the partition function with free energy
  • Applications of Boltzmann statistics
    revisit classical ideal gas, Fermi and Bose gases


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