Chapter 16 - Statistical Thermodynamics 1: The Concepts

• Statistical thermodynamics - The link between individual molecular properties and bulk thermodynamic properties.

The distribution of molecular states

• Population - The average number of molecules that occupy it.
• Principle of equal a priori probabilities - The assumption that all possibilities for the distribution of energy are equally probable.

16.1 Configurations and weights

• Weight of the configuration - The number of ways a general configuration can be achieved.

  

• Stirling's approximation

  

• Boltzmann distribution

  

16.2 The molecular partition function

• Molecular partition function

  

• Common methods used to reach very low temperatures

  • Optical trapping - Where atoms in the gas phase are cooled by inelastic collisions with photons from intense laser beams.
  • Adiabatic demagnetization - Based on that, in the absence of a magnetic field, the unpaired electrons of a paramagnetic material are orientated at random.

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• Thermal wavelength - Decreases with increasing mass and temperature.

The internal energy and the entropy

16.3 The internal energy

• The total energy of the system relative to the energy of the lowest state

  

16.4 The statistical entropy

• Boltzmann formula for the entropy

  

The canonical partition function

16.5 The canonical ensemble

• Canonical ensemble - The imaginary collection of replications of the actual system with a common temperature.

• Microcanonical ensemble - Where the condition of constant temperature is replaced by the requirement that all the systems should have exactly the same energy.

• Grand canonical ensemble - The volume and temperature of each system are the same, but they are open, which means that matter can be imagined as able to pass between the systems; the composition of each one may fluctuate, but the chemical potential is the same in each system.

• Canonical distribution

  

• Canonical partition function (Q) - A function of the temperature.

16.6 The thermodynamic information in the partition function

• The total weight (W) of a configuration of the ensemble - The product of the average weight W of each member of the ensemble.

  

16.7 Independent molecules

• For distinguishable independent molecules

  

• For indistinguishable independent molecules

  

• Sackur-Tetrode equation - It implies that the molar entropy of a perfect gas of high molar mass is greater than one of low molar mass under the same conditions. Used for the entropy of a monoatomic gas.

  

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