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Statistical mechanics, JNH

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Microstates, configurations, weight and entropy

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  • A microstate assigns an energy state to each molecule in a sample. Microstates are usually unknowable as molecules are indistinguishable.
  • A configuration assigns a number of molecules to each energy state. Configurations are knowable. Different microstates can be represented by the same configuration.
  • The weight, W, of a configuration is the number of microstates it represents:
  • Configurations with lower total energy are more likely
  • Of the configurations with the lowest total energy, the one with the highest entropy is most likely

Boltzmann distributions

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  • The configuration with maximum weight (and thus maximum entropy) satisfies the following relation (the Boltzmann distribution):

The Boltzmann distribution of energy levels for molecules in a sample at thermal equilibrium is a manifestation of entropy — more microstates means more disorder, so the most likely configuration is the one with the largest W.

Partition function

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  • The denominator of the Boltzmann distribution is called the partition function and is given the symbol q:
  • Degenerate states (two or more states with the same energy) can be described as a level with a degeneracy gi
  • q can therefore be expressed in terms of levels and degeneracies, rather than states:
  • The Boltzmann distribution can also be expressed in terms of levels and degeneracies:
  • The partition function measures the total number of levels occupied at a given temperature T

Reference energy

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  • It is conventional in statistical mechanics to define the lowest energy state or level of a sample as zero, i.e. ε0 = 0
  • This means statistical mechanics differs in convention from some other fields
  • For example, the vibrational energy of a harmonic oscillator is defined as:
in spectroscopy, but
in statistical mechanics
  • A different choice of reference energy leads to a different value of q, but q is not directly observed
  • The observable quantities statistical mechanics predicts, such as the Boltzmann distribution, are not affected by the choice of reference energy

Vibrational partition function

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Internal energy

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  • The internal energy, U, of a system is related to the partition function
  • The internal energy above that at absolute zero (0 K), UU(0), is the sum of the energies of all the molecules in a system
  • Combining
and
gives
  • You can get away without having to evaluate this tedious summation by using a derivative of the partition function:
  • This means the internal energy can be expressed more simply as
  • Applying this to find the vibrational internal energy gives the following: