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Statistical mechanics, JNH
[edit]Microstates, configurations, weight and entropy
[edit]- 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:
- The entropy of a configuration is a function of its weight, according to Boltzmann's entropy formula:
- 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
[edit]- The configuration with maximum weight (and thus maximum entropy) satisfies the following relation (the Boltzmann distribution):
- β is called thermodynamic beta and is an "inverse temperature":
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
[edit]- 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
[edit]- 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
[edit]- The Maclaurin series for 1/(1−x), a standard result from A-level maths:
- The expression Evib = hνv means the vibrational partition function, qvib can be expressed as a Maclaurin series:
- This is sometimes expressed in terms of vibrational temperature, θ = hν / kB:
Internal energy
[edit]- The internal energy, U, of a system is related to the partition function
- The internal energy above that at absolute zero (0 K), U − U(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: