User:Benjah-bmm27/degree/3/JNH1

Statistical mechanics, JNH

• Statistical mechanics
  • Canonical ensemble
  • Boltzmann factor
  • Boltzmann distribution
  • Root mean square speed
  • Maxwell speed distribution
  • Maxwell–Boltzmann distribution
  • Kinetic theory
  • Collision theory
  • Degrees of freedom (physics and chemistry)
• Partition function (statistical mechanics)

Microstates, configurations, weight and entropy

• 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:

$W={\frac {N!}{n_{1}!n_{2}!n_{3}!...n_{n}!}}\!$

• The entropy of a configuration is a function of its weight, according to Boltzmann's entropy formula:

$S=k_{B}\ln W\!$

• 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

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

${\frac {n_{i}}{N}}={\frac {e^{-\beta \epsilon _{i}}}{\sum _{i}e^{-\beta \epsilon _{i}}}}$

• β is called thermodynamic beta and is an "inverse temperature":

$\beta \equiv {\frac {1}{k_{B}T}}$

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

• The denominator of the Boltzmann distribution is called the partition function and is given the symbol q:

$q=\sum _{i}^{\mbox{states}}e^{-\beta \epsilon _{i}}$

• Degenerate states (two or more states with the same energy) can be described as a level with a degeneracy g_i
• q can therefore be expressed in terms of levels and degeneracies, rather than states:

$q=\sum _{i}^{\mbox{levels}}g_{i}e^{-\beta \epsilon _{i}}$

• The Boltzmann distribution can also be expressed in terms of levels and degeneracies:

${\frac {n_{i}}{N}}={\frac {e^{-\beta \epsilon _{i}}}{\sum _{i}^{\mbox{levels}}g_{i}e^{-\beta \epsilon _{i}}}}$

• The partition function measures the total number of levels occupied at a given temperature T

Reference energy

• 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:

$\epsilon _{v}=h\nu \left(v+{\frac {1}{2}}\right)$ in spectroscopy, but

$\epsilon _{v}=h\nu v$ 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

• The Maclaurin series for 1/(1−x), a standard result from A-level maths:

$\sum _{v=0}^{\infty }x^{n}=1+x+x^{2}+x^{3}+\cdots \!={\frac {1}{1-x}}$

• The expression E_vib = hνv means the vibrational partition function, q_vib can be expressed as a Maclaurin series:

$q_{\mbox{vib}}=\sum _{v=0}^{\infty }\exp {\left({\frac {-\epsilon _{v}}{k_{B}T}}\right)}=\sum _{v=0}^{\infty }\exp {\left({\frac {-h\nu v}{k_{B}T}}\right)}=\sum _{v=0}^{\infty }{\exp {\left({\frac {-h\nu }{k_{B}T}}\right)}}^{v}$

$q_{\mbox{vib}}={\exp {\left({\frac {-h\nu }{k_{B}T}}\right)}}^{0}+{\exp {\left({\frac {-h\nu }{k_{B}T}}\right)}}^{1}+{\exp {\left({\frac {-h\nu }{k_{B}T}}\right)}}^{2}+{\exp {\left({\frac {-h\nu }{k_{B}T}}\right)}}^{3}+...={\frac {1}{1-\exp {\left({\frac {-h\nu }{k_{B}T}}\right)}}}$

• This is sometimes expressed in terms of vibrational temperature, θ = hν / k_B:

$q_{\mbox{vib}}=\left(1-\exp {\left({\frac {-\theta }{T}}\right)}\right)^{-1}$

Internal energy

• 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

$U-U(0)=\sum _{i}{n_{i}\epsilon _{i}}$

and

$n_{i}={\frac {N}{q}}\exp {\left({\frac {-\epsilon _{i}}{k_{B}T}}\right)}$

gives

$U-U(0)={\frac {N}{q}}\sum _{i}{\epsilon _{i}\exp {\left({\frac {-\epsilon _{i}}{k_{B}T}}\right)}}$

• You can get away without having to evaluate this tedious summation by using a derivative of the partition function:

${\frac {\partial q}{\partial T}}={\frac {1}{k_{B}T^{2}}}\sum _{i}{\epsilon _{i}\exp {\left({\frac {-\epsilon _{i}}{k_{B}T}}\right)}}$

• This means the internal energy can be expressed more simply as

$U-U(0)={\frac {Nk_{B}T^{2}}{q}}{\frac {\partial q}{\partial T}}$

• Applying this to find the vibrational internal energy gives the following:

$U_{\mbox{vib}}-U_{\mbox{vib}}(0)=Nh\nu {\frac {\exp {\left({\frac {-h\nu }{k_{B}T}}\right)}}{1-\exp {\left({\frac {-h\nu }{k_{B}T}}\right)}}}$