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":
 * $$\beta \equiv \frac{1}{k_BT}$$

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 gi
 * 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. &epsilon;0 = 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&minus;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 Evib = hνv means the vibrational partition function, qvib 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, &theta; = h&nu; / kB:
 * $$q_{\mbox{vib}} = \left ( 1- \exp{ \left ( \frac{- \theta }{T} \right ) } \right )^{-1}$$

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:
 * $$U_{\mbox{vib}} - U_{\mbox{vib}}(0) = N h \nu \frac{ \exp{ \left ( \frac{- h \nu }{k_B T} \right ) } }{ 1 - \exp{ \left ( \frac{- h \nu }{k_B T} \right ) } }$$