In statistical thermodynamics, the behavior of a macroscopic system is explained in terms of the arrangement of microscopic particles. A large number of particles can exist in various energy levels, and each possible arrangement contributes to the thermodynamic properties of the system.
A quantum state represents a specific microscopic condition of a particle characterized by quantum numbers such as energy, momentum, and spin. When many particles are present, the complete specification of all particles forms a microscopic configuration known as a complexion or microstate.
A macroscopic system may contain a huge number of microstates corresponding to the same observable properties such as pressure, temperature, and energy. These observable properties define the macrostate of the system.
If \( N \) particles are distributed among different energy levels, then the number of possible complexions gives the thermodynamic probability \( W \).
For distinguishable particles distributed among energy levels:
\[ W = \frac{N!}{N_1!N_2!N_3!\cdots} \]
where:
The greater the value of thermodynamic probability, the greater is the likelihood of the system existing in that state.
The combinatory rule is used to determine the number of possible arrangements of particles among available energy states. Suppose \( N \) particles are distributed into different cells.
The first particle can be arranged in \( N \) ways, the second in \( (N-1) \) ways, and so on. Therefore:
\[ N! = N(N-1)(N-2)\cdots 1 \]
If particles are grouped into sets containing:
\[ N_1, N_2, N_3, \ldots \]
then permutations within the same group are indistinguishable. Hence the total number of distinct arrangements becomes:
\[ W = \frac{N!}{N_1!N_2!N_3!\cdots} \]
For energy levels having degeneracy \( g_i \), the expression modifies to:
\[ W = \prod_i \frac{g_i^{N_i}}{N_i!} \]
This relation forms the basis of Maxwell–Boltzmann statistics.
Consider a system having fixed total energy \( E \) and total number of particles \( N \). If particles occupy energy levels \( \epsilon_i \), then:
\[ \sum_i N_i = N \]
and
\[ \sum_i N_i \epsilon_i = E \]
The most probable distribution is obtained by maximizing thermodynamic probability \( W \). Using Stirling’s approximation:
\[ \ln N! \approx N \ln N - N \]
Therefore:
\[ \ln W = \ln N! - \sum_i \ln N_i! \]
Substituting Stirling approximation:
\[ \ln W = N\ln N - N - \sum_i (N_i \ln N_i - N_i) \]
Since:
\[ \sum_i N_i = N \]
we obtain:
\[ \ln W = N\ln N - \sum_i N_i \ln N_i \]
To maximize \( W \), the method of Lagrange multipliers is used:
\[ \delta \left( \ln W + \alpha \sum_i N_i + \beta \sum_i N_i\epsilon_i \right)=0 \]
Differentiating:
\[ -\ln N_i -1 + \alpha + \beta \epsilon_i =0 \]
Hence:
\[ N_i = e^{\alpha -1} e^{-\beta \epsilon_i} \]
or
\[ N_i = A e^{-\beta \epsilon_i} \]
This is the Maxwell–Boltzmann distribution law.
Degeneracy refers to the number of quantum states having the same energy. If an energy level \( \epsilon_i \) possesses \( g_i \) states, then the level is said to be \( g_i \)-fold degenerate.
Including degeneracy, the Maxwell–Boltzmann distribution becomes:
\[ N_i = g_i A e^{-\beta \epsilon_i} \]
where:
Degeneracy increases the probability of occupation because more quantum states are available at the same energy.
The probability of a macrostate is proportional to the number of microstates corresponding to it. Thus:
\[ P \propto W \]
Among all possible distributions, the one having maximum \( W \) is called the most probable distribution.
For a large number of particles, almost all particles remain distributed according to the most probable arrangement. Hence macroscopic properties remain constant.
Boltzmann related entropy to thermodynamic probability by the famous equation:
\[ S = k \ln W \]
where:
Entropy is therefore a measure of molecular disorder or randomness.
In classical mechanics, particles are assumed distinguishable. However, identical particles such as electrons, photons, and atoms are fundamentally indistinguishable in quantum mechanics.
If indistinguishability is ignored, statistical calculations produce excessively large values of thermodynamic probability. To correct this, Gibbs introduced the correction factor:
\[ W = \frac{W'}{N!} \]
where:
Indistinguishability leads to quantum statistics such as:
For dilute gases at high temperature, these reduce to Maxwell–Boltzmann statistics.
Statistical thermodynamics establishes a connection between microscopic particle behavior and macroscopic thermodynamic properties. Concepts such as quantum states, thermodynamic probability, degeneracy, indistinguishability, and most probable distribution form the theoretical basis of Maxwell–Boltzmann statistics and modern thermodynamics.
```htmlIn statistical thermodynamics, the behavior of a macroscopic system is understood in terms of the distribution of particles among various energy states. A quantum state represents a definite arrangement of a particle characterized by quantum numbers and energy. The complete description of all particles in a system constitutes a microscopic description of the system.
A microstate is a particular arrangement of particles among available energy levels. A macrostate is defined by macroscopic properties such as pressure, volume, and total energy without specifying the exact arrangement of particles.
For example, consider two particles distributed between two energy levels:
Possible microstates are:
The macrostate only specifies the number of particles in each energy level, while the microstate specifies the identity and arrangement of each particle.
The total number of microstates corresponding to a macrostate is called the thermodynamic probability or complexion and is represented by \( W \).
For distinguishable particles:
\[ W = \frac{N!}{N_1!N_2!N_3!\dots} \]
where:
The macrostate with the largest thermodynamic probability is the most probable equilibrium state of the system.
The combinatory rule in statistical mechanics determines the number of possible ways in which particles may be distributed among available energy states.
Suppose:
If all particles are distinguishable, the first particle may be placed in any position, followed by the second particle and so on. Therefore, total permutations are:
\[ N! \]
However, interchange of particles within the same energy level does not produce a new arrangement. Therefore, correction factors must be introduced:
\[ N_1!, N_2!, N_3!, \dots \]
Thus, the total number of distinguishable arrangements becomes:
\[ W = \frac{N!}{N_1!N_2!N_3!\dots} \]
Taking logarithm:
\[ \ln W = \ln N! - \sum \ln N_i! \]
Using Stirling approximation:
\[ \ln N! \approx N\ln N - N \]
Therefore:
\[ \ln W = N\ln N - N - \sum (N_i \ln N_i - N_i) \]
Since:
\[ \sum N_i = N \]
Hence:
\[ \ln W = N\ln N - \sum N_i \ln N_i \]
This expression forms the basis for deriving equilibrium distribution laws in statistical thermodynamics.
A thermodynamic system with fixed total energy is called an isolated system. The particles are distributed among various energy levels while satisfying the conservation conditions:
\[ \sum N_i = N \]
and
\[ \sum N_i \epsilon_i = E \]
where:
For equilibrium, the thermodynamic probability must be maximum.
Thus:
\[ \delta (\ln W) = 0 \]
Using:
\[ \ln W = N\ln N - \sum N_i \ln N_i \]
Applying the method of Lagrange multipliers:
\[ \delta \left[ \ln W -\alpha \sum N_i -\beta \sum N_i \epsilon_i \right]=0 \]
Differentiating:
\[ -\ln N_i -1 -\alpha -\beta \epsilon_i =0 \]
Hence:
\[ \ln N_i = -1-\alpha-\beta \epsilon_i \]
Taking exponential:
\[ N_i = Ae^{-\beta \epsilon_i} \]
where:
\[ A=e^{-1-\alpha} \]
This is the Maxwell-Boltzmann distribution law which describes equilibrium distribution of particles among energy states.
In many systems, several quantum states possess the same energy. Such states are called degenerate states and the number of states having identical energy is called degeneracy.
If:
\[ g_i \]
is the degeneracy of the \( i^{th} \) energy level, then each level contains \( g_i \) sublevels.
The probability of occupation increases with degeneracy because more states are available at the same energy.
Hence the Maxwell-Boltzmann distribution becomes:
\[ N_i = g_i A e^{-\beta \epsilon_i} \]
where:
The degeneracy term increases the population of highly degenerate energy levels even if their energy is relatively higher.
The partition function including degeneracy becomes:
\[ Q = \sum g_i e^{-\beta \epsilon_i} \]
The partition function plays a central role in statistical thermodynamics because all thermodynamic properties can be derived from it.
The probability of a macrostate depends upon the number of microstates associated with it. The macrostate possessing maximum thermodynamic probability corresponds to equilibrium and is known as the most probable distribution.
For a very large number of particles:
\[ W_{\text{max}} \gg W_{\text{other}} \]
Hence almost all systems remain in the most probable state.
Classical statistics assumes particles are distinguishable, but real microscopic particles such as electrons, photons, and atoms are fundamentally indistinguishable.
For distinguishable particles:
\[ W = \frac{N!}{N_1!N_2!N_3!\dots} \]
For indistinguishable particles, interchange of particles does not create a new microstate. This significantly reduces the number of possible arrangements.
Indistinguishability leads to the development of:
Maxwell-Boltzmann statistics is valid only when:
The probability of occupation according to Maxwell-Boltzmann statistics is:
\[ \frac{N_i}{N} = \frac{g_i e^{-\beta \epsilon_i}}{Q} \]
where:
\[ Q = \sum g_i e^{-\beta \epsilon_i} \]
This equation gives the fraction of particles occupying a particular energy level at thermal equilibrium.
Statistical thermodynamics connects the microscopic motion of particles with the observable macroscopic properties of matter.```
Hydrogen gas exhibits unusual heat capacity behavior at low temperatures due to the quantization of rotational energy levels. The rotational energy of a diatomic molecule is given by:
\[ E_J = \frac{h^2}{8\pi^2 I}J(J+1) \]
where \(J\) is the rotational quantum number and \(I\) is the moment of inertia.
The rotational partition function is:
\[ q_{rot} = \sum_{J=0}^{\infty}(2J+1)e^{-E_J/kT} \]
At low temperatures only lower rotational states are populated. Hydrogen exists in two nuclear spin modifications:
Thus the partition functions become:
\[ q_{para} = \sum_{J=0,2,4}^{\infty}(2J+1)e^{-E_J/kT} \]
\[ q_{ortho} = 3\sum_{J=1,3,5}^{\infty}(2J+1)e^{-E_J/kT} \]
The total internal energy is:
\[ U = NkT^2\left(\frac{\partial \ln q_{rot}}{\partial T}\right) \]
The heat capacity is obtained as:
\[ C_V = \left(\frac{\partial U}{\partial T}\right)_V \]
At very low temperature, molecules remain mostly in the ground rotational state and rotational contribution to heat capacity becomes very small. As temperature increases, higher rotational levels become populated and heat capacity rises.
The quantization of rotational motion explains why hydrogen behaves differently from classical predictions at low temperatures.
Einstein assumed that every atom in a crystal vibrates independently with the same frequency. Each atom behaves like a quantum harmonic oscillator.
The vibrational energy levels are:
\[ E_n = \left(n+\frac{1}{2}\right)h\nu \]
The vibrational partition function for one oscillator is:
\[ q = \sum_{n=0}^{\infty} e^{-nh\nu/kT} \]
This is a geometric progression:
\[ q = \frac{1}{1-e^{-h\nu/kT}} \]
For one mole of crystal containing \(3N\) oscillators:
\[ Q = q^{3N} \]
The internal energy becomes:
\[ U = 3N\frac{h\nu}{e^{h\nu/kT}-1} \]
Defining Einstein temperature:
\[ \theta_E = \frac{h\nu}{k} \]
the internal energy becomes:
\[ U = 3Nk\frac{\theta_E}{e^{\theta_E/T}-1} \]
Differentiating with respect to temperature:
\[ C_V = \left(\frac{\partial U}{\partial T}\right)_V \]
Therefore:
\[ C_V = 3Nk\left(\frac{\theta_E}{T}\right)^2 \frac{e^{\theta_E/T}} {(e^{\theta_E/T}-1)^2} \]
At high temperature:
\[ C_V \rightarrow 3Nk \]
which agrees with Dulong–Petit law.
At low temperature the heat capacity decreases exponentially.
Einstein’s model successfully introduced quantum theory into solid-state heat capacity calculations.
Debye improved Einstein’s theory by considering lattice vibrations as elastic waves with a continuous distribution of frequencies.
The number of vibrational modes between frequency \(\nu\) and \(\nu+d\nu\) is:
\[ g(\nu)d\nu = \frac{9N\nu^2}{\nu_D^3}d\nu \]
where \(\nu_D\) is the Debye frequency.
The total energy is:
\[ U = \int_0^{\nu_D} \frac{h\nu}{e^{h\nu/kT}-1}g(\nu)d\nu \]
Substituting density of states:
\[ U = \frac{9Nh}{\nu_D^3} \int_0^{\nu_D} \frac{\nu^3 d\nu}{e^{h\nu/kT}-1} \]
Introducing:
\[ x=\frac{h\nu}{kT} \]
and:
\[ \theta_D = \frac{h\nu_D}{k} \]
the expression becomes:
\[ U = 9NkT \left(\frac{T}{\theta_D}\right)^3 \int_0^{\theta_D/T} \frac{x^3dx}{e^x-1} \]
The heat capacity is:
\[ C_V = 9Nk\left(\frac{T}{\theta_D}\right)^3 \int_0^{\theta_D/T} \frac{x^4e^xdx}{(e^x-1)^2} \]
At very low temperatures:
\[ C_V \propto T^3 \]
This successfully explains experimental observations.
At high temperatures:
\[ C_V \rightarrow 3Nk \]
Debye theory remains one of the most successful applications of quantum statistics to solids.
Einstein assumed all oscillators possess identical frequency, whereas Debye considered a continuous range of frequencies.
Einstein heat capacity equation:
\[ C_V = 3Nk\left(\frac{\theta_E}{T}\right)^2 \frac{e^{\theta_E/T}} {(e^{\theta_E/T}-1)^2} \]
Debye heat capacity equation:
\[ C_V = 9Nk\left(\frac{T}{\theta_D}\right)^3 \int_0^{\theta_D/T} \frac{x^4e^xdx}{(e^x-1)^2} \]
At high temperatures both theories predict:
\[ C_V = 3Nk \]
However, at low temperatures:
Debye theory agrees much better with experimental results.
Consider the gaseous reaction:
\[ aA+bB \rightleftharpoons cC+dD \]
The equilibrium constant is related to molecular partition functions by statistical thermodynamics.
The molecular partition function contains translational, rotational, vibrational, and electronic contributions:
\[ q = q_{trans}q_{rot}q_{vib}q_{elec} \]
The equilibrium constant is:
\[ K = \frac{q_C^c q_D^d} {q_A^a q_B^b} e^{-\Delta E_0/RT} \]
where:
The translational partition function contributes strongly to pressure dependence.
Rotational partition function depends upon molecular symmetry and moment of inertia.
Vibrational partition function contributes significantly at high temperature.
Electronic partition functions are important when low-lying electronic states are accessible.
The statistical mechanical treatment provides microscopic interpretation of chemical equilibrium.
Partition functions bridge microscopic molecular behavior with macroscopic thermodynamic equilibrium.
Bose–Einstein statistics is applicable to indistinguishable particles having integral spin. Such particles are known as bosons. In this statistics, any number of particles may occupy the same quantum state. Examples include photons and helium-4 atoms.
Consider a system containing \(N\) indistinguishable bosons distributed among various energy levels. Let the \(i^{th}\) energy level possess energy \(\epsilon_i\) and degeneracy \(g_i\). If \(n_i\) particles occupy this level, then the number of ways of arranging the particles is
\[ W_i = \frac{(n_i + g_i -1)!}{n_i!(g_i-1)!} \]
The total thermodynamic probability becomes
\[ W = \prod_i \frac{(n_i + g_i -1)!}{n_i!(g_i-1)!} \]
Using Stirling’s approximation,
\[ \ln N! \approx N\ln N - N \]
and maximizing \(\ln W\) under the constraints
\[ \sum_i n_i = N \]
and
\[ \sum_i n_i \epsilon_i = E \]
with the method of Lagrange multipliers,
\[ \delta \left[\ln W - \alpha \sum_i n_i - \beta \sum_i n_i\epsilon_i \right]=0 \]
one obtains the Bose–Einstein distribution law:
\[ n_i = \frac{g_i}{e^{\alpha+\beta \epsilon_i}-1} \]
Since
\[ \alpha = -\frac{\mu}{kT}, \qquad \beta = \frac{1}{kT} \]
the final form becomes
\[ n_i = \frac{g_i}{e^{(\epsilon_i-\mu)/kT}-1} \]
Here, \(\mu\) represents the chemical potential. For photons, \(\mu = 0\), therefore
\[ n_i = \frac{g_i}{e^{\epsilon_i/kT}-1} \]
This expression successfully explains black body radiation and the Planck radiation law.
Quantum particles behaving collectively give rise to remarkable macroscopic phenomena.
Fermi–Dirac statistics applies to indistinguishable particles having half-integral spin. These particles are called fermions. Such particles obey the Pauli exclusion principle, according to which no two particles can occupy the same quantum state simultaneously.
For a system of fermions distributed among energy levels, the thermodynamic probability is
\[ W_i = \frac{g_i!}{n_i!(g_i-n_i)!} \]
Hence,
\[ W = \prod_i \frac{g_i!}{n_i!(g_i-n_i)!} \]
Applying Stirling’s approximation and maximizing \(\ln W\) subject to the particle and energy constraints,
\[ \sum_i n_i = N \]
and
\[ \sum_i n_i \epsilon_i = E \]
gives
\[ n_i = \frac{g_i}{e^{\alpha+\beta \epsilon_i}+1} \]
Substituting
\[ \alpha = -\frac{\mu}{kT} \]
and
\[ \beta = \frac{1}{kT} \]
the Fermi–Dirac distribution function becomes
\[ n_i = \frac{g_i}{e^{(\epsilon_i-\mu)/kT}+1} \]
At absolute zero temperature, all energy states below the Fermi energy are completely occupied while those above it remain empty.
The highest occupied energy level at \(0\,K\) is called the Fermi energy.
For an electron gas in metals,
\[ E_F = \frac{h^2}{8m}\left(\frac{3N}{\pi V}\right)^{2/3} \]
The electron gas model successfully explains electrical conductivity, electronic heat capacity, and magnetic properties of metals.
Maxwell–Boltzmann statistics applies to distinguishable particles and forms the basis of classical statistical mechanics. Bose–Einstein and Fermi–Dirac statistics apply to indistinguishable particles and belong to quantum statistics.
The Maxwell–Boltzmann distribution law is
\[ n_i = g_i e^{-(\epsilon_i-\mu)/kT} \]
The Bose–Einstein distribution law is
\[ n_i = \frac{g_i}{e^{(\epsilon_i-\mu)/kT}-1} \]
The Fermi–Dirac distribution law is
\[ n_i = \frac{g_i}{e^{(\epsilon_i-\mu)/kT}+1} \]
At high temperature and low particle density,
\[ e^{(\epsilon_i-\mu)/kT} \gg 1 \]
Thus,
\[ e^{(\epsilon_i-\mu)/kT}\pm1 \approx e^{(\epsilon_i-\mu)/kT} \]
Under these conditions, both Bose–Einstein and Fermi–Dirac statistics reduce to Maxwell–Boltzmann statistics.
Classical behavior emerges naturally when quantum effects become negligible.
At extremely low temperatures, bosons tend to accumulate in the lowest energy state. This phenomenon is known as Bose–Einstein condensation.
For an ideal Bose gas, the total number of particles is
\[ N = \sum_i \frac{g_i}{e^{(\epsilon_i-\mu)/kT}-1} \]
As temperature decreases, the chemical potential approaches zero and a large number of particles occupy the ground state.
The critical temperature for Bose condensation is
\[ T_c = \frac{h^2}{2\pi mk} \left(\frac{N}{2.612V}\right)^{2/3} \]
Below this temperature, macroscopic occupation of the ground state occurs.
Liquid helium-4 behaves as a Bose system because helium-4 atoms possess integral spin. Below \(2.17\,K\), helium undergoes transition into a superfluid phase called helium-II.
Superfluid helium exhibits remarkable properties such as:
These properties arise due to Bose–Einstein condensation of helium atoms.
Hydrogen molecules exist in two forms known as ortho hydrogen and para hydrogen. These forms differ in the orientation of nuclear spins.
In ortho hydrogen, the nuclear spins are parallel, whereas in para hydrogen they are antiparallel.
The rotational energy levels of a diatomic molecule are
\[ E_J = \frac{h^2}{8\pi^2I}J(J+1) \]
where \(J\) is the rotational quantum number and \(I\) is the moment of inertia.
The rotational partition function is
\[ q_r = \sum_{J=0}^{\infty}(2J+1)e^{-E_J/kT} \]
For para hydrogen, only even values of \(J\) are allowed:
\[ q_p = \sum_{J=0,2,4}^{\infty}(2J+1)e^{-E_J/kT} \]
For ortho hydrogen, only odd values of \(J\) are allowed:
\[ q_o = 3\sum_{J=1,3,5}^{\infty}(2J+1)e^{-E_J/kT} \]
The factor 3 arises because ortho hydrogen possesses three spin states.
At ordinary temperatures, the ortho to para ratio approaches \(3:1\), while at very low temperatures para hydrogen becomes dominant.
The equilibrium between ortho and para forms significantly affects the heat capacity and thermodynamic behavior of hydrogen gas.
Nuclear spin symmetry plays a decisive role in the thermodynamic behavior of hydrogen molecules.
Statistical thermodynamics establishes a connection between microscopic properties of particles and macroscopic thermodynamic behavior of matter. The macroscopic properties such as pressure, entropy, internal energy and free energy arise from the collective behavior of atoms and molecules.
The probability distribution of particles among available energy states forms the foundation of statistical mechanics.
Macroscopic thermodynamics becomes understandable when viewed through microscopic molecular behavior.
Maxwell–Boltzmann statistics applies to classical particles which are distinguishable and do not obey quantum restrictions.
Consider a system containing \(N\) particles distributed among different energy levels. If \(n_i\) particles occupy the \(i^{th}\) energy state having degeneracy \(g_i\), then the thermodynamic probability is
\[ W = \frac{N!}{\prod_i n_i!}\prod_i g_i^{n_i} \]
Taking logarithm,
\[ \ln W = \ln N! - \sum_i \ln n_i! + \sum_i n_i \ln g_i \]
Using Stirling approximation,
\[ \ln N! \approx N\ln N - N \]
therefore,
\[ \ln W = N\ln N - N - \sum_i (n_i\ln n_i - n_i) + \sum_i n_i \ln g_i \]
The most probable distribution is obtained by maximizing \(W\) subject to the conditions
\[ \sum_i n_i = N \]
and
\[ \sum_i n_i\epsilon_i = E \]
Using Lagrange multipliers,
\[ \delta \left[ \ln W - \alpha \sum_i n_i - \beta \sum_i n_i\epsilon_i \right]=0 \]
Differentiation gives
\[ \ln n_i = \ln g_i - \alpha - \beta \epsilon_i \]
Hence,
\[ n_i = g_i e^{-\alpha} e^{-\beta \epsilon_i} \]
Since
\[ \beta = \frac{1}{kT} \]
the Maxwell–Boltzmann distribution law becomes
\[ n_i = g_i e^{-\alpha} e^{-\epsilon_i/kT} \]
or
\[ \frac{n_i}{g_i} \propto e^{-\epsilon_i/kT} \]
This equation shows that particles tend to occupy lower energy states more preferentially.
For an ideal gas, the kinetic energy of a molecule is
\[ \epsilon = \frac{1}{2}mv^2 \]
Substituting into the Maxwell–Boltzmann distribution expression, the number of molecules possessing velocity between \(v\) and \(v+dv\) becomes
\[ dn = 4\pi N \left( \frac{m}{2\pi kT} \right)^{3/2} v^2 e^{-mv^2/2kT} dv \]
This equation is known as the Maxwell speed distribution law.
The most probable velocity is obtained by differentiating the distribution function with respect to velocity.
\[ \frac{d}{dv} \left( v^2 e^{-mv^2/2kT} \right)=0 \]
Solving,
\[ v_{mp} = \sqrt{\frac{2kT}{m}} \]
The average velocity is
\[ v_{avg} = \sqrt{\frac{8kT}{\pi m}} \]
The root mean square velocity becomes
\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]
Thus,
\[ v_{rms} > v_{avg} > v_{mp} \]
Gas molecules possess a distribution of velocities rather than a single uniform speed.
The partition function is one of the most important quantities in statistical thermodynamics. It connects microscopic molecular states with macroscopic thermodynamic properties.
For a system having discrete energy levels, the molecular partition function is defined as
\[ q = \sum_i g_i e^{-\epsilon_i/kT} \]
where
For \(N\) independent particles,
\[ Q = \frac{q^N}{N!} \]
where \(Q\) is the canonical partition function.
The partition function acts as a measure of the number of accessible states available to the system.
The partition function is the central quantity in statistical thermodynamics. It provides a direct connection between microscopic energy levels and macroscopic thermodynamic properties.
For a system having discrete energy levels, the molecular partition function is defined as
\[ q = \sum_i g_i e^{-\epsilon_i/kT} \]
where
For a system containing \(N\) independent particles, the canonical partition function becomes
\[ Q = \frac{q^N}{N!} \]
The factorial term \(N!\) removes overcounting arising from indistinguishable particles.
The partition function acts as a bridge between microscopic molecular states and observable thermodynamic behavior.
Consider a particle confined in a three-dimensional box of lengths \(L_x\), \(L_y\), and \(L_z\).
According to quantum mechanics, the translational energy levels are
\[ \epsilon = \frac{h^2}{8m} \left( \frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2} \right) \]
where
The translational partition function is
\[ q_{trans} = \sum_{n_x} \sum_{n_y} \sum_{n_z} e^{-\epsilon/kT} \]
Substituting the energy expression,
\[ q_{trans} = \sum \exp \left[ -\frac{h^2}{8mkT} \left( \frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2} \right) \right] \]
For macroscopic systems, the quantum numbers are very large, therefore summation can be replaced by integration:
\[ q_{trans} = \int_0^\infty \int_0^\infty \int_0^\infty e^{-a(n_x^2+n_y^2+n_z^2)} dn_xdn_ydn_z \]
where
\[ a = \frac{h^2}{8mkTL^2} \]
Using the Gaussian integral,
\[ \int_0^\infty e^{-ax^2}dx = \frac{1}{2}\sqrt{\frac{\pi}{a}} \]
the translational partition function becomes
\[ q_{trans} = \left( \frac{2\pi mkT}{h^2} \right)^{3/2}V \]
where \(V=L_xL_yL_z\).
Thus, translational partition function increases with temperature and volume.
For a rigid diatomic molecule, the rotational energy levels are
\[ E_J = \frac{h^2}{8\pi^2I}J(J+1) \]
where
Each rotational level possesses degeneracy
\[ g_J = 2J+1 \]
Hence, the rotational partition function becomes
\[ q_{rot} = \sum_{J=0}^{\infty} (2J+1) e^{-E_J/kT} \]
Substituting \(E_J\),
\[ q_{rot} = \sum_{J=0}^{\infty} (2J+1) \exp \left[ -\frac{h^2J(J+1)} {8\pi^2IkT} \right] \]
At high temperature, the spacing between rotational levels becomes very small, therefore summation may be replaced by integration:
\[ q_{rot} = \int_0^\infty (2J+1) e^{-BJ(J+1)/kT}dJ \]
where
\[ B=\frac{h^2}{8\pi^2I} \]
Solving the integral gives
\[ q_{rot} = \frac{8\pi^2IkT}{\sigma h^2} \]
where \(\sigma\) is the symmetry number.
For homonuclear molecules,
\[ \sigma = 2 \]
while for heteronuclear molecules,
\[ \sigma = 1 \]
For a harmonic oscillator, the vibrational energy levels are
\[ E_v = \left( v+\frac{1}{2} \right)h\nu \]
where
The vibrational partition function becomes
\[ q_{vib} = \sum_{v=0}^{\infty} e^{-E_v/kT} \]
Substituting \(E_v\),
\[ q_{vib} = \sum_{v=0}^{\infty} e^{-(v+1/2)h\nu/kT} \]
Separating terms,
\[ q_{vib} = e^{-h\nu/2kT} \sum_{v=0}^{\infty} e^{-vh\nu/kT} \]
This forms a geometric progression:
\[ 1+x+x^2+x^3+\dots = \frac{1}{1-x} \]
where
\[ x=e^{-h\nu/kT} \]
Hence,
\[ q_{vib} = \frac{e^{-h\nu/2kT}} {1-e^{-h\nu/kT}} \]
At ordinary temperatures, vibrational contributions are usually small because \(h\nu \gg kT\).
Electronic energy levels are widely separated. The electronic partition function is
\[ q_{elec} = \sum_i g_i e^{-\epsilon_i/kT} \]
Taking the ground state energy as zero,
\[ q_{elec} = g_0 + g_1e^{-\epsilon_1/kT} + g_2e^{-\epsilon_2/kT} +\dots \]
Since excited electronic energies are very large,
\[ e^{-\epsilon_i/kT}\approx 0 \]
Therefore,
\[ q_{elec}\approx g_0 \]
Hence only the ground electronic state contributes significantly at ordinary temperatures.
The nuclear partition function is
\[ q_{nuc} = \sum_i g_i e^{-\epsilon_i/kT} \]
Nuclear energy level separations are extremely large, therefore
\[ e^{-\epsilon_i/kT}\approx 0 \]
Thus,
\[ q_{nuc}\approx 1 \]
The nuclear contribution to thermodynamic properties is generally negligible.
The total partition function is the product of all independent molecular motions:
\[ q = q_{trans} q_{rot} q_{vib} q_{elec} q_{nuc} \]
This relation assumes separability of different modes of motion.
Every molecular motion contributes independently to the thermodynamic behavior of matter.
The Helmholtz free energy is related to the canonical partition function by
\[ A=-kT\ln Q \]
Derivation:
From statistical thermodynamics,
\[ P_i = \frac{e^{-\epsilon_i/kT}}{Q} \]
The total entropy is
\[ S = -k\sum_i P_i\ln P_i \]
Substituting \(P_i\),
\[ S = -k\sum_i P_i \left( -\frac{\epsilon_i}{kT}-\ln Q \right) \]
Therefore,
\[ S = \frac{U}{T}+k\ln Q \]
Using
\[ A=U-TS \]
we obtain
\[ A=-kT\ln Q \]
The internal energy is
\[ U = kT^2 \left( \frac{\partial \ln Q}{\partial T} \right)_V \]
Derivation:
Starting from
\[ Q = \sum_i e^{-\epsilon_i/kT} \]
Differentiating with respect to temperature,
\[ \frac{\partial \ln Q}{\partial T} = \frac{1}{Q} \frac{\partial Q}{\partial T} \]
Since
\[ \frac{\partial Q}{\partial T} = \sum_i \frac{\epsilon_i}{kT^2} e^{-\epsilon_i/kT} \]
therefore,
\[ \frac{\partial \ln Q}{\partial T} = \frac{1}{kT^2} \frac{\sum_i \epsilon_i e^{-\epsilon_i/kT}}{Q} \]
But,
\[ U = \sum_i P_i\epsilon_i \]
Hence,
\[ U = kT^2 \left( \frac{\partial \ln Q}{\partial T} \right)_V \]
Entropy is given by
\[ S = k\ln Q + kT \left( \frac{\partial \ln Q}{\partial T} \right)_V \]
Using
\[ A=U-TS \]
and substituting expressions for \(A\) and \(U\), the entropy relation is obtained.
Pressure is related to partition function by
\[ P = kT \left( \frac{\partial \ln Q}{\partial V} \right)_T \]
Derivation:
Since
\[ A=-kT\ln Q \]
and
\[ P = -\left( \frac{\partial A}{\partial V} \right)_T \]
therefore,
\[ P = kT \left( \frac{\partial \ln Q}{\partial V} \right)_T \]
The heat capacity at constant volume is
\[ C_V = \left( \frac{\partial U}{\partial T} \right)_V \]
Substituting internal energy expression,
\[ C_V = \frac{\partial}{\partial T} \left[ kT^2 \left( \frac{\partial \ln Q}{\partial T} \right)_V \right] \]
For an ideal monatomic gas,
\[ q_{trans} = \left( \frac{2\pi mkT}{h^2} \right)^{3/2}V \]
The canonical partition function becomes
\[ Q = \frac{q_{trans}^N}{N!} \]
Using Stirling approximation,
\[ \ln N! = N\ln N-N \]
Therefore,
\[ \ln Q = N\ln q_{trans}-N\ln N+N \]
Substituting \(q_{trans}\),
\[ \ln Q = N \ln \left[ \left( \frac{2\pi mkT}{h^2} \right)^{3/2} V \right] -N\ln N+N \]
Using thermodynamic relations:
\[ U = \frac{3}{2}NkT \]
\[ H = \frac{5}{2}NkT \]
\[ C_V = \frac{3}{2}Nk \]
\[ C_P = \frac{5}{2}Nk \]
\[ PV=NkT \]
The entropy becomes
\[ S = Nk \left[ \ln \left( \frac{V}{N} \left( \frac{2\pi mkT}{h^2} \right)^{3/2} \right) +\frac{5}{2} \right] \]
All thermodynamic properties of an ideal gas emerge naturally from the translational partition function.
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