Rotational molecular partition function

The rotational molecular partition function is a measure of the number of thermally accessible rotational energy states available to a single non-interacting molecule at a specific temperature.

 

Linear rotors

Consider a heteronuclear diatomic molecule with rotational energies given by eq45: . Substituting this equation into the rotational component of eq257, noting that each energy level is -fold degenerate, gives:

At low temperatures, is calculated by substituting experimental values of into eq270. At higher temperatures, such as typical laboratory conditions at room temperature, many rotational energy levels become thermally accessible and populated because . Since the exponential term varies smoothly with  at room temperature when , we can simplify eq270 by approximating the sum as an integral:

Substituting and into eq271 yields , which evaluates to:

Since has units of Kelvin, eq272 can also be expressed as:

where is known as the characteristic rotational temperature.

Eq273 is valid only when , which is generally true at room temperature. For this reason, it is often referred to as the high-temperature approximation to the rotational partition function.

As discussed in the article on the effect of nuclear statistics on rotational states, nuclear spins dictate which rotational states are populated for symmetrical molecules such as O2 and CO2, resulting in half of all possible rotational states being inaccessible. For such molecules, eq273 becomes:

In general, the rotational partition function for all linear rotors can be written as:

where the symmetry number is 1 for non-symmetrical linear molecules like HCl and 2 for for symmetrical linear molecules such as O2 and CO2.

 

Question

Is unique to rotational partition functions?

Answer

Yes. The symmetry number appears specifically in rotational partition functions. Although we have explained it using quantum mechanics, can also be understood within the framework of classical physics.

At high temperatures, the spacing between rotational energy levels becomes very small compared to , making the discrete quantum levels effectively continuous. The rotational partition function is then approximated classically as an integral over phase space, i.e. over all possible orientations and angular momenta. Here, each energy state corresponds to a particular configuration of positions and momenta. However, for symmetric molecules like CO₂, certain rotational operations (such as a 180° rotation around the molecular axis) produce configurations that are physically indistinguishable. To avoid overcounting these indistinguishable orientations, the symmetry number is introduced as a correction factor in the rotational partition function.

 

 

Spherical rotors

The rotational partition function for spherical rotors is derived using the same logic as that for linear rotors. Since the rotational energy levels of a spherical rotor is given by the same expression as that for a linear rotor, the rotational partition function of a spherical rotor, noting that each energy level is -fold degenerate, is given by:

Similar to linear rotors, at low temperatures is calculated by substituting experimental values of into eq276.  At higher temperatures, we apply the high-temperature approximation, with eq276 becoming:

The factor in the function increases quadratically with , while decreases exponentially. It follows that the graph of versus forms a skewed bell curve with a maximum. If , the contribution of to the integral is negligible at low and becomes significant only at higher (see diagram below). Therefore, eq277 can be simplified to:

 

Question

Prove that for .

Answer

We shall begin with the identity (see this article for derivation).

So,

 

Eq278 evaluates to:

Like linear rotors, the symmetry number is added to prevent overcounting of rotational states, giving:

 

Question

How do we determine the value of for non-linear molecules?

Answer

As mentioned in the previous article, the symmetry number is a correction factor to prevent overcounting indistinguishable classical energy states. A straightforward way to determine is to identify the rotational subgroup to which the non-linear molecule belongs. The order of this group, which is the number of unique proper rotational symmetry operations that map the molecule onto itself, is equal to . For example, CH4 belongs to the tetrahedral point group , whose rotational subgroup has an order of 12 (including 11 rotational symmetries and the identity), so .

 

 

Symmetric rotors

The energy equation of a symmetric rotor depends on three quantum numbers: . Therefore, the rotational partition function for symmetric rotors involves three summations:

or equivalently,

 

Question

Is the total degeneracy of a symmetric rotor or ?

Answer

The rotational partition function sums over all possible rotational states. The total degeneracy of a symmetric rotor is . Here, comes from the number of values for a given , which is accounted for by the most inner sum. The additional factor of 2 comes from the quantum number , which ranges from to . For , the energy levels associated with are degenerate. However, these -dependent energy levels must be summed over explicitly, as not all values of correspond to the same energy.

 

The double summation in eq282 involves fixing a value of and summing the allowed values of . Since runs from to , with for each , the minimum allowed value of for a given is . This allows us to change the order of summation in eq282 by fixing a value for and then summing over the allowed values of :

Substituting into eq283 and using the high-temperature approximation gives:

Using the earlier argument that if , the contribution of to the 2nd integral in eq284 is negligible at low and becomes significant only at higher , we have

Since , the 2nd integral in eq285 evaluates to and eq285 becomes:

Using the identity , we have

where we have added the symmetry number (e.g. for NH3), with and .

 

Question

What is the expression for the rotational partition function of an asymmetric rotor?

Answer

An asymmetric rotor has three distinct moments of inertia: . Its rotational energy is characterised by three rotational constants: , and . If two of the moments of inertia are equal, the molecule becomes a symmetric rotor. It follows that the rotational partition function of asymmetric rotor is:

which reduces to eq286 when .

 

 

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Electronic molecular partition function

The electronic molecular partition function is a measure of the number of thermally accessible electronic energy states available to a single non-interacting molecule at a specific temperature.

It is defined by eq255:

where

is the degeneracy of the -th electronic level,
is the energy of the -th electronic state,
is the Boltzmann constant,
is the absolute temperature.

Expanding the above equation yields:

where the ground electronic energy level, , is the lowest electronic energy level with rotational quantum number and vibrational quantum number .

For many diatomic molecules under standard laboratory conditions (e.g. room temperature), the electronic partition function simplifies significantly. This is because the energy gap between the ground state and the first excited electronic state is often much larger than , leading to an exponentially small population in the excited states. In other words, for at temperatures up to 10000 K, giving:

Since the ground electronic level for most diatomic molecules, as well as monatomic species with filled subshells, is nondegenerate, .

Some exceptions to this generalisation include O2 and NO. The ground electronic level of O2 consists of two unpaired electrons, each occupying one of two degenerate  antibonding molecular orbitals, resulting in a triplet state. Therefore, . Furthermore, O2 has an excited electronic state that becomes significantly populated above 1500 K, so at higher temperatures.

In its ground electronic state, NO has a single unpaired electron occupying one of two degenerate antibonding molecular orbitals. These orbitals are formed by the overlap of the and atomic orbitals of N and O. Consequently, the unpaired electron can have an orbital angular momentum projection of either +1 or -1, giving rise to two degenerate orbital states. Additionally, the unpaired electron is characterised by a doublet state (spin degeneracy of 2). Due to spin-orbit coupling, each of the two orbital states is further split into two fine-structure levels, with a small energy separation of approximately 0.015 eV (or 121 cm-1). Therefore, at , and it rapidly approaches 4 as the temperature increases.

Since the electronic partition function is tied directly to electronic energy levels, it reflects the electronic structure of the molecule. Spectroscopic data from techniques such as UV-Vis spectroscopy or photoelectron spectroscopy provide the necessary input for estimating the energy levels , and hence computing . Self-consistency methods can also be used to numerically evaluate with accuracy, especially in systems where experimental data are lacking.

In practice, becomes crucial in systems where excited electronic states are thermally accessible, such as:

    • High-temperature systems (e.g. combustion, plasma),
    • Photochemically excited molecules,
    • Transition metals and radicals, which often have low-lying excited states,
    • Astrophysical and atmospheric systems where thermal populations can include excited states.

In these cases, the electronic partition function influences calculations of internal energy, entropy, heat capacity and equilibrium constants.

 

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Gibbs paradox

The Gibbs paradox refers to the apparent contradiction between statistical and classical thermodynamics where the entropy change for mixing identical gases is nonzero unless quantum indistinguishability is considered.

In classical thermodynamics, the entropy of mixing two gases should be zero if the gases are identical (nothing changes), and positive if the gases are different (mixing different gases leads to a greater number of possible microstates). To analyse whether statistical thermodynamics aligns with this, let’s consider a system of single-species, non-interacting, distinguishable particles. The canonical partition function is given by eq166:

where

is the molecular partition function
is the number of particles

Substituting eq166 into eq159 gives:

If the partition between two containers, each with identical gas particles occupying a volume , is removed, the entropy of mixing is:

 

Question

Show that .

Answer

From eq257, we have . The translational part of the molecular partition function of the gas is proportional to (see eq266), while the remaining components (rotational, vibrational and electronic) are independent of . So,

In other words, is independent of , and so, .

 

Therefore,

Eq341 suggests an increase in entropy simply from removing a partition between two identical gases, even though nothing physical has changed (the final state is macroscopically the same as the initial one). This is not possible and leads to the Gibbs paradox.

To resolve the paradox, the partition function must be divided by . Substituting eq167 into eq159 gives:

So,

Using Stirling’s approximation, . Therefore, the entropy of mixing of two identical gases is zero only if the partition function correctly accounts for indistinguishability by dividing by .

 

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Canonical partition function

The canonical partition function  is a statistical sum over all microstates in the canonical ensemble, providing the normalisation needed to determine the probability that each state is occupied in a system in thermal equilibrium at a fixed temperature.

It mathematically given by eq135. To explain why it is called a partition function, we need to evaluate in eq133, which defines the functional form of the probability distribution for a microstate in any system in thermal equilibrium with a heat bath.

Consider a thermal bath in contact with two distinct systems, and , where system has volume and particle number , while system has a different volume and particle number . The microstates of system are governed by a constant , such that , and those of system are governed by a constant , with . Similarly, the combined system is governed by a single constant, , with the total energy of a microstate for the combined system equal to the sum of the energies of the individual microstates, ​. According to the derived probability distribution, the probability of a microstate of the combined system is:

On the other hand, the probability of finding the combined system in a specific state is also the product of the probabilities of finding each individual system in its corresponding state:

For eq140 and eq141 to be consistent for all possible values of ​ and ​, we must have ​. As explained in the previous article, eq133 is derived under the postulate that all microstates with equal energy have equal probability of occurring in any system with fixed volume, composition and temperature. This implies that ​, and possibly ​, may depend on ​, ​ and ​. However, the two systems have different ​ and ​, yet share the value of ​. Therefore, ​ cannot depend on ​ or ​. Since the only parameter common to both systems is the temperature of the thermal bath, ​ can be a function of temperature alone: ​.

 

Question

Explain why is a function of and only.

Answer

, the energy corresponding to each microstate, is determined by solving the Schrodinger equation , where is the wavefunction that describes the quantum state resulting from the positions and momenta of all particles in the system. Therefore, is a function of and only.

 

Building on the above general argument that is solely a function of temperature, consider the special case of two systems with the same , and , as in the canonical ensemble. Applying the same reasoning here, we again find that the systems share the same , reinforcing that depends only on temperature. Thus, regardless of whether the systems have identical or differing volumes and particle numbers, as long as they are in thermal equilibrium with the same heat bath, is a function solely of temperature, independent of the systems’ internal structure or composition.

To determine the exact expression for , we refer to eq120, which is based on the postulate that the macroscopic properties of a system are given by the ensemble average. In other words, the equation applies to the average of any mechanical property :

Consider the system’s pressure . When we change the volume of the system, the energy associated with a microstate of the system also changes. The work done on the system is , where is the pressure contributed by the -th microstate of the system. Since the work done on the system corresponds to a mechanical change in the system’s internal energy, we can also write , or equivalently,

Substituting eq143 back into eq142 gives:

Using the chain rule, the partial differentiation of eq135 yields:

Substituting eq133 and eq145 into eq144 results in:

The partial derivative of with respect to is:

Substituting eq147 into eq134 gives:

It follows that:

Assuming that the function is continuous over the domain of interest, we can interchange the derivatives:

Substituting eq146 into eq149 yields:

Comparing eq150 with the thermodynamic identity given by eq127 results in:

Using the chain rule and substituting it into eq151 gives:

Substituting into eq152 and using the reciprocal identity yields:

Since and are functions of only, the chain rule form of eq153 can be simplified to , or equivalently, . When integrated, this results in , or , where is a constant. Therefore,

where is a constant.

Substituting eq154 back into eq135 gives the explicit form of the canonical partition function:

As discussed, the probability of finding a system, which is in thermal equilibrium with a constant temperature heat bath, in a specific microstate with energy is given by . Here, the numerator represents the unnormalised statistical weight of microstate . This weight becomes a meaningful probability only after dividing by , which represents the total statistical weight of all accessible microstates. In this sense, enables the partitioning of the total probability of 1 across all microstates, in proportion to their energy. This is why it is called a partition function.

 

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Statistical thermodynamics (overview)

Statistical thermodynamics is a theory that connects the microscopic behaviour of atoms and molecules with the macroscopic laws of thermodynamics.

It uses the principles of probability and statistics to predict the average behaviour of a large number of particles, allowing us to understand how the properties of individual atoms and molecules give rise to observable thermodynamic quantities such as temperature, pressure and entropy. Rather than tracking every particle individually, it focuses on the distribution of particles among various energy states and uses this information to compute macroscopic properties to explain the laws of thermodynamics.

Statistical thermodynamics has a wide range of applications in chemistry, including explaining and predicting chemical equilibrium and the transition state theory; deriving expressions for heat capacity, internal energy and entropy based on molecular motions; and understanding phenomena like melting, boiling, and magnetisation by analysing how microscopic interactions change with temperature and pressure.

To fully comprehend the theory, one must first understand the concept of a canonical ensemble.

 

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Molecular partition function

The molecular partition function  is a mathematical expression that describes how a non-interacting particle in a system is distributed among different energy levels at a given temperature.

It naturally emerges when analysing the canonical partition function of a system composed of non-interacting, distinguishable particles. Under these conditions, the total partition function factorises into a product of single-particle partition functions, allowing the definition of the molecular partition function as the canonical partition function of an individual particle.

Mathematically, it is given by

where is the -th energy level of the particle and is the Boltzmann constant.

It is also equivalent to the denominator of the Boltzmann distribution, which gives the probability of finding a particle in the -th energy state at thermal equilibrium:

 

Question

What are non-interacting molecules?

Answer

Non-interacting molecules behave independently and do not interact with one another through forces such as chemical bonding, electrostatic attraction/repulsion or van der Waals forces. The energy levels of each molecule are determined solely by its own internal structure (electronic, vibrational, rotational and translational states), and not altered by the presence of nearby molecules. Ideal gases, or very dilute gases where interactions are negligible, can be considered systems containing non-interacting molecules.

 

We know that is a function of temperature , but why is it called a partition function? Consider the case when is very high. In this limit, each term in the sum approaches 1. Conversely, when , only the term with (the ground state) is equal to 1. This means that tells us how many energy states are accessible to a molecule at a given temperature. When it appears in the Boltzmann distribution, it then reveals how the total probability is partitioned among the available energy levels.

Although the molecular partition function is defined for a single molecule, it can be used to describe the behaviour of a collection of non-interacting molecules in a system. This is because, when the molecules do not interact, the probability that any one molecule occupies a particular energy state given by eq50, also represents the fraction of all molecules found in that state at thermal equilibrium. This situation is analogous to rolling a four-sided biased die, where each face has a different probability of appearing. In this analogy, the probability of a particular outcome corresponds to the probability of a molecule occupying a specific energy state. Since each independent roll follows the same probability distribution (e.g. 40% for 1, 30% for 2, 17% for 3 and 13% for 4), the likelihood of any outcome is the same for a single roll or for many rolls. Likewise, at equilibrium, the probability that a single molecule is in a given energy state is equal to the fraction of the total number of molecules in that state.

To elaborate on this concept, consider an equally spaced array of energy levels (0, , , …, ) like those of a harmonic oscillator, with

Let . Since for (see this article for details), eq252 and eq251 become

and

respectively, where .

Most diatomic molecules are in their vibrational ground state at room temperature. For example, if and , the state populations (according to eq254) are , and . At higher temperatures, e.g. , more vibrational levels become populated, with , , , , , , and . These results are illustrated in the diagram below.

Notably, a higher value of corresponds to a larger number of significantly populated states. Therefore, the molecular partition function provides a measure of how many energy states are thermally accessible at a given temperature and, together with the Boltzmann distribution, partitions the total probability among the available energy levels.

 

Question

If the molecular partition function is related to the Boltzmann distribution, which is based on distinguishable particles, is it applicable to distinguishable or indistinguishable particles since it can also be used to describe a collection of particles?

Answer

The molecular partition function is defined for a single non-interacting molecule and does not itself assume distinguishability or indistinguishability. When extended to a system of non-interacting molecules, the probability that a molecule occupies a particular state  remains , which is the same fraction of molecules in that state regardless of whether the molecules are considered distinguishable or indistinguishable. This is because each molecule independently follows the same single-particle probability distribution. The distinction between distinguishable and indistinguishable particles only affects the form of the canonical partition function , not the calculation of these fractions.

 

The molecular partition function can also be expressed as:

where refers to the number of states having the same energy (degenerate states), and the sum is now over degenerate states, not individual states.

 

Question

Use an example to show that eq250 and eq255 are equivalent.

Answer

 

Energy Level,

Degeneracy, Contributing States

0

2

A, B

3

C, D, E

1

F

 

Using eq250,

Using eq255,

 

Finally, the total energy of a molecule in state  can be approximated as:

where

is the translational energy of the molecule in state
is the rotational energy of the molecule in state
is the vibrational energy of the molecule in state
is the electronic energy of the molecule in state

The validity of eq256 is based on the following assumptions:

    1. The four types of energy are independent of each other. For instance, coupling between vibrational and rotational motion is small and can be neglected.
    2. The Born-Oppenheimer approximation is applied, which assumes that electronic motion can be treated independently of nuclear motion (translation, rotation, vibration) because electrons move much faster than nuclei. This results in the total wavefunction of the molecule being approximated as the product of four wavefunctions: .
  1. Substituting eq256 into eq250 gives

Since the sum is over all states , we have

This factorisation of the molecular partition function is extremely useful in statistical mechanics because it allows us to treat each mode of motion — translational, rotational, vibrational, and electronic — independently. Instead of summing over all possible combinations of quantum states for the entire molecule (which can be computationally intractable), we can compute simpler, separate partition functions for each type of motion. This greatly simplifies calculations of thermodynamic properties such as internal energy, entropy and heat capacity. It also helps to explain the transition state theory.

 

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Thermodynamic functions involving the molecular partition function

Thermodynamic functions involving the molecular partition function describe the macroscopic thermodynamic properties of solids and ideal gases composed of non-interacting particles in the canonical ensemble.

These functions are derived using thermodynamic functions involving the canonical partition function :

Thermodynamic Property Function Derivation
Pressure eq146
Internal energy

eq148

eq157

Entropy eq159
Helmholtz energy eq159a
Enthalpy eq159b
Constant-volume heat capacity eq159c
Gibbs energy eq159d
Chemical potential eq159e

For example, substituting eq166 into eq157 yields the internal energy for a system of  non-interacting, distinguishable particles (such as those in a solid):

Here, is the total single-particle partition function (i.e. eq257), with contributions from translational, rotational, vibrational and electronic degrees of freedom. Eq320 assumes that all components of , including , are defined using absolute energy levels — for example, the vibrational molecular partition function includes the zero-point energy (i.e. eq290). However, if is instead defined relative to the zero-point energy (i.e. eq292), then we need to add back the zero-point energy term giving:

 

Question

Show how eq320 is applicable to a system in which particles can occupy two states with absolute energies 0 and .

Answer

From eq250, . Therefore,

As , , which implies that all particles are in the ground state. As , . This is also physically intuitive. At very high temperatures, thermal energy far exceeds the energy gap . The two energy states are equally populated, and the average energy per particle is . The total internal energy is therefore  times this average energy.

 

The above Q&A demonstrates that although the molecular partition function is defined for a single particle, thermodynamic functions derived from it describe the macroscopic properties of a system. In the case of an ideal gas (composed of non-interacting, indistinguishable particles), substituting eq167 into eq157 also results in eq320. Similarly, for non-interacting, indistinguishable particles, each of the expressions for pressure , enthalpy and constant-volume heat capacity in terms of the molecular partition function is the same. They are given by:

 

Question

Compare the expressions for pressure derived from the canonical partition function and the molecular partition function.

Answer

Comparing eq322 and eq146,

Aspect

Dependence

Canonical (full system) partition function

Molecular (single-particle) partition function

Assumed interactions

Valid for both interacting and non-interacting particles

Assumes non-interacting, indistinguishable particles

Application Real gases

Ideal gases

 

However, the statistical thermodynamic expressions for entropy , as well as for Helmholtz energy, Gibbs energy and chemical potential, differ between distinguishable and indistinguishable particles due to the Gibbs paradox. For an ideal gas, substituting eq167 into eq159 results in:

It follows that the Helmholtz energy expression for an ideal gas is:

For Gibbs energy, we have . Hence,

 

Question

Show that .

Answer

Substituting into eq326 yields , where . For an ideal gas, , where is the number of moles and , with  being the Avogadro constant. So,

where .

Using Stirling’s approximation and ,

where is the molar molecular partition function.

Dividing eq328 through by gives:

 

Substituting eq167 into eq159e gives the expression for the chemical potential of an ideal gas:

 

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Statistical thermodynamic derivation of the ideal gas law

The ideal gas law is derived from the statistical thermodynamic expression for pressure using the canonical partition function of non-interacting, indistinguishable particles.

Substituting eq257 into eq322 gives:

Since only the translational molecular partition function is dependent on , we have

Substituting eq266 into eq350 yields:

With , where is the Avogadro constant and is the number of moles, we have the ideal gas law:

where the universal gas constant .

This derivation demonstrates how macroscopic thermodynamic laws are consistent with the microscopic principles of quantum mechanics, as described by statistical thermodynamics. It reinforces the concept that, in an ideal gas, the particles are non-interacting point masses whose energy is solely due to translational motion.

 

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Equilibrium constant in statistical thermodynamics

The equilibrium constant in statistical thermodynamics relates to the ratio of molecular partition functions for products and reactants. It reflects how molecular energy levels and their populations determine macroscopic equilibrium.

Consider the reaction

where is the stoichiometric coefficient of the species in the reversible reaction. Note that the stoichiometric coefficients for reactants are negative by convention, while those for products are positive.

The reaction Gibbs energy for the reaction is:

Substituting eq329 into eq380 gives,

where the reaction zero-point energy change is .

Eq381 can be rewritten as:

Substituting eq8 into eq382 and rearranging yields:

 

Question

Show that eq383 is consistent with the classical thermodyamic equilibrium constant .

Answer

As is derived using (see this article), we need to show that the statistical thermodynamics expression for the chemical potential of an ideal gas can be transformed into . Substituting eq257 into eq330 yields:

Substituting eq266 into eq384 gives:

where .

Substituting the ideal gas law into eq385 gives:

or equivalently,

From eq386, we can define the standard chemical potential as . Then,

where is the activity of an ideal gas.

 

 

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Internal energy in statistical thermodynamics

The internal energy of a thermodynamic system consisting of non-interacting particles can be represented statistically using the molecular partition function .

It is equivalent to the product of the mean molecular energy and the number of non-interacting particles in the system. From eq320, where , we have:

As mentioned in an earlier article, is the total molecular partition function (i.e. eq257), with contributions from translational, rotational, vibrational and electronic degrees of freedom. Eq360 assumes that all components of , including , are defined using absolute energy levels — for example, the vibrational molecular partition function includes the zero-point energy (i.e. eq290). However, if is instead defined relative to the zero-point energy (i.e. eq292), then we need to add back the zero-point energy term giving:

where .

 

Monatomic ideal gas

For a monatomic ideal gas, each atom is a point mass with no rotational or vibrational degrees of freedom. This implies that there is only a single rotational ground state and a single vibrational ground state, each with energy equal to zero. Therefore, . Furthermore, as explained in the derivation of eq297, for an atom not subjected to extreme temperatures. Hence, eq361 simplifies to:

Substituting eq266 into eq362 and differentiating gives:

Comparing eq363 with eq305, . Substituting , where is the Avogadro constant and is the number of moles, into eq363 yields:

where the universal gas constant is .

 

Diatomic ideal gas

For a diatomic ideal gas, is again given by eq363. is derived by substituting eq275 into and differentiating to give:

Comparing eq364 with eq309, .

For , we substitute eq293 into and differentiating to give:

where .

At high but not extreme temperatures, , which allows us to expand as a Taylor series () to give:

Comparing eq366 with eq313, .

Similar to an atom,  for a diatomic molecule (see derivation of eq297). Hence,

which is consistent with eq316.

The total internal energy of a system of a diatomic ideal gas not subjected to extreme temperatures is therefore:

 

Polyatomic ideal gas

The translational component of the internal energy of a polyatomic ideal gas is again:

As mentioned in this article, each normal mode of vibration of a polyatmomic molecule behaves approximately like a separate harmonic oscillator, with the total vibrational partition function given by eq294:

where for non-linear molecules,  for linear molecules, and is the number of atoms per molecule.

Substituting into of of and differentiating gives:

It follows that at high but not extreme temperatures,

for a linear rotor is derived by substituting eq275 into and differentiating to yield:

for a non-linear rotor (including spherical, symmetric and asymmetric rotors), is derived by substituting either eq280, eq286 or eq287 into and differentiating to give:

Once again, for a polyatomic molecule not subjected to extreme temperatures. Therefore, the total internal energy of a system of a polyatomic ideal gas not subjected to extreme temperatures is:

 

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