# Ideal gas

In physics and physical chemistry, an ideal gas is a specific idealized model of a real gas . This is based on the assumption that there are a large number of particles in disorderly motion and that only hard, elastic collisions with one another and with the walls are considered as interactions between the particles. Although this model represents a strong simplification, it can be used to understand many thermodynamic processes in gases and to describe them mathematically.

In quantum mechanics, a distinction is made between the ideal Bosegas and the ideal Fermigas .

## Model of the ideal gas

In the model of the ideal gas of classical physics, all gas particles are assumed to be non-expansion mass points that can move freely through the volume available to them. In essence, several idealizations apply, from which numerous other properties are derived:

• The gas particles are free, they do not exert any forces of attraction or repulsion on one another. Only elastic collisions take place between wall and particle or particle and particle, in which momentum is exchanged.
• The particles themselves do not occupy any volume in their space. It follows implicitly from this that the energy of the particles is only stored in the translational movement (X, Y, Z movement directions in space).

Real particles can also store energy in their own rotation; in molecules , oscillations of the atoms against each other can also be excited. In the Van der Waals equation , the two idealizations are corrected by appropriate terms in order to extend the properties of ideal gases to real gases.

By free it is meant that the particles do not feel any forces . However may (and must) the particles to each other and to the wall of the volume discharged . A gas particle moves in a straight line at a constant speed until an impact deflects it in a different direction and can accelerate or decelerate it.

The model must accept impacts. If no impacts were allowed, then on the one hand the gas could not be locked in a volume because it did not notice the wall, and on the other hand each gas particle would retain its initial velocity for all time. The latter would prevent the energy of the gas from being evenly distributed over all degrees of freedom on average . However, such a system cannot be in thermodynamic equilibrium , which is an imperative for the applicability of the main thermodynamic laws. The collisions, the particles move only a short path length free . In order for collisions to occur, a collision cross-section must be assumed. More precise models show that the (average) collision cross-section is to be set as a function of temperature ( Sutherland constant ), which is to be understood by the dependence of the collision process on the energy of the two particles.

## thermodynamics

Formula symbol meaning
State variables
${\ displaystyle p}$ pressure
${\ displaystyle V}$ volume
${\ displaystyle N}$ Number of particles (particle number)
${\ displaystyle T}$ Absolute temperature
${\ displaystyle n}$ Amount of substance
${\ displaystyle m}$ Dimensions
${\ displaystyle U}$ Inner energy
${\ displaystyle F}$ Free energy
Constants
${\ displaystyle k _ {\ mathrm {B}}}$ Boltzmann's constant
${\ displaystyle R}$ Universal gas constant
${\ displaystyle R_ {s}}$ specific gas constant
${\ displaystyle h}$ Planck's quantum of action

### Equations of state

The thermal equation of state for describing an ideal gas is called the general gas equation . It was first derived from various individual empirical gas laws . Later, the Boltzmann statistics allowed a direct justification based on the microscopic description of the system made up of individual gas particles.

The general gas equation describes the interdependencies of the state variables of the ideal gas. In the literature it is usually given in one of the following forms:

${\ displaystyle p \ cdot V = n \ cdot R \ cdot T \ qquad p \ cdot V = N \ cdot k _ {\ mathrm {B}} \ cdot T \ qquad p \ cdot V = m \ cdot R_ {s} \ cdot T,}$

where denotes the universal gas constant and represents the specific gas constant . With the help of this equation and the main principles of thermodynamics , the thermodynamic processes of ideal gases can be described mathematically. ${\ displaystyle R = 8 {,} 314 \, 462 \, 618 \ dots \, \ mathrm {J} \ cdot \ mathrm {mol} ^ {- 1} \ cdot \ mathrm {K} ^ {- 1}}$${\ displaystyle R_ {s}}$

In addition to the thermal, there is also the caloric equation of state in thermodynamics. For the ideal gas (without internal degrees of freedom ) this reads :

${\ displaystyle U = {\ frac {3} {2}} \ cdot n \ cdot R \ cdot T \ qquad U = {\ frac {3} {2}} \ cdot N \ cdot k _ {\ mathrm {B} } \ cdot T.}$

However, thermal and caloric equations of state are dependent on each other, which is what the second law of thermodynamics calls.

### Properties of ideal gases

An ideal gas has a number of special properties, all of which can be deduced from the general gas equation and the main principles of thermodynamics. The general gas equation is the compact summary of various laws:

The same volumes of ideal gases contain the same number of molecules at the same pressure and temperature.

The amount of substance as a measure of the number of particles (atoms or molecules) is measured in the international unit mole . ${\ displaystyle n}$

${\ displaystyle 1 \ \ mathrm {mol} \ approx 6 {,} 022 \ cdot 10 ^ {23} \ \ mathrm {particle}.}$

The mole is therefore a multiple of the unit.

The volume of an ideal gas with an amount of substance under standard conditions (according to DIN 1343) ${\ displaystyle n = 1 {\ rm {mol}}}$

( and )${\ displaystyle p = 1 \ \ mathrm {atm} = 1 {,} 01325 \ cdot 10 ^ {5} \ \ mathrm {Pa}}$${\ displaystyle T = 0 \ ^ {\ mathrm {o}} \ mathrm {C} = 273 {,} 15 \ \ mathrm {K}}$

results from the general gas equation:

${\ displaystyle V = {\ frac {1 \ \ mathrm {mol} \ cdot 8 {,} 314 \, 463 \ \ mathrm {J \ cdot mol ^ {- 1} \ cdot K ^ {- 1}} \ cdot 273 {,} 15 \ \ mathrm {K}} {1 {,} 01325 \ cdot 10 ^ {5} \ \ mathrm {Pa}}} = 0 {,} 022 \, 413 \, 97 \ \ mathrm {m ^ {3}} = 22 {,} 413 \, 97 \ \ mathrm {dm ^ {3}} = 22 {,} 413 \, 97 \ \ mathrm {L}.}$

The molar mass (mass of 1 mol) corresponds to the mass of an amount of gas that is contained at 0 ° C and in a volume of 22.413 97 liters (measurable from the weight difference of a gas-filled and then evacuated piston). ${\ displaystyle \ mathrm {M}}$${\ displaystyle 1 \ \ mathrm {atm} = 1 {,} 01325 \ \ mathrm {bar}}$

Law of Boyle - Mariotte
At constant temperature, the pressure is inversely proportional to the volume: ${\ displaystyle p \ sim V ^ {- 1} \ qquad (T \ \ mathrm {const.})}$
Law Amontons
At constant volume, the pressure increases like the absolute temperature: ${\ displaystyle p \ sim T \ qquad (V \ \ mathrm {const.})}$

This law is the basis for the Jolly gas thermometer .

Law of Gay-Lussac
At constant pressure the volume increases like the absolute temperature: ${\ displaystyle V \ sim T \ qquad (p \ \ mathrm {const.})}$

### Molar volume of an ideal gas

The molar volume of an ideal gas V m0 is a fundamental physical constant that represents the molar volume of an ideal gas under normal conditions , i.e. H. at normal pressure p 0  = 101.325 kPa and normal temperature T 0  = 273.15 K. It is calculated using the universal gas constant R as:

{\ displaystyle {\ begin {aligned} V _ {\ mathrm {m} 0} = {\ frac {R \, T_ {0}} {p_ {0}}} \ approx 22 {,} 413 \, 97 \ cdot 10 ^ {- 3} \, {\ frac {\ mathrm {m} ^ {3}} {\ mathrm {mol}}} \ approx \ mathrm {22 {,} 414} \, {\ frac {\ mathrm { \ ell}} {\ mathrm {mol}}} \ end {aligned}}}

Since R has an exact value due to the definition of the units of measurement , one can also specify V m0 exactly.

### Thermodynamic quantities

In general, the following applies to an ideal gas:

• Heat capacity (monatomic):${\ displaystyle C_ {V} = {\ tfrac {3} {2}} Nk _ {\ mathrm {B}} \, \ quad C_ {p} = {\ tfrac {5} {2}} Nk _ {\ mathrm { B}}}$
• Adiabatic coefficient (monatomic):${\ displaystyle \ kappa = {\ tfrac {5} {3}}}$
• Entropy change :${\ displaystyle \ Delta S = C_ {V} \ ln \! \ left ({\ frac {T_ {2}} {T_ {1}}} \ right) + Nk _ {\ mathrm {B}} \ ln \! \ left ({\ frac {V_ {2}} {V_ {1}}} \ right)}$
• isobaric volume expansion coefficient :${\ displaystyle \ gamma = {\ frac {1} {T}}}$
• isothermal compressibility :${\ displaystyle \ kappa _ {T} = {\ frac {1} {p}}}$
• isochoric stress coefficient :${\ displaystyle \ beta = {\ frac {1} {T}}}$

Under normal conditions the following applies to an ideal gas:

• molar volume :${\ displaystyle V_ {M} = 0 {,} 022414 \, {\ frac {\ mathrm {m} ^ {3}} {\ mathrm {mol}}} = 22 {,} 414 \, {\ frac {\ mathrm {l}} {\ mathrm {mol}}}}$
• Volume expansion coefficient: ${\ displaystyle \ gamma _ {0} = {\ frac {1} {273 {,} 15}} \, \ mathrm {K} ^ {- 1}}$
• isothermal compressibility: ${\ displaystyle \ kappa _ {0} = {\ frac {1} {101325}} \, \ mathrm {Pa} ^ {- 1}}$

## Ideal gas mixture

Reversible cycle : separation of a gas mixture (yellow) by means of partially permeable membranes (red or blue plate) into a component A (green) and a component B (brown). The entropy remains constant during this process.

Opposite the time-lapse representation of the reversible segregation of an ideal gas mixture by means of partially permeable ( semipermeable ) membranes. The left (red) membrane is permeable to components (green) and impermeable to components (brown), while conversely the right (blue) membrane is impermeable to components and permeable to components . The pistons have the same dimensions and move at the same speed. The total work done by the external forces (red arrows on the cylinders) is zero. Experience has shown that when ideal gases are mixed, no heat of mixing occurs and the same applies to reversible separation. Neither work nor heat is exchanged with the environment. Since the process is reversible, the entropy remains constant. ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$

Designates the change in volume per stroke, the pressure, the entropy and the temperature of the mixture, and , , , and , , , the corresponding quantities of the component or , then: ${\ displaystyle V}$${\ displaystyle p}$${\ displaystyle S}$${\ displaystyle T}$${\ displaystyle V_ {A}}$${\ displaystyle p_ {A}}$${\ displaystyle S_ {A}}$${\ displaystyle T_ {A}}$${\ displaystyle V_ {B}}$${\ displaystyle p_ {B}}$${\ displaystyle S_ {B}}$${\ displaystyle T_ {B}}$${\ displaystyle A}$${\ displaystyle B}$

• ${\ displaystyle V = V_ {A} = V_ {B}}$
• ${\ displaystyle p = p_ {A} + p_ {B}}$( Dalton's law )
• ${\ displaystyle S = S_ {A} + S_ {B}}$
• ${\ displaystyle T = T_ {A} = T_ {B}}$

Similarly, for a multicomponent ideal gas mixture:

${\ displaystyle S (T, V) = \ sum _ {i} S_ {i} (T, V),}$

when denotes the entropy of the mixture and the entropy of the separate -th component at temperature and volume . ${\ displaystyle S (T, V)}$${\ displaystyle S_ {i} (T, V)}$${\ displaystyle i}$${\ displaystyle T}$${\ displaystyle V}$

Ideal gases of the same temperature superimpose one another in a common volume without influencing one another, whereby the pressure (Dalton's law), the thermodynamic potentials (entropy, internal energy, enthalpy) and the heat capacities of the individual components add up to the corresponding values ​​of the mixture.

### Entropy of mixing of an ideal gas mixture

Diffusion experiment: Different gases mix by themselves through diffusion and the entropy increases.

The illustration on the right shows how diffusion of two originally separate gases creates a uniform mixture. The temperatures and pressures of the initially separated gases (green or brown) are the same. By turning the upper of the two cylindrical containers, which are in contact with one another with their flat sealing surfaces (1), the closed volumes and the closed volume are combined. The gases contained therein diffuse into one another (2) until finally, without any external influence, a uniform ( homogeneous ) mixture is created in which every component is evenly distributed over the entire volume (3). If the gases behave like ideal gases, no mixing heat occurs during this diffusion process and the following applies: ${\ displaystyle T_ {1} = T_ {2}}$${\ displaystyle p_ {1} = p_ {2}}$${\ displaystyle V_ {1}}$${\ displaystyle V_ {2}}$${\ displaystyle V = V_ {1} + V_ {2}}$${\ displaystyle V}$

• ${\ displaystyle p_ {1} V_ {1} = n_ {1} RT_ {1} \ qquad; \ qquad p_ {2} V_ {2} = n_ {2} RT_ {2}}$
• ${\ displaystyle p = p_ {1} = p_ {2}}$
• ${\ displaystyle T = T_ {1} = T_ {2}}$
• ${\ displaystyle V = V_ {1} + V_ {2}}$

where and denote the number of moles of the separated gases. ${\ displaystyle n_ {1}}$${\ displaystyle n_ {2}}$

The entropy of mixing corresponds to the change in entropy when the gases expand from their original volumes or to the common mixture volume : ${\ displaystyle V_ {1}}$${\ displaystyle V_ {2}}$${\ displaystyle V}$

${\ displaystyle \ Delta S = n_ {1} \ cdot R \ cdot \ ln {\ frac {V} {V_ {1}}} + n_ {2} \ cdot R \ cdot \ ln {\ frac {V} { V_ {2}}}}$

or with , and : ${\ displaystyle n_ {1}}$${\ displaystyle n_ {2}}$${\ displaystyle n = n_ {1} + n_ {2}}$

${\ displaystyle \ Delta S = R \ cdot \ left (n_ {1} \ cdot \ ln {\ frac {n} {n_ {1}}} + n_ {2} \ cdot \ ln {\ frac {n} { n_ {2}}} \ right)}$

For a multi-component ideal gas mixture, the following applies analogously:

${\ displaystyle \ Delta S = R \ cdot \ sum _ {i} \ left (n_ {i} \ cdot \ ln {\ frac {n} {n_ {i}}} \ right) \ qquad {\ rm {with }} \ qquad n = \ sum _ {i} n_ {i}.}$

This formula applies if the separated gases do not contain identical components, even for chemically very similar gases such as ortho- and para-hydrogen . It also applies approximately to the entropy of mixing of real gases, the more precisely the better they fulfill the ideal gas equation. If the two partial volumes and but identical gases contain, so no diffusion takes place on combination and there is no mixing entropy. So it is impermissible to let the gases become more and more similar and finally identical in a continuous border crossing - see Gibbs' paradox . ${\ displaystyle V_ {1}}$${\ displaystyle V_ {2}}$

#### Reversible mixture of gases

Reversible mixing and demixing of a gas.

Not every mixture of gases is irreversible. The graphic shows a thought experiment in which a piston on the left has a semipermeable (semi-permeable) wall that only lets gas A through, but represents a barrier for gas B. The middle wall of the cylinder is also semi-permeable, but this time for gas B. The right part is evacuated so that no force acts on the piston from this side.

• Arrangement 1 : The mixture of gas A and gas B fills the chamber. The total pressure is made up of the partial pressures of gas A and gas B.
• Arrangement 2 : The piston was moved so that gas A can enter the resulting space through the left wall of the piston. Gas B in turn enters the right-hand room through the semi-permeable wall. Part of the mixture remains in the middle chamber and continues to act on the piston with the sum of the partial pressures of gas A and B.

As can be seen from the equilibrium of forces on the piston (here also pressure equilibrium, since the effective piston area is the same in all three sub-spaces), the forces are canceled out by the gas pressures. The two partial pressures of the pure gases are equal to the total pressure of the mixture in every piston position. In practice, this structure will cause friction on the piston and seals as well as losses when the gases flow through the semi-permeable walls. However, there are no physical limits, so that these losses can be reduced as desired and theoretically reduced to zero. Semipermeable walls are also known for at least some gases, for example glowing platinum, which is permeable to hydrogen but blocks oxygen . The mixing and demixing of the two gases is thus completely reversible, since the piston is in equilibrium of the compressive forces in every position and in the theoretical limit case can be moved into any position without loss and without work.

## Statistical description

While the equations of state are introduced as pure empirical equations in thermodynamics, these can be obtained directly from the microscopic description of the system as a collection of individual gas particles using the means of statistical physics . Apart from the assumptions of the model itself described above, no further approximation is required. The possibility of an exact mathematical description is one of the main reasons why the ideal gas is widely used as the simplest gas model and serves as a starting point for better models.

### Sum of states of the ideal monatomic gas

Here the statistical description of the ideal gas is to be made with the help of the canonical ensemble (for an alternative derivation in the microcanonical ensemble - Sackur-Tetrode equation ). To do this, we consider a system of particles in a volume at constant temperature . All thermodynamic relations can be calculated from the canonical partition function, which is defined as follows: ${\ displaystyle N}$${\ displaystyle V}$${\ displaystyle T}$

${\ displaystyle Z (T, V, N) = \ sum _ {r} e ^ {- \ beta E_ {r}}, \ quad \ beta = {\ frac {1} {k _ {\ mathrm {B}} T}}.}$

It is

${\ displaystyle r \ in \ left \ {(\ mathbf {r} _ {1}, \ ldots, \ mathbf {r} _ {N}; \ mathbf {p} _ {1}, \ ldots, \ mathbf { p} _ {N}) | \ mathbf {r} _ {i} \ in V, \ mathbf {p} _ {i} \ in \ mathbb {R} ^ {3}, i = 1, \ ldots, N \ right \}}$

a state of the system and

${\ displaystyle E_ {r} = E (\ mathbf {r} _ {1}, \ ldots, \ mathbf {r} _ {N}; \ mathbf {p} _ {1}, \ ldots, \ mathbf {p } _ {N})}$

the associated energy. r i is the position and p i is the momentum of the -th particle. For free, non-interacting particles, the energy is independent of the location of the particles and is the sum of the kinetic energies of the particles: ${\ displaystyle i}$

${\ displaystyle E (\ mathbf {r} _ {1}, \ ldots, \ mathbf {r} _ {N}; \ mathbf {p} _ {1}, \ ldots, \ mathbf {p} _ {N} ) = E (\ mathbf {p} _ {1}, \ ldots, \ mathbf {p} _ {N}) = \ sum _ {k = 1} ^ {N} {\ frac {\ mathbf {p} _ {k} ^ {2}} {2m}}.}$

Instead of evaluating the sum of states directly, it can be more easily calculated using an integral over the phase space .

${\ displaystyle Z (T, V, N) = {\ frac {1} {N!}} {\ frac {1} {h ^ {3N}}} \ int _ {V} \! \ mathrm {d} ^ {3} \ mathbf {r} _ {1} \ cdots \ int _ {V} \ mathrm {d} ^ {3} \ mathbf {r} _ {N} \ int _ {\ mathbb {R} ^ { 3}} \! \ Mathrm {d} ^ {3} \ mathbf {p} _ {1} \ cdots \ int _ {\ mathbb {R} ^ {3}} \ mathrm {d} ^ {3} \ mathbf {p} _ {N} \; \ exp \ left (- \ beta \ sum _ {k = 1} ^ {N} {\ frac {p_ {k} ^ {2}} {2m}} \ right). }$

The additional factor takes into account the indistinguishability of the gas particles. The spatial integrations can be carried out elementarily, since the integrand does not depend on the location; this gives the exponentiated volume . Furthermore, the exponential function breaks down into individual factors for each momentum component, whereby the individual Gaussian integrals can be evaluated analytically: ${\ displaystyle 1 / N!}$${\ displaystyle V ^ {N}}$

${\ displaystyle Z (T, V, N) = {\ frac {1} {N!}} \ left ({\ frac {V} {h ^ {3}}} \ right) ^ {N} {\ Bigg (} \ underbrace {\ int _ {- \ infty} ^ {\ infty} \! \ mathrm {d} p \, e ^ {- p ^ {2} \ beta / 2m}} _ {= {\ sqrt { 2m \ pi / \ beta}} =: h / \ lambda} {\ Bigg)} ^ {3N}.}$

Ultimately, we get the canonical partition function of the ideal gas

${\ displaystyle Z (T, V, N) = {\ frac {1} {N!}} \ left ({\ frac {V} {h ^ {3}}} \ right) ^ {N} \ left ( {\ frac {2 \ pi m} {\ beta}} \ right) ^ {\ frac {3N} {2}} = {\ frac {1} {N!}} \ left ({\ frac {V} { \ lambda ^ {3}}} \ right) ^ {N},}$

where in the last step the thermal wavelength

${\ displaystyle \ lambda = h {\ sqrt {\ frac {\ beta} {2 \ pi m}}} ​​= {\ frac {h} {\ sqrt {2 \ pi mk _ {\ mathrm {B}} T}} }}$

was introduced. The partition function has the property that it can also be calculated directly from the partition function of a single particle:

${\ displaystyle Z (T, V, N) = {\ frac {1} {N!}} \ left [Z (T, V, 1) \ right] ^ {N}.}$

This peculiarity is inherent in every ideal system in statistical physics and is an expression of the lack of interactions between the gas particles. A better gas model that wants to take these interactions into account must therefore also be dependent on at least the 2-particle partition function.

### Derivation of thermodynamic relations for the monatomic ideal gas

The thermodynamic potential assigned to the canonical partition function is the free energy

{\ displaystyle {\ begin {aligned} F (T, V, N) & = - k _ {\ mathrm {B}} T \ ln Z \\ & = k _ {\ mathrm {B}} T \ left (\ ln N! -N \ ln V + 3N \ ln \ lambda \ right) \\ & \ approx -Nk _ {\ mathrm {B}} T \ ln \ left ({\ frac {Ve} {N \ lambda ^ {3} }} \ right). \ end {aligned}}}

For large numbers of particles, the faculty can with the Stirling formula develop . ${\ displaystyle \ ln (N!) \ approx N \ ln (N) -N = N \ ln \ left ({\ tfrac {N} {e}} \ right)}$

All thermodynamic relations can now be derived from the free energy:

${\ displaystyle {\ begin {pmatrix} -S \\ - p \\\ mu \ end {pmatrix}} = {\ begin {pmatrix} \ partial _ {T} \\\ partial _ {V} \\\ partial _ {N} \ end {pmatrix}} F (T, V, N).}$

In addition, the internal energy is linked to the free energy via . ${\ displaystyle U}$${\ displaystyle F}$${\ displaystyle U = F + TS}$

#### entropy

The entropy of the ideal gas is:

{\ displaystyle {\ begin {aligned} S (T, V, N) & = - \ left ({\ frac {\ partial F} {\ partial T}} \ right) _ {V, N} = - k_ { \ mathrm {B}} \ left (\ ln N! -N \ ln V + 3N \ ln \ lambda \ right) -3Nk _ {\ mathrm {B}} T \ underbrace {{\ frac {\ partial} {\ partial T}} \ ln \ lambda} _ {= {\ frac {-1} {(2T)}}} \\ & = - k _ {\ mathrm {B}} \ left (\ ln N! -N \ ln V + 3N \ ln \ lambda \ right) + {\ frac {3} {2}} Nk _ {\ mathrm {B}} \\ & \ approx Nk _ {\ mathrm {B}} \ ln \ left ({\ frac { V} {N \ lambda ^ {3}}} \ right) + {\ frac {5} {2}} Nk _ {\ mathrm {B}}. \ End {aligned}}}

#### Thermal equation of state

The thermal equation of state results from

${\ displaystyle p (T, V, N) = - \ left ({\ frac {\ partial F} {\ partial V}} \ right) _ {T, N} = k _ {\ mathrm {B}} T { \ frac {N} {V}},}$

which can be converted into the familiar form of the ideal gas equation

${\ displaystyle \ left.pV = Nk _ {\ mathrm {B}} T \ right ..}$

#### Chemical potential

The chemical potential of the ideal gas is given by

${\ displaystyle \ mu (T, V, N) = \ left ({\ frac {\ partial F} {\ partial N}} \ right) _ {T, V} = {\ frac {F (T, V, N + 1) -F (T, V, N)} {1}} = - k_ {B} T \ ln \ left ({\ frac {V} {(N + 1) \ lambda ^ {3}}} \ right).}$

#### Caloric equation of state

The caloric equation of state (the internal energy as a function of temperature, volume and number of particles) can be determined from the equations and . ${\ displaystyle F = U-TS}$${\ displaystyle S = - {\ tfrac {\ partial F} {\ partial T}}}$

{\ displaystyle {\ begin {aligned} U (T, V, N) & = F + TS = FT {\ frac {\ partial F} {\ partial T}} = T ^ {2} \ left ({\ frac {1} {T ^ {2}}} F - {\ frac {1} {T}} {\ frac {\ partial F} {\ partial T}} \ right) = T ^ {2} {\ frac { \ partial} {\ partial T}} \ left (- {\ frac {F} {T}} \ right) \\ & = k _ {\ mathrm {B}} T ^ {2} {\ frac {\ partial} {\ partial T}} \ left (- {\ frac {F} {k _ {\ mathrm {B}} T}} \ right) = k _ {\ mathrm {B}} T ^ {2} {\ frac {\ partial \ ln Z} {\ partial T}} = - {\ frac {\ partial} {\ partial \ beta}} \ ln Z \\ & = - {\ frac {\ partial} {\ partial \ beta}} \ left (- {\ frac {3N} {2}} \ ln \ beta \ right) = {\ frac {3N} {2 \ beta}} = {\ frac {3} {2}} Nk _ {\ mathrm {B. }} T. \ end {aligned}}}

This ultimately results

${\ displaystyle U = {\ frac {3} {2}} Nk _ {\ mathrm {B}} T.}$

The remarkable thing about this equation is that the internal energy is independent of the volume. It follows e.g. B. that in the isothermal expansion of an ideal gas, the heat supplied is completely converted into work.

## Scope

Among the real gases , the light noble gases and hydrogen come closest to this state, especially at low pressure and high temperature , since they have a vanishingly small size compared to their mean free path . The velocity distribution of the particles in an ideal gas is described by the Maxwell-Boltzmann distribution .

The lower the pressure and the higher the temperature, the more a real gas behaves like an ideal one. A practical measure for this is the “normalized” distance between the current temperature and the boiling point: For example, the boiling point of hydrogen is 20 K; at room temperature that is about 15 times that, which means almost ideal behavior. In contrast, with water vapor of 300 ° C (573 K) the distance from the boiling point (373 K) is only about one and a half times - far from ideal behavior.

The critical point must be used here as a quantitative comparison variable: A real gas behaves like an ideal one if its pressure is low compared to the critical pressure or its temperature is high compared to the critical temperature.

Ideal gases are not subject to the Joule-Thomson effect , from which one can conclude that their internal energy and enthalpy are independent of pressure and volume. The Joule-Thomson coefficient is therefore always zero for ideal gases, and the inversion temperature ( ) has no discrete value , i.e. it extends over the entire temperature range. ${\ displaystyle T_ {i} \ gamma = 1}$

## Extensions

### Ideal polyatomic gas

If one would like to describe polyatomic gas particles, i.e. molecules , with the ideal gas model, this can be done by expanding the caloric equation of state

${\ displaystyle U = {\ frac {f} {2}} Nk _ {\ mathrm {B}} T.}$

It indicates the number of degrees of freedom per particle. In addition to the three degrees of translational freedom, molecules have other degrees of freedom for rotations and vibrations . Each oscillation has two degrees of freedom because the potential and kinetic degrees of freedom of an oscillation are separate degrees of freedom. ${\ displaystyle f}$

For example, a diatomic gas has a total of 7 degrees of freedom, namely

• three degrees of translational freedom,
• two degrees of freedom of rotation for rotations about axes perpendicular to the line connecting the molecular atoms and
• two degrees of freedom for the one possible oscillation of the molecular atoms to each other.

Since the rotation and oscillation frequencies of molecules are quantized in nature , a certain minimum energy is required to excite them. Often, under normal conditions, the thermal energy is only sufficient to stimulate rotation in a diatomic molecule. In this case the vibrational degrees of freedom are frozen and the diatomic gas effectively only has five degrees of freedom. At even lower temperatures, the degrees of freedom of rotation also freeze, so that only the three degrees of translational freedom remain. For the same reason, the theoretically existing third degree of freedom of rotation for rotations around the connecting line does not occur in practice, since the energies required for this are sufficient to dissociate the molecule . Here again there would be monatomic gas particles. In the case of non-rod-shaped molecules made up of more than two atoms, however, the third degree of freedom of rotation and other degrees of freedom of vibration usually occur.

### Relativistic ideal gas

If the temperatures become so high that the mean speeds of the gas particles can be compared with the speed of light , the relativistic increase in mass of the particles must be taken into account. This model can also be described theoretically well, but a real gas is usually a plasma at very high temperatures , i.e. H. the previously electrically neutral gas particles are present separately as electrons and ions . Since the interaction between electrons and ions is much stronger than between neutral particles, the model of an ideal gas can only provide limited information about the physics of hot plasmas.

### Ideal quantum gas

Every kind of matter is ultimately made up of elementary particles that are either fermions or bosons . With fermions and bosons, the so-called exchange symmetry must always be taken into account, which changes the statistical description of the system. A pure ideal gas is basically always either an ideal Fermigas or an ideal Bose gas . However, the quantum nature of a gas only becomes noticeable when the mean free path of the gas particles is comparable to or smaller than their thermal wavelength. This case therefore becomes more important at low temperatures or very high pressures.

Ideal quantum gases have found a very wide range of applications. For example, the conduction electrons in metals can be excellently described by the ideal Fermi gas. The cavity radiation and Planck's law of radiation of a black body can be excellently explained by the ideal photon gas - which is a special (massless) ideal Bosegas. Ideal Bose gases can also show a phase transition to Bose-Einstein condensates at very low temperatures .

### Van der Waals gas

Real gases are better described by the so-called van der Waals gas , which takes into account the van der Waals forces that are always present between the gas particles and their own volume. The Van der Waals equation modifies the ideal gas equation by two corresponding additional terms. In the statistical description, this equation can be obtained through the so-called virial expansion .

### Perfect gas

Ideal gases are called perfect gases, which have a constant heat capacity that is not dependent on pressure and temperature.