# Ideal gas

An ideal gas is defined in physics and physical chemistry a certain idealized conceptual model of a real gas. In going from a plurality of particles in a random motion, and attracts the interaction of the particles with each other and only collisions with the walls into consideration. Although this model is an oversimplification, can be understood with him many thermodynamic processes of gases and describe mathematically.

In quantum mechanics, a distinction is the ideal Bose gas and the ideal Fermi gas.

- 3.1 mixing entropy of an ideal gas mixture

- 4.1 partition function of the ideal gas
- 4.2 Derivation of thermodynamic relations 4.2.1 entropy
- 4.2.2 Thermal equation of state
- 4.2.3 Chemical potential
- 4.2.4 Caloric equation of state

- 6.1 Ideal polyatomic gas
- 6.2 Relativistic ideal gas
- 6.3 Ideal quantum gas
- 6.4 Van der Waals gas
- 6.5 Perfect Gas

## Model of an ideal gas

In the model of an ideal gas of classical physics all gas particles are assumed to be extensionless point masses, which can move freely through the volume available to them. With free it is meant that the particles do not feel any forces. However, may ( and must) be push the particles to each other and to the wall of the volume. A gas particle thus moves in a straight line at a constant speed until a shock can steer it in another direction and speed it up or slow down.

Accepting shock is necessary for the model. Yet if no shocks, so you might the gas for a non lock up in a volume, as it did not notice the wall, and on the other would retain each gas particle for all times its initial speed. The latter would prevent the energy of the gas could be distributed uniformly in the middle of all degrees of freedom. However, such a system can not be in thermodynamic equilibrium, which is a mandatory requirement for the applicability of thermodynamics sets. Through the shock, the particles move only a short path length freely. Thus it comes to collisions, a collision cross -section must be assumed. More detailed models show that the ( average ) collision cross section depends on the temperature to be set ( Sutherlandkonstante ), what is meant by the dependence of the collision process of the energy of the two particles.

## Thermodynamics

### Equations of state

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

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

Wherein the universal gas constant and is called the specific gas constant. Using this equation and the laws of thermodynamics can be described mathematically, the thermodynamic processes of ideal gases.

In addition to thermal, there is still the caloric in thermodynamics equation of state. This is for the ideal gas (without internal degrees of freedom ):

However, thermal and caloric equation of state are dependent on each other, which is called the second law of thermodynamics.

### Ideal gas properties

An ideal gas has a number of special properties that can be all from the ideal gas law and the laws of thermodynamics inferred. The general gas equation is a compact summary of various laws:

The amount of substance as a measure of the number of particles (atoms or molecules ) is measured in the international unit of mol.

The mole is therefore a multiple of the unit.

The volume of an ideal gas with a mole under standard conditions ( according to DIN 1343 )

Turns out to be from the general gas equation:

The molar mass ( mass of 1 mol) thus corresponds to the mass of a volume of gas that is contained at 0 ° C and in a volume of 22.414 liters ( measured from the weight difference of a gas-filled and then evacuated envelope ).

This law is the basis for the gas thermometer Jolly.

### Molar volume of an ideal gas

The molar volume of an ideal gas Vm0 is a fundamental physical constant indicating the molar volume of an ideal gas under normal conditions, i.e. at atmospheric pressure P0 = 101.325 Pa and atmospheric temperature T0 = 273.15K. The value after current measurement accuracy: 22.413 968 (20).

It is calculated on the universal gas constant R to

### Thermodynamic sizes

In general, for an ideal gas:

- Heat capacity ( monatomic ):
- Adiabatic ( monatomic ):
- Entropy:
- Isobaric volume expansion coefficient:
- Isothermal compressibility:
- Isochoric tension coefficient:

In normal conditions for an ideal gas:

- Molar volume:
- Volume expansion coefficient:
- Isothermal compressibility:

## Ideal gas mixture

Opposite, the time-lapse view of the reversible segregation of an ideal gas mixture by partially permeable ( semipermeable ) membranes. The left (red ) membrane is permeable to component ( green) and impermeable to component ( brown), while conversely the right (blue ) membrane is impermeable and permeable component by component. The pistons have the same dimensions and move at the same speed. The whole of the external forces (red arrows on the cylinders ) work done is zero. Experience shows that occurs when the mixture of ideal gases no heat of mixing and the same is true for the reversible segregation. It is neither work nor heat exchanged with the surroundings. Since the process is reversible, the entropy remains constant.

- ( Dalton's law )

Analogously, for a multicomponent ideal gas mixture:

If the entropy of the mixture and the entropy of the separate th component at the temperature and the volume respectively.

Ideal gases same temperature overlap in a common volume, without mutual interference, whereby adding the pressure ( Dalton 's law), the thermodynamic potential ( entropy, internal energy, enthalpy ) and the heat capacities of the individual components to the corresponding quantities of the mixture.

### Entropy of mixing of an ideal gas mixture

The illustration shows how a uniform mixture is formed by diffusion of two originally separate gases. The temperatures and pressures of the gases initially separated (green or brown) are equal. By rotation of the upper of the two cylindrical containers, which lie with their flat sealing surfaces to each other (1), the confined volumes and the closed volume are combined. The gases contained therein diffuse into each other (2 ) until it is finally formed by itself without any external action, a uniform (homogeneous ) mixture in which each component is uniformly distributed throughout the volume (3). Behavior of the gases as ideal gases, as occurs in this diffusion process, no heat of mixing and we have:

Where and are the mole numbers of the separate gases, respectively.

The entropy of mixing corresponds to the entropy change upon expansion of the gases from their original volumes or on the common mixture volume:

Or, and:

For a multicomponent ideal gas mixture is considered analogously:

This formula is valid when the separate gases do not contain identical ingredients, even for chemically very similar gases such as ortho-and para-hydrogen. It applies approximately also for the entropy of mixing of real gases and indeed more accurate, the better fulfill this the ideal gas equation. If the two partial volumes but identical and contain gases, we find when matching instead of no diffusion and also there is no entropy of mixing. So it is not permissible to leave less in a continuous border crossing the gases always similar and eventually become identical - see Gibbs paradox.

## Statistical Description

While in thermodynamics, the state equations are introduced as mere empirical equations, these can be obtained with the methods of statistical physics directly from the microscopic description of the system as a collection of individual gas particles. In addition to the above assumptions of the model itself, no further approximation is needed here. The possibility of exact mathematical description is the main reason why the ideal gas is widely used as the simplest gas model and serves as a starting point for better models.

### Partition function of the ideal gas

Here, the statistical description of the ideal gas with the help of the canonical ensemble is to take place ( for an alternative derivation of the microcanonical ensemble - Sackur - Tetrode equation). This purpose we consider a system of particles in a volume at constant temperature. All thermodynamic relations can be derived from the canonical partition function calculated, which is defined as follows:

It is

A state of the system and

The corresponding energy. ri is the location and the momentum of the pi - th particle. For free, non-interacting particles, the energy is independent of the location of the particles and is calculated as the sum of the kinetic energies of the particles:

Instead of evaluating the partition function directly, it can be calculated easily by an integral over the phase space.

The additional factor takes into account the indistinguishability of the gas particles. The local space integrations can be performed elemental, since the integrand does not depend on the location; it is obtained the potentiated volume. Furthermore, the exponential decays into individual factors for each pulse component, wherein, the individual Gaussian integrals evaluated analytically:

Finally, one obtains for the canonical partition function of the ideal gas

Where in the last step, the thermal wavelength

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

This feature is any ideal system in statistical physics on its own and is an expression of the lack of interactions between the gas particles. A better gas model, which will take into account these interactions, therefore, must also be a function of at least the two -particle partition function.

### Derivation of thermodynamic relations

The canonical partition function associated thermodynamic potential is the free energy

For large numbers of particles can the faculty with the Stirling formula develop.

From the free energy can now be derived all thermodynamic relations:

#### Entropy

The entropy of an ideal gas is:

#### Thermal equation of state

The thermal equation of state is given by

Which can be accommodated by changing the well-known form of the ideal gas equation

#### Chemical potential

The chemical potential of an ideal gas is given for large numbers of particles by

#### Caloric equation of state

The caloric state equation ( the internal energy as a function of temperature, volume and number of particles ) can be determined from equations and.

This eventually results in

## Scope

Under the real gases, the noble gases and hydrogen coming slightly this state, the next, particularly at low pressure and high temperature, because they have a negligibly small extent compared to its mean free path. The velocity distribution of the particles in an ideal gas is described by the Maxwell -Boltzmann distribution.

The lower the pressure, the higher the temperature is, the stronger the behavior of a real gas behaves like an ideal. A practical measure of this is the " normalized " distance between the actual temperature of boiling point: For example, the boiling point of hydrogen at 20 K; at room temperature which is about 15 times, which means a nearly ideal behavior. In contrast, in steam of 300 ° C (573 K), the distance from the boiling point ( 373 K), only about a half times - far from ideal behavior.

As a quantitative comparison variable here is the critical point must be taken into account: A real gas behaves like an ideal when its pressure against the critical pressure or the temperature is small in size compared to the critical temperature.

Ideal gases are not subject to the Joule -Thomson effect, from which one can conclude that its internal energy and enthalpy are independent of its pressure and volume. Therefore, the Joule-Thomson coefficient is at an ideal gas is always zero, and the inversion temperature ( Tiγ = 1) does not have a discrete value of, ie extends over the entire temperature range.

## Extensions

### Ideal polyatomic gas

If you want the ideal gas model polyatomic gas particles, ie molecules describe, this can be done by extending the caloric equation of state

In this case, indicates the number of degrees of freedom per particle. Molecules have three translational degrees of freedom in addition to the additional degrees of freedom for rotations and vibrations. Each vibration serves two degrees of freedom because of the potential and kinetic degree of freedom of oscillation are separate degrees of freedom.

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

- Three translational degrees of freedom,
- Two rotational degrees of freedom for rotation about axes perpendicular to the line of the molecular atoms and
- Two vibration degrees of freedom for a possible vibration of the molecule atoms to each other.

As in nature, the rotational and vibrational frequencies of molecules are quantized, a certain minimum energy is required to excite them. Often, under normal conditions, only the thermal energy suffices to excite rotations in a diatomic molecule. In this case, the vibration degrees of freedom are frozen and the diatomic gases effectively has only five degrees of freedom. At even lower temperatures and the rotational degrees of freedom freeze, so that only the three translational degrees of freedom remain. For the same reason the theoretically existing third rotational degree of freedom for rotation around the connecting line in practice does not occur, since the energy required for this purpose were sufficient to dissociate the molecule. This would then lie again before monatomic gas particles. In non rod-shaped molecules of more than two atoms of the third rotational degree of freedom together with additional degrees of freedom, however, occurs in the rule.

### Relativistic ideal gas

If the temperatures are such that the mean velocities of the gas particles with the speed of light can be compared, the relativistic mass increase of the particles must be considered. This model can also be described well in theory, but a real gas usually at very high temperatures is already a plasma, ie the previously electrically neutral gas particles are stored separately as electrons and ions. Since the interaction between electrons and ions but is much stronger than between neutral particles, the conceptual model of an ideal gas can provide only limited information about the physics of hot plasmas.

### Ideal quantum gas

Each kind of matter ultimately consists of elementary particles, which are either fermions or bosons. For fermions and bosons, the so-called exchange symmetry must be considered always what changes the statistical description of the system. A pure ideal gas is basically so always either an ideal Fermi gas or an ideal Bose gas. The quantum nature of a gas is, however, only noticeable when the mean free path of gas particles is comparable or smaller than their thermal wavelength. Thus, this case will win at low temperatures or very high pressures in importance.

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

### Van-der- Waals gas

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

### Perfect gas

As a perfect gas perfect gases are referred to, which have a constant heat capacity, which is not dependent on pressure and temperature.