Ideal gas law

Thermal ideal gas, often referred to as imprecise general gas equation, describes the relationship between the thermal state variables of an ideal gas. It combines the individual experimental results and derived from these gas laws to a general equation of state.

• 3.1 Derivation of the kinetic theory of gases
• 3.2 Derivation of the special cases 3.2.1 The combination of the laws of Amontons and Gay-Lussac
• 3.2.2 Actual derivation of gas equation

The thermal equation of state

The equation describes the state of an ideal gas with respect to the state variables pressure p, volume V, temperature T and amount of substance n and particle number N and mass m. The equation can be formulated in different mutually equivalent forms, all these forms to describe the state of the system under consideration in the same manner and clearly. Your first formulation comes from Émile Clapeyron in 1834.

Extensive forms:

Intensive form:

The individual symbols stand for the following sizes:

• Rm - or universal molar gas constant
• Rs - personal or specific gas constant
• ρ - density
• Vm - molar volume
• V - specific volume
• N - number of particles
• N - amount of substance
• M - mass
• M - Molar Mass
• T - Temperature (absolute, in Kelvin)
• P - pressure

The equation represents the limiting case of all -zero thermal condition equations for density, that is, for vanishing pressure at a sufficiently high temperature. In this case can be the internal volume of the gas molecules and the cohesion - the attractive force between the molecules - negligible. The equation is a good approximation for many gases, such as air wasserdampfungesättigte at normal conditions.

1873 Advanced Johannes Diderik van der Waals gas law for the Van der Waals equation, which may be the intrinsic volume of the gas particles and the attraction between them as opposed to the general gas equation taken into account and should serve as an approximation to well real gases. Another approximation of real gases is the expansion of Virialgleichungen, the general gas equation is identical to a termination of the series expansion in the first member. In general, the general gas equation is suitable as an approximate solution for weak real gases at low intermolecular interactions, low pressures and high temperatures ( large molar volumes ). Especially ideal gases have on no Joule- Thomson effect.

Special cases

There are several special cases of the general gas law, which establish a relationship between two variables, while all other variables are held constant. Explains not only empirically derived these relationships between the state variables of a gas are by its particle nature, so by the kinetic theory of gases.

Boyle- Mariotte

The law of Boyle -Mariotte, even Boyle's Law or Boyle's law, and often abbreviated as Boyle's law, testified that the pressure of ideal gases at constant temperature ( isothermal change of state) and a constant amount of substance is inversely proportional to the volume. Increasing the pressure to a gas packet, the volume is reduced by the increased pressure. If you reduce the pressure, it expands. This law was discovered independently by two physicists, the Irishman Robert Boyle ( 1662) and the Frenchman Edme Mariotte ( 1676 ).

For T = const and n = const applies:

Gay- Lussac

The first law of Gay- Lussac, also gay Lussacsches Law, Law of Charles and Charles 's law, states that the volume of ideal gas at constant pressure ( isobaric change of state) and a constant amount of substance is directly proportional to temperature. A gas thus expands when heated and contracts when cooled. This relationship was recognized in 1787 by Jacques Charles and 1802 by Joseph Louis Gay -Lussac.

For p = const and n = const applies:

The actual law of Gay- Lussac ( the above is only the part that is usually referred to as the Law of Charles) is:

Here, T0 is the temperature at the zero point of the Celsius scale, ie 273.15 K or 0 ° C. Demhingegen T is the desired temperature, one must be careful to use the same unit as at T0. Similarly, V is the volume, T, T0 and V0, the volume of the volume expansion coefficient of γ0 at T0, where for ideal gases generally γ = 1 / T.

From this equation, one can deduce that there must be an absolute zero temperature, since the equation for this a volume of zero predicts and the volume can not be negative. Their empirical basis is therefore also the basis for the absolute Kelvin temperature scale, as could be here over-determined by extrapolation of the zero temperature.

Law of Amontons

The law of Amontons, often second law of Gay- Lussac, testified that the pressure of ideal gases at constant volume ( isochoric change of state) and a constant amount of substance is directly proportional to temperature. At a heating of the gas, therefore, the pressure increases and at a cooling it is lower. This relationship was discovered by Guillaume Amontons.

For V = const and n = const applies:

Similar to the law of Gay- Lussac this is also true:

Act of Uniformity

The law of homogeneity indicates that an ideal gas through and through homogeneous, that is, uniformly, is that it has so everywhere the same density. When in a large tank with a homogeneous material, for example with a gas at a point V1 is a subset included this contains the same amount of material as a subset of the same volume V1 elsewhere. Dividing the total molar amount into two equal volumes, so they contain the same amount of substance, namely half of the original. It follows:

The volume is proportional at constant pressure and constant temperature to substance.

For T = const and p = const:

These laws apply to all homogeneous materials, as long as the temperature and pressure remain unchanged, and, indeed, for ideal gases.

Avogadro's law

The law of Avogadro states that two equal volumes of gas that are under the same pressure and have the same temperature, include the same number of particles. This is true even when the volumes containing various gases. Of course, it is also true for the case where the composition is in the two volumes is the same; therefore follows from the law of Avogadro, the relation V ~ n for T = const and p = const. But in addition, it also means that a gas packet in a certain volume, a certain number of particles has, independent of the type of fabric. However, there are some exceptions, when, for example less or too many particles in a gas pack.

The law of Avogadro was discovered in 1811 by Amedeo Avogadro.

It can also be formulated as follows: The molar volume is identical for a given temperature and at a given pressure for all ideal gases. Measurements have shown that one mole of an ideal gas at 0 ° C = 273.15 K and 1013.25 hPa pressure occupies a volume of around 22.4 dm ³.

A very important consequence of the law is that the gas constant is the same for all ideal gases.

Derivations

Derivation from the kinetic theory of gases

The kinetic theory of gases states that gases are composed of many individual atoms or molecules, which each in itself have a mass and a speed. The average kinetic energy of all the particles is proportional to the temperature of the gas. The following applies:

Wherein the mean square velocity of the particles. It is seen that the molecules move, at a higher temperature of the gas at higher speeds. In this case, not all particles have the same speed, but it takes a statistical distribution of velocities ( Maxwell -Boltzmann distribution).

If the gas is enclosed in a container with the volume, so are often faced gas molecules against the wall of the container and are reflected. Characterized the particles are per unit time and per panel transmitting a given pulse on the wall. It interacts with the particle collisions in any part of the wall a force which we understand as the gas pressure.

This pressure is the greater, the faster the particles. On the one hand increases at high particle velocities, the rate hit with the gas molecules on the wall as they pass through the container space faster. Secondly, the impact against the wall violently and it grows while the transmitted pulse are. Is the particle density is increased, the probability of encounter with the molecules at the wall increases. From such considerations one can derive this equation for the pressure:

Pressing the average kinetic energy of the gas by the temperature, so this results in the ideal gas equation:

However, this equation is valid only for gases with low particle density and at sufficiently high temperature. In this derivation is in fact neglected that attractive forces between the particles act that weaken the Teilchendruck against the wall. In addition, the molecules themselves have a volume and the gas can not be compressed any, because the particles pushing each other aside. The description of such a real gas overcomes the van der Waals equation.

Derivation of the special cases

The combination of the laws of Amontons and Gay-Lussac

The laws of Amontons and Gay-Lussac, both of which were found in time before the gas equation, can be summarized, for example, through the thought experiment of a two-step change of state, which is in this case generally assumes a constant amount of substance.

First, we consider an isochoric change of state according to the law of Amontons. The starting point is the state 1 with p1, V1 and T1. Endpoint state 2 with p2, V2 ( = V1) and T2.

It follows an isobaric change of state according to the law of Gay- Lussac from state 2 after state 3 with p3 ( = p2), V3 and T3.

Substituting now the expression for T2 from the above equation into the expression for T2 from the lower equation and the set, which is p3 = p2 and V2 = V1 must be taken into account, we obtain as a result of the relationship:

Actual derivation of gas equation

As a last step you have to determine the constant in the right-hand term of the above expression. Assuming that a mole at 273.15 K and 101.325 kilopascals occupies exactly 22.414 liters, so the law of Avogadro is valid, then one can also assume that n moles of an ideal gas occupy exactly n · 22.414 liters. Substituting this into the above equation, we obtain:

Multiplying the equation with T3 and isolated at the same time n from the break, we get:

The left fraction is a constant number, we define this as the gas constant R and calculates its value.

You stroke now the indices, we obtain the desired general gas equation:

Derivation from the thermal equation of state

The basis for the derivation of the thermal equation of state:

The individual symbols stand for the following sizes:

• κ - compressibility
• γ - coefficient of volume expansion
• Vm - molar volume (vm = V / n)

For an ideal gas particularly applies:

Thereby, the thermal equation of state is simplified to the following form:

It is now possible from a state 1 to a state 2 integrate ( definite integral ) and thereby obtains the equation of state:

To match two of the state variables, ie they have not changed from state 1 to state 2, these can be shortened and thereby obtained the respective special cases. The universal gas constant R must in this case be determined experimentally and is not derived from the integration constant. An exemplary experimental setup can be read in the article the gas constant.