# Vapors behave as an ideal gas

## Thermal equation of state of ideal gases

The **thermal equation of state of ideal gases**, also for short as **ideal**, **universal** or **general gas equation** or as **Gas law** describes exactly the behavior and properties of an ideal gas. It combines all individual experimental results and the gas laws derived from them to form a general equation of state.

### General equation of state

The general equation of state describes the state of the ideal gas in terms of the state variables pressure*p*, Volume *V.*, Temperature*T* and amount of substance*n* or number of particles*N* or mass *m*. The equation can be formulated in different, mutually equivalent forms, all of these forms describing the state of the system under consideration in the same way and unambiguously.

Extensive forms:

Intense form:

The individual symbols represent the following values:

The ideal gas equation represents the limiting case of all thermal equations of state for vanishing density that means for vanishing pressure at a sufficiently high temperature. In this case, the intrinsic volume of the gas molecules and the cohesion - the attractive force between the molecules - can be neglected.

The ideal gas equation is a good approximation for many gases such as air that is not saturated with water vapor, even under normal conditions. From the ideal gas equation it follows that the internal energy of an ideal gas is independent of pressure and volume and only depends on the temperature. It consists only of the kinetic energy of the heat movement of the molecules.

In 1873 Johannes Diderik van der Waals extended the gas law to the van der Waals equation, which, in contrast to the general gas equation, takes into account the intrinsic volume of the gas particles and the attraction between them and can therefore also be used as an approximation for clearly real gases. Another approximate solution for real gases is the series expansion of the virial equations, whereby the general gas equation is identical to a break in the series expansion after the first term. In general, the general gas equation is suitable as an approximate solution for weakly real gases with low intermolecular interactions, low pressures and high temperatures (large molar volumes). In particular, ideal gases here have no Joule-Thomson effect.

### Special cases

There are various special cases of the general gas law which establish a connection between two quantities while all other quantities are kept constant. These relationships between the state variables of a gas are explained and not only empirically derived through its particle character, i.e. through the kinetic gas theory.

### Boyle-Mariotte's Law

The **Boyle-Mariotte's Law**, also **Boyle-Mariotte law** or **Boyle Mariotte Law** and often with **Boyle's law** abbreviated, says that the pressure of ideal gases at constant temperature and constant amount of substance is inversely proportional to the volume. If the pressure on a gas package is increased, the volume is reduced by the increased pressure. If you reduce the pressure, it expands. This law was discovered independently by two physicists, the Irish Robert Boyle (1662) and the French Edme Mariotte (1676): (isothermal)

For *T* = const and *n* = const applies:

### Gay-Lussac's law

The **first law of Gay-Lussac**, also **Gay-Lussac's law**, **Charles' law** or **Charles's law**, states that the volume of ideal gases at constant pressure and constant amount of substance is directly proportional to temperature. A gas expands when it is heated and contracts when it cools. This connection was recognized by Jacques Charles in 1787 and by Joseph Louis Gay-Lussac in 1802.

For *p* = const and *n* = const applies:

The actual law of Gay-Lussac (the above is just the part that is mostly referred to as the *Charles' law* labeled) reads:

- With

Here is *T _{0}* the temperature at the zero point of the Celsius scale, i.e. 273.15 K or 0 ° C. On the other hand is

*T*the temperature you are looking for, taking care to use the same unit as for

*T*

_{0}to use. Is analog

*V.*the volume at

*T*,

*V.*

_{0}the volume at

*T*

_{0}and

*γ*

_{0}the volume expansion coefficient at

*T*

_{0}, where for ideal gases in general

*γ*= 1/

*T*applies.

From this equation it can be concluded that there must be an absolute temperature zero, since the equation for this predicts a volume of zero and the volume cannot become negative. Their empirical basis is therefore also the basis for the absolute Kelvin temperature scale, since the temperature zero point could be determined by extrapolation.

### Law of Amontons

The **Law of Amontons**, often too **2. Gay-Lussac's law**, states that the pressure of ideal gases with constant volume and constant amount of substance is directly proportional to the temperature. When the gas is heated, the pressure increases and when the gas cools, it decreases. This connection was discovered by Gay-Guillaume Amontons.

For *V.* = const and *n* = const applies:

Analogous to the *Gay-Lussac's law* also applies here:

### Law of Uniformity

The law of homogeneity states that an ideal gas is homogeneous through and through, that is, uniform, that it has the same density everywhere. If in a large container with a homogeneous substance, for example with a gas, a partial amount in one place *V.*_{1} is included, it contains the same amount of substance as a subset with the same volume *V.*_{1} elsewhere. If you divide the total amount of substance into two equal volumes, they contain the same amount of substance, namely half of the original. It follows:

At constant pressure and constant temperature, the amount of substance is proportional to the volume, or vice versa:

At constant pressure and constant temperature, the volume is proportional to the amount of substance.

For *T* = const and *p* = const applies:

- .

These laws apply to all homogeneous substances as long as temperature and pressure remain unchanged, and also to ideal gases.

### Law of Avogadro

Avogadro's law states that two gas volumes of the same size, which are under the same pressure and which have the same temperature, also include the same number of particles. This is true even if the volumes contain different gases. It goes without saying that it also applies in the event that the composition is the same in the two volumes; therefore the relation V ~ n for follows from Avogadro's law *T* = const and *p* = const. In addition, it also means that a gas packet in a certain volume also has a certain number of particles, which is independent of the type of substance. However, there are certain exceptions, for example if there are fewer or too many particles in a gas packet.

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

It can also be formulated like this: the molar volume is identical for all ideal gases at a certain temperature and at a certain pressure. Measurements have shown that one mole of an ideal gas at 0 ° C = 273.15 K and 1013.25 hPa pressure takes up a volume of around 22.4 dm³.

A very significant consequence of the law is: **The gas constant is identical for all ideal gases**.

### Derivations

### Derivation from the kinetic gas theory

The kinetic gas theory says that gases are composed of many individual atoms or molecules, each of which has a mass *m* and a speed *v* to have. The mean kinetic energy of all particles is proportional to the temperature of the gas. The following applies:

,

in which is the root mean square velocity of the particles. You can see that the higher the temperature of the gas, the molecules move at higher speeds. Not all particles have the same speed, but a statistical distribution of the speeds occurs (Maxwell-Boltzmann distribution).

Is the gas in a container with the volume *V.* enclosed, gas molecules repeatedly hit the wall of the container and are reflected. As a result, the particles transmit a certain momentum to the wall per unit of time and per wall surface. With the particle collisions, a force acts on every part of the wall, which we call gas pressure *p* understand.

This pressure *p* is greater, the faster the particles are. On the one hand, at high particle velocities, the rate at which the gas molecules hit the wall increases because they cross the container space more quickly. On the other hand, the bumps against the wall *more violent* and the impulse transmitted thereby increases. Will the particle density *N* / *V.* increases, so does the likelihood of molecules hitting the wall. From such considerations one can derive this equation for the pressure:

.

If the mean kinetic energy of the gas particles is expressed by the temperature, the ideal gas equation results:

.

However, this equation only applies to gases with a low particle density and at a sufficiently high temperature. This derivation neglects the fact that forces of attraction act between the particles, which weaken the particle pressure against the wall. In addition, the molecules themselves have a volume and the gas cannot be compressed at will because the particles displace one another. The description of such a real gas copes with the van der Waals equation.

### Derivation from the special cases

#### The combination of the laws of Amontons and Gay-Lussac

The laws of Amontons and Gay-Lussac, both of which are temporal *in front* of the gas equation can be summarized, for example, by the thought experiment of a two-stage change of state, whereby one generally assumes a constant amount of substance.

First, consider an isochoric change in state after the *Law of Amontons*. The starting point here is state 1 with *p _{1}*,

*V.*and

_{1}*T*. The end point is state 2 with

_{1}*p*,

_{2}*V.*(=

_{2}*V.*) and

_{1}*T*.

_{2}An isobaric change of state follows after *Gay-Lussac's law* from state 2 to state 3 with *p _{3}* (=

*p*),

_{2}*V.*and

_{3}*T*.

_{3}If one now sets the expression for *T _{2}* from the above equation into the expression for

*T*from the equation below and converts, whereby one

_{2}*p*and

_{3}= p_{2}*V.*must take into account, the result is the relationship:

_{2}= V_{1}#### Actual derivation of the gas equation

The last step is to find the constant in the right term of the above expression. If one assumes that one mole at 273.15 Kelvin and 101.325 kilopascals takes up exactly 22.414 liters, i.e. that Avogadro's law is valid, one can also assume that *n* Moles of an ideal gas occupy exactly n 22.414 liters. Plugging this into the above equation gives:

Multiply the equation by *T*_{3} and isolates that at the same time *n* from the fraction we get:

The fraction on the left is a constant number, which is defined as the gas constant *R.* and calculates its numerical value.

If you delete the indices, you get the general gas equation you are looking for:

### Derivation from the thermal equation of state

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

The individual symbols stand for the following quantities:

In the case of an ideal gas, the following applies in particular:

This simplifies the thermal equation of state to the following form:

One can now use the equation of state for a state**1** to a state **2** integrate (definite integral) and thereby obtain:

If two of the state variables are the same, i.e. if they have not changed from state 1 to state 2, they can be shortened and the respective special cases are obtained. The universal gas constant*R.* must be determined experimentally and is not derived from the constant of integration. An exemplary experimental setup can be found in the article on the gas constant.

Category: equation of state

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