An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of negligible volume, with no intermolecular forces. Additionally, the constituent atoms or molecules undergo perfectly elastic collisions with the walls of the container. Real gases do not exhibit these exact properties, although the approximation is often good enough to describe real gases. The approximation breaks down at high pressures and low temperatures, where the intermolecular forces play a greater role in determining the properties of the gas. There are basically three types of ideal gas:

* the classical or Maxwell-Boltzmann ideal gas,
* the ideal quantum Bose gas, composed of bosons, and
* the ideal quantum Fermi gas, composed of fermions.

The thermodynamic properties of an ideal gas can be described by two equations: The equation of state of a classical ideal gas is given by the ideal gas law.

The internal energy of an ideal gas is given by:

where is a constant (e.g. equal to 3/2 for a monatomic gas) and: (with SI units appended)

* U is internal energy (joule)
* P is the pressure (pascal)
* V is the volume (cubic meter)
* n is the amount of gas (mole)
* R is the ideal gas constant (joule per kelvin per mole)
* T is the absolute temperature (kelvin)
* N is the number of particles
* k is the Boltzmann constant (joule per kelvin per particle),
* nR=Nk is the amount of energy in the gas per Kelvin (joule per kelvin).

The probability distribution of particles by velocity or energy is given by the Boltzmann distribution.

The ideal gas law is an extension of experimentally discovered gas laws. Real fluids at low density and high temperature, approximate the behavior of a classical ideal gas. However, at lower temperatures or a higher density, a real fluid deviates strongly from the behavior of an ideal gas, particularly as it condenses from a gas into a liquid or solid.

Ideal Gas Law:

Substituting in variables, the formula is:

Explanation and Discussion:

The Ideal Gas Law may be the largest and most complex of the gas laws. This is in part because of the number of variables in the equation, and in part to the abstraction of an “ideal” gas that the law is built on. The Ideal Gas Law is also designed as a sort of umbrella for Boyle’s, Charles’, and Avogadro’s laws.

First, we’ll go over the parts of the equation, PV=nRT. P is pressure. Pressure can be in either atmospheres (atm) or kilopascals (kPa). V is volume in liters (L). n is the number of moles of the gas. Because moles of a substance are determined by mass divided by molecular mass, it can create an interesting variant we will discuss later. R is the Ideal Gas Constant. Depending on whether atmosphers or kilospascals were used, the value is either 0.0821 L-atm/mol-K or 8.31 L-kPa/mol-K, respectively. Temperature is in absolute degrees Kelvin.

An interesting aspect of the Ideal Gas Law is its flexibility. It contains elements that allow you to solve for other quantities, such as density or molecular mass. To solve for molecular mass:

PV=nRT - start with the equation
PV=mass/mol. mass x RT - change moles to mass(m) in grams divided by molecular mass in grams
mol. mass x PV = mRT - multiply by molecular mass
molecular mass = mRT/PV - divide by pressure and volume.

We can also see density in that last equation, m/V (grams/liter). The same equation, but with density(d) in place of mass per volume (m/V), is:
molecular mass = dRT/P

To solve just for density, the equation would become:
density = (molecular mass x pressure)/(constant x temperature)

So far, we have been skirting the concept of an ideal gas. What exactly is an ideal gas? An ideal gas is one that exactly conforms to the kinetic theory. The kinetic theory, as stated by Rudolf Clausius in 1857, has five key points. These are:

1. Gases are made of molecules in constant, random movement. Gases like Argon have 1-atom molecules.
2. The large portion of the volume of a gas is empty space. The volume of all gas molecules, in comparison, is negligible.
3. The molecules show no forces of attraction or repulsion.
4. No energy is lost in collision of molecules; the impacts are completely elastic.
5. The temperature of a gas is the average kinetic energy of all of the molecules.

Non-Ideal Behavior

The Kinetic Theory makes several assumptions about an ideal gas. These cause problems because real gases are not ideal. The main causes of error are related to pressure and temperature.

Pressure
At high pressures, the behavior of real gases changes dramatically from that predicted by the Ideal Gas Law. Under 10 atmospheres of pressure or less, Ideal Gas Law predictions are very close to real amounts and do not generate serious error.

Temperature
When the temperature of a gas is close to its liquefaction point, the behavior is very different from Ideal Gas Law predictions. With increasing temperatures, the Ideal Gas Law predictions become close to real values.

The ideal gases have molecular volume and show no attraction between molecules at any distance; real gas molecules have volume and show attraction at short distances. Let us first consider what pressure does. Pressure at high degrees will bring the molecules very close together. This causes more collisions and also allows the weak attractive forces to come into play. With low temperatures, the molecules do not have enough energy to continue on their path to avoid that attraction.

The Van der Waals Equation

In order to overcome the errors in the Ideal Gas Law, Johannes van der Waals developed an equation to predict the behavior of real gases. Johannes van der Waals’ equation included corrections for the finite volume of the molecules of the gas and the attractive forces between the molecules. Two new constants, a and b, were added. The corrected equation is:

The correction nb subtracts the volume of the molecules. b is measured in liters/mole. The correction with a reflects the strength of attraction and is measured in liters2-atmosperes per moles2.

The equation is generally put in the form:

Values of a and b are different for each gas. The values of a and b generally increase with increased mass of the molecule and complexity of the molecule.
Pressure x Volume = Moles x Ideal Gas Constant x Temperature