Ideal Gases

The **ideal gas** is a simplified model of a gas in which intermolecular interactions between gas particles become negligible. Although ideal gases do not occur in nature, they have proven to be reliable approximations of real gases, especially dilute gases.

## Equations of State of Ideal Gases

Ideal gases demonstrate several consistent relationships between qualities such as pressure, temperature, quantity of particles, and volume.

* Pressure (*P*) varies directly with temperature (*T*) when quantity of particles and volume are held constant:
	+ *P = kT*
* Pressure (*P*) also varies directly with quantity of particles (*N*) when temperature and volume are held constant:
	+ *P = kN*

* However, pressure (*P*) varies *in*directly with volume (*V*) when temperature and quantity of particles are held constant:
	+ *P = k*/*V*

* When all these naturally occurring trends are combined, pressure can be defined by the expression:
	+ *P = k*(*NT*/*V*)

### Boltzmann Constant

The **Boltzmann constant**, represented in the above equations as *k*, is a constant that relates the average kinetic energy within a quantity gas particles to the gas's temperature, and is expressed in SI units as:

* *k =* 1.38E-23 J/K

### Universal Gas Constant

The **universal gas constant**, represented as *R*, is derived by multiplying the Boltzmann constant by Avogadro's number (6.022E23 molecules/mole), and instead relates the average kinetic energy within a number of moles of gas (NOT individual particles) to the gas's temperature. It is expressed in SI units as:

* *R* = 8.31 J/(molK)

### The Ideal Gas Law

The **Ideal Gas Law**, or the equation of state for an ideal gas, is most commonly expressed thusly:

* *PV* = *nRT*

In this case, *n* is defined as the number of **moles** of gas particles. This more common form of the similar equation found above replaces the values *N* and *k* with *n* and *R* because it is generally agreed that it is easier to work with moles than individual particles because moles are larger groups of particles, and are therefore easier to work with than the incredibly large numbers of particles found in even small amounts of gas. In summary, although the Ideal Gas Law can be expressed in multiple ways, the equation shown here is the most popular and is widely considered to be the most user-friendly of all variations.

## Boyle's Law

The equation above that states that pressure varies indirectly with volume is known as **Boyle's Law**, and can also be expressed by the statement that, when number of particles and temperature are held constant, the product of the gas's initial pressure and volume is equal to the product of the gas's final pressure and volume, and those products are therefore equal to a constant:

* *PiVi = PfVf*

One can demonstrate this consistent relationship by graphing the change in pressure with respect to volume at different temperatures. The resulting curves are called **isotherms**, and they are commonly used as a graphical representation of the predictable nature of ideal gases.

## Charles's Law

Another constant ratio that is commonly referred to in the context of ideal gases is **Charles's law**, which is analogous to Boyle's Law, only it relates volume and temperature and, as opposed to the other relationships demonstrated in this article, assumes number of particles and *pressure* are held constant. It is expressed as:

* *Vi*/*Ti = Vf*/*Tf*

This can be demonstrated graphically by depicting the change in volume with respect to temperature at various temperatures. Instead of taking the form of a curve as an isotherm does, this relationship is linear. Incidentally, most other relationships referred to in this article are linear as well.

## Sources and Further Research

Walker, James S. *Physics: Fourth Edition: AP Edition*. Pearson Education, Inc., 2010.

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