# Kinetic Theory and Ideal Gases

This article is a topic within the subject Higher Physics 1A.

## Contents |

## Ideal Gases

^{[1]}Gases have very weak (almost negligible) interatomic forces, and so behave very differently to solids and liquids. Importantly, gases have no standard atomic spacing and will expand in volume to fill their container.

The relationship between gas properties is quite complicated, so at low pressures the simpler **ideal gas model** is used. The properties describing ideal gases are moles, pressure, volume and temperature.

#### Moles

The amount of gas is measured in moles. One mole contains N_{A} (Avogadro's number) of atoms. The number of moles (n) is equal to mass (m) divided by molar mass (M).

**n = m/M**

#### Pressure

Gases exert pressure on their container (due to gas atoms striking the walls of the container). Pressure (P) is force (F) divided by area (A).

**P = F/A**

### Ideal Gas Law

The ideal gas law describes the relationship between the properties of an ideal gas. The product of pressure (P) and volume (V) is equal to the product of the number of moles (n), temperature (T), and the gas constant (R).

**PV = nRT**

Alternately, the ideal gas law may be expressed in terms of the Boltzmann constant (k_{B}) and the number of particles (N). Note that nR = Nk_{B}.

**PV = Nk _{B}T**

### Molecular model of Ideal Gases

^{[2]}The following assumptions are made about gases when we treat them as **ideal**:

- There are many molecules in the gas, and the size of the molecules is negligible compared to the separation between molecules.
- The particles move randomly, obeying Newton's laws of motion.
- The only inter-particle interactions are short-range forces during collisions. Collisions (with walls or other particles) are elastic.
- All molecules are identical.

At low pressures (around atmospheric pressure), most gases can be approximated well using the **ideal gas model**.

## Kinetic Theory of Temperature

^{[3]}Ideal gases store energy as the kinetic energy of individual gas molecules. This means that gas temperature can be expressed in terms of the kinetic energy of particles. The equation below shows that the average translational kinetic energy of a gas particle (½mv_{avg}^{2}) is proportional to the gas temperature (the constant of proportionality being (3/2)k_{B}).

**(1/2)mv _{avg}^{2} = (3/2)k_{B}T**

This may be proved by considering ideal gas particles in a box colliding with the walls and exerting a pressure.^{[4]}^{[5]}

The total translational kinetic energy of a gas may be found by multiplying the average energy with the number of particles.

**K _{total, trans} = N(1/2)mv_{avg}^{2} = N(3/2)k_{B}T = (3/2)nRT**

### Root-mean-square speed

^{[6]}When measuring the speed of molecules, **root-mean-square** speed is often used. As the name implies, root-mean-square speed is the square root of the mean particle speed.

**v _{rms} = sqrt(v_{avg})**

## Equipartition of Energy

^{[7]}^{[8]}The total energy of a particle depends on its movement. It will have translational kinetic energy in the x,y,z directions, and may also rotate and vibrate (other forms of kinetic energy). Each way in which the kinetic energy is stored is known as a **degree of freedom**. The more complex a particle is, the more degrees of freedom it has.

- All particles have 3 degrees of freedom, due to translational kinetic energy in the x,y,z directions.
- p\Poly-atomic particles can vibrate, as the distance between atoms changes (imagine the bonds are springs, which expand and contract to cause vibration). Monatomic molecules (including ideal gas particles) cannot vibrate. The amount that a poly-atomic particle vibrates depends on temperature (as does the number of vibrational degrees of freedom).
- Poly-atomic particles can rotate on up to 3 axes, providing up to 3 degrees of freedom. Quantum effects mean that atoms are effectively featureless, though, so rotation only counts if the alignment of the particle changes. This means a monatomic particle
*cannot*rotate; any rotation is undetectable. A diatomic particle can only rotate on two axes (perpendicular to the bond), and more complex particles can rotate on 3 axes.

The total number of degrees of freedom (f) is proportional to the energy of the gas. Each degree of freedom will add (1/2)Nk_{B}T to the total energy, giving the following equation by the equation:

**E _{int} = (f/2)Nk_{B}T = (f/2)nRT**

## End

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## References

**Textbook** refers to Serway & Jewett, *Physics for Scientists and Engineers* (Brooks/Cole , 8th ed, 2010)

**(Slides)** refers to those distributed by Angstmann, E (2012) on UNSW Blackboard.