# Kinetic Molecular Theory of Gases

The kinetic molecular theory of gases (KMT or simply kinetic theory of gases) is a theoretical model that explains the macroscopic properties of a gas using statistical mechanics. These properties include the pressure, volume, and temperature of a gas, as well at its viscosity, thermal conductivity and mass diffusivity. While it’s basically an adaptation of the ideal gas law, the kinetic molecular theory of gases predicts the behavior of most real gases under normal conditions, so it has practical applications. The theory finds use in physical chemistry, thermodynamics, statistical mechanics, and engineering.

### Kinetic Molecular Theory of Gases Assumptions

The theory makes assumptions about the nature and behavior of gas particles. Essentially, these assumptions are that the gas behaves as an ideal gas:

• The gas contains many particles so applying statistics is valid.
• Each particle has negligible volume and is distant from its neighbors. In other words, each particle is a point mass. Most of a gas’s volume is empty space.
• Particles don’t interact. That is, they are not attracted or repelled by each other.
• Gas particles are in constant random motion.
• Collisions between gas particles or between particles and a container wall are elastic. In other words, molecules don’t stick to each other and no energy gets lost in the collision.

Based on these assumptions, gases behave in a predictable manner:

• Gas particles move randomly, but they always travel in a straight line.
• Because gas particles move and strike their container, the volume of the container is the same as the volume of the gas.
• The pressure of the gas is proportional to the number of particles colliding with the container walls.
• Particles gain kinetic energy as temperature increases. Increasing kinetic energy increases the number of collisions and the pressure of a gas. So, pressure is directly proportional to absolute temperature.
• Particles do not all have the same energy (speed), but because there are so many of them, they have an average kinetic energy that is proportional to the temperature of the gas.
• The distance between individual particles varies, but there is an average distance between them, called the mean free path.
• The chemical identity of the gas does not matter. So, a container of oxygen gas behaves exactly the same as a container of air.

The ideal gas law summarizes the relationships between the properties of a gas:

PV = nRT

Here, P is pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant, and T is the absolute temperature.

### Gas Laws Relating to the Kinetic Theory of Gases

The kinetic theory of gases establishes relationships between different macroscopic properties. These special cases of the ideal gas law occur when you hold certain values constant:

• P α n: At constant temperature and volume, pressure is directly proportional to the amount of gas. For example, doubling the number of moles of a gas in a container doubles its pressure.
• V α n (Avogadro’s law): At constant temperature and pressure, volume is directly proportional to the amount of gas. For example, if you remove half the particles of a gas, the only way pressure remains the same is if volume decreases by half.
• P α 1/V (Boyle’s law): Pressure increases as volume decreases, assuming the amount of gas and its temperature remains unchanged. In other words, gases are compressible. When you apply pressure without changing temperature, molecules don’t move faster. As volume decreases, particles travel a shorter distance to container walls and strike it more often (increased pressure). Increasing volume means particles travel further to reach container walls and strike it less often (decreased pressure).
• V α T (Charles’ law): Gas volume is directly proportional to absolute temperature, assuming constant pressure and amount of gas. In other words, if you increase temperature, a gas increases its volume. Lowering the temperature decreases its volume. For example, double gas temperature doubles its volume.
• P α T (Gay-Lussac’s or Amonton’s law): If you hold mass and volume constant, pressure is directly proportional to temperature. For example, tripling temperature triples its pressure. Releasing the pressure on a gas lowers its temperature.
• v α (1/M)½ (Graham’s law of diffusion): The average velocity of gas particles is directly proportional to molecular weight. Or, comparing two gases, v12/v22= M2/M1.
• Kinetic energy and velocity: The average kinetic energy (KE) relates to the average velocity (root mean square or rms or u) of gas molecules: KE = 1/2 mu2
• Temperature, molar mass, and RMS: Combining the equation for kinetic energy and the ideal gas law relates the root mean square velocity (u) to absolute temperature and molar mass: u = (3RT/M)½
• Dalton’s law of partial pressure: The total pressure of a mixture of gases equals the sum of the partial pressures of the component gases.

### Example Problems

#### Doubling the Amount of Gas

Find the new pressure of a gas if it starts at 100 kPa pressure and the amount of gas changes from 5 moles to 2.5 moles. Assume temperature and volume are constant.

The key is determining what happens to the ideal gas law at constant temperature and volume. If you recognize P α n, then you see reducing the number of moles by half also decreases pressure by half. So, the new pressure is 100 ÷ 2 = 50 kPa.

Otherwise, rearrange the ideal gas law and set the two equations equal to each other:

P1/n1 = P2/n2 (because V, R, and T are unchanged)

100/5 = x/2.5

x = (100/5) * 2.5

x = 50 kPa

#### Calculate RMS Speed

If molecules have speeds of 3.0, 4.5, 8.3, and 5.2 m/s, find the the average speed and rms speed of molecules in the gas.

The average or mean of the values is simply their sum divided by how many values there are:

(3.0 + 4.5 + 8.3 + 5.2)/4 = 5.25 m/s

However the root mean square speed or rms is the square root of the sum of the square of the speeds divided by the total number of values:

u = [(3.02 + 4.52 + 8.32 + 5.22)/4] ½ = 5.59 m/s

#### RMS Speed from Temperature

Calculate the RMS speed of a sample of oxygen gas at 298 K.

Since the temperature is in Kelvin (which is absolute temperature), no unit conversion is necessary. However, you need the molar mass of oxygen gas. Get this from the atomic mass of oxygen. There are two oxygen atoms per molecule, so you multiply by 2. Then, convert from grams per mole to kilograms per mole so the units mesh with those for the ideal gas constant.

MM = 2 x 18.0 g/mol = 32 g/mol = 0.032 kg/mol

u = (3RT/M)½ = [(3)(8.3145 J/K·mol)(298 K) / (0.032 kg/mol)] ½

Remember, a joule is a kg⋅m2⋅s−2.

u = 482 m/s

### References

• Chapman, Sydney; Cowling, Thomas George (1970). The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases (3rd ed.). London: Cambridge University Press.
• Grad, Harold (1949). “On the Kinetic Theory of Rarefied Gases.” Communications on Pure and Applied Mathematics. 2 (4): 331–407. doi:10.1002/cpa.3160020403
• Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. (1964). Molecular Theory of Gases and Liquids (rev. ed.). Wiley-Interscience. ISBN 978-0471400653.
• Maxwell, J. C. (1867). “On the Dynamical Theory of Gases”. Philosophical Transactions of the Royal Society of London. 157: 49–88. doi:10.1098/rstl.1867.0004
• Williams, M. M. R. (1971). Mathematical Methods in Particle Transport Theory. Butterworths, London. ISBN 9780408700696.