Kinetic Molecular Theory

Let's think about a container with a fixed volume that is filled with pure monatomic helium (He). When we lower its temperature, the pressure exerted by the helium gas in the container decreases, also affecting the kinetic energy of the gas molecules. The molecules of helium are so far apart from each other that it is assumed that no intermolecular forces are present to hold molecules together. The volume of monatomic helium gas is also negligible because they are so small.

Monoatomic helium (He) is a great example of a gas molecule that behaves very close to what is expressed by the Kinetic Molecular Theory. But, what does it mean to behave according to this theory? Let's dive into the assumptions of Kinetic Molecular Theory!

In this article, we will talk about the definition of the kinetic molecular theory, talk about the postulates of the kinetic molecular theory, and look at some examples.

Kinetic Molecular Theory – Definition

The Kinetic Molecular Theory aims to explain the behavior of gases. Gases that behave ideally are known as ideal gases.

Postulates of Kinetic Molecular Theory of Gases

The kinetic molecular theory makes the following assumptions about ideal gases:

1. Gases are made up of particles that are in constant, random motion.
2. Gas particles have kinetic energy, and the amount of kinetic energy depends on the temperature of the gas.
3. The collisions between gas particles are elastic, so there is no transfer of energy or loss of energy.
4. Particles are very small so they occupy no volume
5. There are no attraction or repulsion (intermolecular forces) present, so gas particles will move in a straight line until they collide with the walls of the container/other gas particles.

Now, let's break down these five postulates and examine each one of them individually!

Postulate 1: Gases are made up of particles that are in constant, random motion

When we look at the basic properties of gases, we know that gases take the shape and volume of the container, gases can be compressed and they exert a force on the container, this is called pressure.

This pressure is coming from the collisions between the walls and the gas molecules. Inside a container, gas particles move in constant, random, straight-line motion, colliding with the walls of the container and between gas particles. This constant movement prevents gas particles from staying still in one area of the container and helps gas particles to spread throughout the container.

Graham's Law

Think about a balloon inflated with helium. After a while, the balloon will start shrinking. This is because the rubber contains very small holes that allow gas molecules to escape. So, when considering gases, we also have to talk about the gas properties of diffusion and effusion.

And, as you could expect, there is also a law that explains this behavior of gases! This law is called Graham's Law.

The formula for Graham's Law is:

$\frac{r_1}{r_2}=\sqrt{\frac{M_2}{M_1}}$

Where,

• r₁ = the rate of effusion of gas A
• r₂ = the rate of effusion of gas B
• M₁ = molar mass of gas A
• M₂ = molar mass of gas B

Let's look at an example that involves calculating the ratio of effusion between two gases, using the formula for graham's law!

Root-mean-square speed

Gases also have a very unique sort of speed that is used to describe the collision of gas particles, while considering speed and direction. The average velocity of gas particles is called the root-mean-square speed ($v_{\mathit{rms}}$) and is represented by the following equation:

$v_{\mathrm{rms}}=\sqrt{\frac{3\mathrm{RT}}{\mathrm{M}}}$

Where,

• R = gas constant (R = 8.3145 J/K·mol)
• T = temperature of a gas in Kelvin (K)
• M = molar mass of gas in Kg/mol

Postulate 2: Gas particles have kinetic energy.

Gas particles at higher temperatures have higher kinetic energy. So, the higher the kinetic energy, the more collisions will occur between the gas particles and/or the walls of the container.

When dealing with kinetic energies, we can use the following equation to calculate the kinetic energy of particles:

$K\mathit{.}E\mathit{}\mathit{=}\frac{1}{2}m{v}^2$

Where,

• m = mass
• v = velocity

Let's apply this formula to a simple example!

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution shows how temperature affects the velocity of ideal gases. The probability distribution of the speed of gas particles is given by the following equation:

$\rho \left(v\right)=4\mathrm{\pi }{\left(\frac{\mathrm{M}}{2\mathrm{\pi RT}}\right)}^{3/2}\mathit{}{v}^2{\mathrm{e}}^{-\mathrm{M}{v}^2/2\mathrm{RT}}$

The key to understanding the Maxwell-Boltzmann distribution is knowing that:

• Gases that are at the same temperature will have the same distribution.
• If temperature increases, the kinetic energy of gases also increases.

For the scope of AP chemistry, you won't need to use the equation above to solve calculations. However, you do need to be familiar with what a Maxwell-Boltzmann distribution curve looks like! So, Let's interpret a typical Maxwell-Boltzmann distribution curve and some concepts that you might come across in your exams.

Maxwell-Boltzmann distribution curve, Isadora Santos - StudySmarter Originals.

The distribution curve has three different speeds: probable speed, mean speed, and root-mean-square speed. The probable speed shows the largest number of molecules with that speed. The mean speed is the average speed of gas molecules. The root-mean-square speed is the average velocity of gas particles.

Probable speed ${\mathrm{V}}_{\mathrm{p}}=\sqrt{\frac{2\mathrm{RT}}{\mathrm{M}}}$

Mean speed $\stackrel{-}{\mathrm{V}}=\sqrt{\frac{8\mathrm{RT}}{\mathrm{M}}}$

Root-mean-square speed ${\mathrm{V}}_{\mathrm{rms}}=\sqrt{\frac{3\mathrm{RT}}{\mathrm{M}}}$

Both temperature and molar mass affect the shape of the distribution curve. When temperature increases, the molecules move with a faster velocity. The higher the velocity, the broader the distribution curve will be. When molar mass increases, the molecules moving at faster velocities decrease. The lower the molar mass, the broader the distribution curve. A broader curve means that there is a larger range for the velocities of the individual gas molecules.

For example, in the distribution curve below, we can see that since He has the smallest molar mass, they have the highest velocity compared to Xe, which is a very heavy gas.

Maxwell-Boltzmann distribution of noble gases at room temperature

Postulate 3: The collisions between gas particles are elastic.

The third postulate of the kinetic molecular theory states that when gas particles collide, no energy is lost or transferred from one gas particle to another. So, the total kinetic energy before collision will be the same as the total kinetic energy after the collision.

Postulate 4: Gas particles are very small so their volume is insignificant.

Inside a container, there is a lot of empty distance so the distance between gas particles is large! (compared to a gas particle)

So, the fourth postulate of the kinetic molecular theory states that ideal gases occupy no volume since their particles are so small compared to the volume in which it is being contained.

Postulate 5: Gas particles have no attractive or repulsive forces.

According to the kinetic molecular theory, gases contain no intermolecular forces holding them together.

Intermolecular forces can be:

• Ion-dipole forces - attractive forces between an ion and a dipole (polar) molecule.
• Dipole-dipole forces - attractive forces that exist between polar molecules.
• Hydrogen bonding - attractive forces that exist between molecules with hydrogen that are bonded to nitrogen, oxygen, or fluorine.
• London dispersion forces - weak attractive forces that are present in all molecules. London dispersion forces are the only type of intermolecular force that is seen in nonpolar molecules.

_References:

_{Arbuckle, D., & Albert.io. (2022, March 01). The Ultimate Study Guide to AP® Chemistry. Retrieved April 5, 2022, from https://www.albert.io/blog/ultimate-study-guide-to-ap-chemistry/}

_{Atkins, P. (2017). Atkins' physical chemistry. New York, NY: Oxford University Press.}

_{Hill, J. C., Brown, T. L., LeMay, H. E., Bursten, B. E., Murphy, C. J., Woodward, P. M., & Stoltzfus, M. (2015). Chemistry: The Central Science, 13th edition. Boston: Pearson.}

_{Moore, J. T., & Langley, R. (2021). McGraw Hill: AP Chemistry, 2022. New York: McGraw-Hill Education.}

_{Zumdahl, S. S., Zumdahl, S. A., & DeCoste, D. J. (2017). Chemistry. Boston, MA: Cengage.}