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.

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Jetzt kostenlos anmeldenLet'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.

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

**Ideal gases** are gases that behave according to the kinetic molecular theory.

To learn more about the behavior of ideal gases, check out the article "Ideal Gas Law"!

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

- Gases are made up of particles that are in constant, random motion.
- Gas particles have kinetic energy, and the amount of kinetic energy depends on the temperature of the gas.
- The collisions between gas particles are elastic, so there is no transfer of energy or loss of energy.
- Particles are very small so they occupy no volume
- 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!

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.

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. **

**Graham's law** states that, at a constant temperature and pressure, *the rates of effusion of gases are inversely proportional to the square root of their molar masses. *In other words, the greater the molar mass, the slower the speed of the gas.

The formula for Graham's Law is:

$\frac{{r}_{1}}{{r}_{2}}=\sqrt{\frac{{M}_{2}}{{M}_{1}}}$

Where,

- r
_{1}= the rate of effusion of gas A - r
_{2 }= the rate of effusion of gas B - M
_{1}= molar mass of gas A - M
_{2}= molar mass of gas B

**Which of the following gases will have the highest and the lowest rates of effusion? H _{2}, CO_{2}, and PF_{5.}**

First of all, we need to calculate the molar masses of each of those gases. Then, we compare their molar masses. The gas with the smaller molar mass will have the greatest rate of effusion, while the heavier gas will have the lowest rate of effusion!

${\mathrm{H}}_{2}=2\times 1.008\mathrm{g}\mathrm{H}=2.016\mathrm{g}/\mathrm{mol}\phantom{\rule{0ex}{0ex}}{\mathrm{CO}}_{2}=12.011\mathrm{g}\mathrm{C}+\hspace{0.17em}(2\times 15.999\mathrm{g}\mathrm{O})=44.009\mathrm{g}/\mathrm{mol}\phantom{\rule{0ex}{0ex}}{\mathrm{PF}}_{5}=30.974\mathrm{g}\mathrm{P}+5\times 18.998\mathrm{g}\mathrm{F})=125.966\mathrm{g}/\mathrm{mol}$

So, from the calculated molar masses, we can say that H_{2} has the highest effusion rate, while PF_{5} has the lowest rate of effusion!

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

**Calculate the ratio of the rates of effusion of helium (He) to methane (CH _{4}).**

First, find the molar masses for He and CH_{4}**:**

$\mathrm{He}=4.0026\mathrm{g}/\mathrm{mol}\phantom{\rule{0ex}{0ex}}{\mathrm{CH}}_{4}=12.011\mathrm{g}\mathrm{C}+\hspace{0.17em}(4\times 1.008\mathrm{g}\mathrm{H})=16.043\mathrm{g}/\mathrm{mol}$

Now, we can plug these molar masses into graham's law equation and find the ratio of helium to methane!

$\frac{\mathrm{rate}\mathrm{of}\mathrm{effusion}\mathrm{of}\mathrm{He}}{\mathrm{rate}\mathrm{of}\mathrm{effusion}\mathrm{of}{\mathrm{CH}}_{4}}=\sqrt{\frac{molarmassofC{H}_{4}}{molarmassofHe}}=\sqrt{\frac{16.046g/mol}{4.0026g/mol}}=2.002$

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

**What would be the root-mean-square speed of oxygen gas (O _{2}) at 50°C?**

Notice that we were given the temperature in celsius. First, we need to convert 50°C to Kelvin

$\mathrm{K}=\xb0\mathrm{C}+273\phantom{\rule{0ex}{0ex}}\mathrm{K}=50\xb0\mathrm{C}+273=323\mathrm{K}$

Next, calculate the molar mass of O_{2 } in Kg/mol:

Finally, we can plug all of these variables into the root-mean-square-velocity equation!

${u}_{\mathit{rms}}=\sqrt{\frac{3\mathrm{RT}}{\mathrm{M}}}=\sqrt{\frac{3\times 8.314\mathrm{J}/\mathrm{K}\xb7\mathrm{mol}\times 323\mathrm{K}}{0.031998\mathrm{kg}/\mathrm{mol}}}=502\mathrm{m}/\mathrm{s}$

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.

- The higher the number of gas particles, the greater the number of collisions.
- If you have two gases at the same temperature, they will also have the same
*average kinetic energy*.

**Kinetic energy** is the energy of motion.

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!

**You throw a rock into a lake. The rock weighs 0.0156 kg and travels at 6.21 m/s. Calculate its kinetic energy.**

This is an easy one. The question already gave us mass and velocity, so all we have to do is use the equation above to find the kinetic energy of the rock!

$\mathrm{K}.\mathrm{E}=\frac{1}{2}m{v}^{\mathit{2}}=\frac{1}{2}\times 0.0156\mathrm{kg}\times 6.21\mathrm{m}/{\mathrm{s}}^{2}=0.301\mathrm{J}$

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.

The **Maxwell-Boltzmann distribution** illustrates the distribution of the kinetic energy of gas particles at a given temperature.

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.

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.

**Calculate the most probable speed of F _{2} molecules at a temperature of 335 K.**

First, we need to calculate the molar mass of F_{2} in kg/mol:

${\mathrm{M}}_{{\mathrm{F}}_{2}}=2\times 18.998\mathrm{g}\mathrm{O}=37.996\mathrm{g}/\mathrm{mol}{\mathrm{F}}_{2}\phantom{\rule{0ex}{0ex}}37.996\mathrm{g}/\mathrm{mol}\times \frac{1\mathrm{kg}}{{10}^{3}\mathrm{g}}=0.037996\mathrm{kg}/\mathrm{mol}{\mathrm{F}}_{2}$

${\mathrm{V}}_{\mathrm{p}}=\sqrt{\frac{2\mathrm{RT}}{\mathrm{M}}}=\sqrt{\frac{\left(2\right)(8.314\mathrm{J}/\mathrm{mol}\xb7\mathrm{K})(335\mathrm{K})}{0.037996\mathrm{kg}/\mathrm{mol}}}=383\mathrm{m}/\mathrm{s}$

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.

**Elastic collisions** are collisions where the internal kinetic energy is conserved (no energy is lost).

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.

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

**Intermolecular forces **are forces of attraction between molecules that influence the physical properties of molecules.

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.

If you want to learn more about the intermolecular forces, go to the article "Intermolecular Forces".

- The
**kinetic molecular theory**aims to explain the behavior of**ideal gases**. - The
**five postulates of the kinetic molecular theory**are: 1) Gases are made up of particles that are in constant, random, point-like motion, 2) Gas particles have kinetic energy, 3) The collisions between gas particles are elastic, 4) Particles are very small so they occupy no volume and 5) There are no intermolecular forces present between gas molecules. - According to
**Graham's Law**, the greater the molar mass, the slower the speed of the gas. - The
**Maxwell-Boltzmann distribution**shows the distribution of the kinetic energy of gas particles at a given temperature.

_{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.}

The Kinetic molecular theory is a theory used to describe the behavior of ideal gases.

The postulates of the kinetic molecular theory (KMT) are:

Gases are made up of particles that are in constant, random, point-like motion.

Gas particles have kinetic energy, and the amount of kinetic energy depends on the temperature of the gas.

The collisions between gas particles are elastic, so there is no transfer of energy or loss of energy.

Particles are very small so they occupy no volume

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.

Condensation is the process of turning a gas into a liquid by cooling the temperature. When the temperature decreases, the molecules also slow down and allow the intermolecular forces to infleunce the movement of the gas molecules and convert them into a liquid.

The kinetic molecular theory explains condensation because it states that gas particles contain kinetic energy. When the temperature decreases, kinetic energy also decreases.

________ are gases that behave according to the kinetic molecular theory.

Ideal gases

The following are postulates of the kinetic molecular theory of Gases, except:

Gases are made up of particles that are in constant, random, point-like motion.

_____ is the movement of a gas mixture from high to low concentration.

Diffusion

_____ is the rate at which gas is able to escape through a hole in the container.

Effusion

True or false: **Graham's law** states that, at a constant temperature and pressure, *the rates of effusion of gases are inversely proportional to the square root of their molar masses. *

True

The gas with the smaller molar mass will have the _____ rate of effusion.

Highest

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