Kinetic Theory of Gases

We have seen how smoke from a forest fire covers up the entire area like a blanket, though the starting point of the fire might be far away; with time it engulfs the entire area. This is the unique property of gases, which is to flow and occupy the available space. Gas is known as the third state of matter. This is that state of matter which has the capability to flow and move freely. This has the highest randomness among all the other states of matter, like solids and liquids. In this article, we will be going to learn more about the equations that govern this state of matter i.e. gases.

The randomness in gases is due to the flow of molecules in the space in which the gas has been kept and due to which we see the randomness in other words we call it disordered behaviour. This randomness is due to the motion in the molecules of gases which comes from the molecular kinetic energy.

Fig. 1. A gas disperses into the surrounding air. Gases diffuse into their surrounding region, this is why they occupy the volume they are contained in.

The kinetic theory of gases was a theory that was developed in various stages over the past few centuries. Some date its origins to the time of the ancient Greeks. Since then, a number of scientists have made contributions to the theory. In this explanation, we will introduce the kinetic theory of gases and describe and explain its key principles and assumptions.

The kinetic theory of gases: Definition

The kinetic theory of gases explains the macroscopic properties of gas — such as its pressure and its temperature—in terms of the behavior of the molecules that make it up. In physics, we study the behavior of gases, which are governed by a system of equations called gas laws.

Kinetic theory of gases principles:

To understand the properties of a gas we need to have a standard method of measuring the volume of the gases to do so we would like to introduce the concept of the mole. A mole is one of the seven SI base units of the physical measuring system.

Now we have another important question: “How many atoms or molecules are there in a mole?”

The answer to this question has been determined experimentally, which has a value of

$$N_A\;=\;6.02\;\times\;10^{23}\;\mathrm{mol^{-1}},$$

where mol⁻¹ represents the inverse mole or “per mole,” and mol is the abbreviation for mole.

This number \(N_A\) has been named Avogadro’s number after Italian scientist Amedeo Avogadro (1776–1856), who suggested that all gases occupy the same volume under the same conditions of temperature and pressure and contain the same number of atoms or molecules.

Now from Avogadro’s number, we can determine that the number of moles \(n\) which constitutes any gas is equal to the ratio of the number of molecules \(N\) in the sample to the number of molecules \(N_A\) in 1 mol as follows:

$$n=\frac{N}{N_A}.$$

It has been observed that if we put 1 mol of different gases in containers of identical volume and at the same temperature their pressure will also be equal, thus gases at lower densities follow the ideal gas equation or the perfect gas law.

$$pV\;={\;nRT}.$$

Where p is the absolute (not gauge) pressure, n is the number of moles of gas present, and T is the temperature in kelvins. The symbol R is a constant called the universal gas constant that has the same value for all gases.

The value is the gas constant \(R\;=\;8.314\mathrm{J/(mol K)}\).

It is assumed and observed that at lower density this law holds true for any specific gas because under this condition real gases tend to behave like ideal gases. This gives us another important equation that has been derived from equation 1 and can be written in terms of Boltzmann's constant \(k_B\), which is defined as:

$$\\\begin{array}{rcl}k\;=\frac R{N_A}\\k&=&\frac{8.31{\frac{\mathrm J}{\mathrm{mol}\;\mathrm K}}\;}{6.023\;\mathrm x\;10^{23}\;\frac1{\mathrm{mol}}}\\k&=&1.38\;\mathrm x\;10^{-23}\;\;{\frac{\mathrm J}{\mathrm K}}\end{array}$$

Boltzmann constant (k) = universal gas constant (R) divided by Avogadro's number (N_A ) gives us the value of the Boltzmann constant.

The equation above enables us to arrive at the following equation,

$$R\;=\;kN_A.$$

With the equation

$$n=\frac{N}{{N}_A},$$

we can write the following equation,

$$nR\;=\;Nk.$$

Substituting the above equation into the ideal gas equation we get,

$$pV\;=\;NkT.$$

Note the difference between the two expressions shown above, for the ideal gas law equation or the perfect gas law, involves the number of moles \(n\), and the equation above involves the number of molecules \(N\).)

Kinetic theory of gases assumptions:

Just like any physical theory, there are a number of assumptions that underlie the kinetic theory of gases which must be understood in order to get a comprehensive view of the kinetic theory of gases. Here are the assumptions of the kinetic theory of gases:

• The kinetic theory of gases models gases as consisting of an ensemble of a large number of very small particles. The gas molecules are assumed to be so small that the sum of the volumes of the particles is negligible compared to the volume that the gas as a whole occupies.
• The kinetic theory of gases is a statistical description of gases. We assume that we are justified in using a statistical treatment because the number of particles is so large. We often refer to this assumption as the thermodynamic limit.
• The particles that make up the gas are moving very quickly and are constantly bumping into each other. We say that these collisions are perfectly elastic which means that the total amount of kinetic energy of the particles before and after each collision is conserved and the gas particles can be considered to be rigid spheres.
• Interactions between the gas particles are negligible. The only forces experienced by the particles are due to their collisions with one another.

Fig. 3. The kinetic theory of gases models gases as consisting of a large ensemble of identical rigid spheres which are moving at a very high relative speed to one another and colliding with each other frequently, exchanging energy and momentum as they do. Wikimedia Commons

If we want to model more specific gas systems we can apply more specific assumptions such as keeping certain state variables of the gaseous system constant.

At constant temperature (Isothermal):

Suppose also that we allow the gas to expand from an initial volume V_i to a final volume V_f while we keep the temperature T of the gas constant. At a constant temperature, such a process is called an isothermal expansion (and the reverse is called an isothermal compression). The equation shown below represents an ideal gas in an isothermal process.

$$W\;=\;nRT\;ln(\frac{V_f}{V_i})$$

The symbol \(\ln\) denotes the natural logarithm, having a base \(e\).

At constant volume and constant pressure (isobaric):

Work done by gas can be a constant-volume process and a constant-pressure process. If the volume of the gas is constant, then,

$$W\;=\;p\triangle V.$$

In a constant volume scenario, the change in volume is 0 hence, the work done is zero:

$$W\;=\;0$$

(at constant volume). So the work done is zero. The volume changes while the pressure \(P\) of the gas is held constant,

$$\begin{array}{rcl}W&=&p(V_f-\;V_i\;)\;\\W&=&p\triangle V\end{array}$$

Kinetic theory of gases: formula

Here we provide a comprehensive list of the formulae that you might need to know for your studies of the topic of the kinetic theory of gases.

$$n\;=\;\frac N{N_A}$$

$$pV\;=nRT$$

$$R\;=\;kN_A$$

$$nR\;=\;Nk$$

$$pV\;=\;NkT$$

$$W\;=\;nRT\;ln(\frac{V_f}{V_i})$$

$$W\;=\;p\triangle V$$

Where \(N\) is the number of gas particles, \(P\) is the pressure of the gas, \(V\) is the volume of the gas, \(T\) is the temperature of the gas, \(n\) is the number of moles of gas, \(\Delta\;V\) is the change in volume of the gas, \(N_A\) is Avagadro's number, \(R\) is the gas constant and \(k\) is Boltzmann's constant.