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  • Kinetic theory of gases

"All matter is composed of molecules (atoms) that are in continuous chaotic thermal motion".

The Kinetic Theory of Gases relate the macroscopic properties of the gas (pressure, volume, temperature) to the microscopic properties (mechanical energy of the molecules that make up the gas).

The Microscopic Model of Matter

The currently accepted model assumes that:
  • All matter is made of very small particles (atoms).
  • The constituent particles of matter exert forces of interaction between them.
  • The number of particles is very large.
  • The movement of the particles is: continuous and random.

Microscopic Model of Perfect Gas

In this model, the forces of interaction between molecules are considered negligible except during collisions. That is, we only consider the forces of contact. It is also considered that the average distance between atoms and molecules is often greater than the size of the molecules themselves. The monatomic gases are well described by this model, since it is possible to consider that their molecules have mass but not volume. (Perfect gas and ideal gas are used interchangeably in this text.)

Ideal gas confined in a container. The accepted scientific model for a gas is illustrated in the figure. It is considered that the molecules that make up the gas (blue balls) are always in motion and in random directions. It is also considered that the molecules are so small that they occupy virtually no volume, but here they are represented larger for the sake of visualization.

Definitions for the study of gases are:

It is the smallest particle of a chemical element. The combination of them generates the different substances.
It is the smallest stable particle that has the basic chemical properties of a given substance.
Molecular Mass \((M)\)
It is also called relative mass. It is the sum of the atomic mass of each atom of the molecule. The atomic mass unit is Dalton and its symbol is the \(u\), \([M] = u\).
Number of mol
The number \(n\) molecules (mol) of a mass quantity \(m\) can be found using the formula: $$n = \frac{m}{M},$$ where \(M\) is the molecular weight.
Boltzmann Constant \((k_B)\)
\(k_B = \frac{R}{N_A} = 1.38 \times 10^{-23} J/K\)
Gas Constant \((R)\)
\(R = 8.31~J/mol\)

Molecular Model of an Ideal Gas Pressure

In our model, the pressure exerted by a gas on a given surface is due to the collisions of the gas molecules against this surface. As it would be impossible to treat each molecule separately from the gas, in this case it is important to work with averages such as:

Average Kinetic Energy of a Molecule
$$E_{k_m} = \frac{1}{2} M \left\langle v^{2} \right\rangle,$$ where \(M\) is the mass of a molecule and \(\left\langle v^{2} \right\rangle\) is the quadratic average speed;
Root Mean Square Speed \((V_{rms})\)
$$v_{rms}=\sqrt{\frac{v_{1}^{2}+v_{2}^{2}+\cdots+v_{n}^{2}}{N}} \text{, where:}$$ $$v_{1}^{2}+v_{2}^{2}+\cdots+v_{n}^{2}$$ is the sum of the velocities of all the molecules of the system.
The pressure due to the \(n\) molecules of an ideal gas contained in a volume \(V\) is given by: $$P = \left ( \frac{2}{3} \frac{N}{V} \right ). \frac{1}{2} M \left\langle v^{2} \right\rangle.$$ Note that this pressure is directly proportional to the number of molecule and the kinetic energy of them. As more energetic the molecules get, more collisions happen at the walls, and greater the pressure will be. This pressure is inversely proportional to the volume, since in a bigger volume, greater is the space the molecules have to travel, and fewer collisions will happen with the walls.

Molecular Interpretation of Temperature

Temperature is a measure of the average molecular kinetic energy.

Heavier molecules move at a slower speed than lighter molecules; that is, at the same temperature, the lighter the faster. The absolute temperature of an ideal gas is related to the average kinetic energy of translation per molecule, through the expression: $$T = \frac{2}{3} \frac{E_{k_m}}{k_B},$$ where \(k_B\) is the Boltzmann constant and \(E_{k_m}\) is the energy of the center of mass.

This means that different gases at the same temperature have the same average kinetic energy per molecule.

Average Kinetic Energy Per Molecule
The average kinetic energy per molecule is independent of the nature of the gas and is given by the formula: $$E_{K_m} = \frac{3}{2} k_B T,$$ where \(T\) is the gas temperature and \(k_B\) is the Boltzmann constant. At a given temperature \(T\), all gas molecules, regardless of their mass, have the same average translational kinetic energy. When we measure the temperature of a gas, we are measuring the average translational kinetic energy of its molecules.

Joule the Ideal Gas Law

"The internal energy of a given quantity of ideal gas is the exclusive function of its temperature."

For a gas in a transition between two thermodynamic states, internal energy change \((\Delta U)\) is always accompanied by temperature change \((\Delta T)\). The total energy \(U\) of \(N\) molecules (or \(n\) mol) of a monatomic gas is: $$U = \frac{3}{2} N k_B T,$$ or $$U = \frac{3}{2} n R T.$$

The variation of the internal energy of \(n\) mols of any ideal gas that undergoes a temperature variation \(\Delta T\), for any process, is: $$\Delta U = n~C_V~\Delta T,$$ where \(C_v\) is the molar heat capacity at constant volume. And its value depends on the type of molecule:


Also, the thermal capacity at constant pressure \(C_p\) can be defined. For an ideal gas, it is related to the thermal capacity at constant volume: $$C_P - C_V = R.$$

Molecular Kinetic Interpretation of Some Quantities

In short, we can interpret different magnitudes as follow:
Continuous effect of molecular collisions on the surface.
Thermal energy
Overall mechanical energy associated with molecular agitation.
Work on molecular scale (molecule to molecule interaction on conduction, photon to molecule interaction on irradiation)
Proportional to the average translational kinetic energy of molecules.
Compressibility (expandability)
The intermolecular space can be reduced or increased with a change of temperature, pressure and volume.
Random motion of molecules.
When molecules approach one another, molecular interaction becomes important and molecules become more cohesive.
The fastest molecules within the liquid can "escape" from the region occupied by the liquid.
Vapor pressure (maximum pressure)
Dynamic balance between the molecules escaping from the liquid and the ones who rejoin it.


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