- Gases are composed of very small molecules and their number of molecules is very large.
- These molecules are elastic.
- They are negligible size compare to their container.
- Their thermal motions are random.

To begin, let’s visualize a rectangular box with length L, areas of ends A₁ and A₂. There is a single molecule with speed v_x traveling left and right to the end of the box by colliding with the end walls.

3D Demonstration of Ideal Gas

The time between collisions with the wall is the distance of travel between wall collisions divided by the speed.

1.

The frequency of collisions with the wall in collisions per second is

2.

According to Newton, force is the time rate of change of the momentum

3.

The momentum change is equal to the momentum after collision minus the momentum before collision. Since we consider the momentum after collision to be mv, the momentum before collision should be in opposite direction and therefore equal to -mv.

4.

The average force on the wall is equal to the momentum change per collision times the frequency of collisions.

5.

The pressure, P, exerted by a single molecule is the average force per unit area, A. Also V=AL which is the volume of the rectangular box.

6.

Let’s say that we have N molecules of gas traveling on the x-axis. The pressure will be

7.

To simplify the situation we will take the mean square speed of N number of molecules instead of summing up individual molecules. Therefore, equation #7 will become

8. 

Earlier we are trying to simplify the situation by only considering that a molecule with mass m is traveling on the x axis.  However, the real world is much more complicated than that. To make a more accurate derivation we need to account all 3 possible components of the particle’s speed, v_x, v_y and v_z.

9.

Since there are a large number of molecules we can assume that there are equal numbers of molecules moving in each of co-ordinate directions.

10.

Our final equation becomes

11.

However to simplify the equation further, we define the temperature, T, as a measure of thermal motion of gas particles. The only energy involve in this model is kinetic energy.

12.

To combine the two equations we solve kinetic energy equation #12 for mv²

13.

Since the temperature can be obtained easily with simple measurement. we will now replace the result of kinetic equation #13 with with a constant R times the temperature, T

14.

Because a molecule is too small and therefore impractical we will take the number of molecules and divide by the Avogadro’s number, N_A

15.

Sources:
1. Significant of PV=nRT
2. Kinetic Theory of Gas

Motion of Molecules

According to the Theory:

1. The gas consists of very small particles, each of which has a mass or weight in SI units.

2. The number of molecules is large such that statistical treatment can be applied.

3. Molecules are in constant and random motion.

4. The rapidly moving particles constantly collide with each other and with the walls of the container.

5. The  collisions of gas particles with the walls of the container holding them are perfectly elastic.

6. The interactions among molecules are negligible. They exert no forces on one another except during collisions.

7. The total volume of the individual gas molecules added up is negligible compared to the volume of the container.

8.  The molecules are perfectly spherical in shape, and elastic in nature.

9. The average kinentic energy of the gas particles depends only on the temperature of the system.

10. The time during collision of molecule with the container’s wall is negligible as comparable to time between successive collisions.

11. The equation of motion of the molecules are time-reversible.

Sources:
1. Kinetic Theory
2. The Motion of Molecules

In 1811, Avogadro proposed the following law:

“Equal volumes of ideal or perfect gases, at the same temperature and pressure, contain the same number of particles, or molecules.”

This Law is later confirmed experimentally. With the basis of Avogadro’s Laws, it became possible to compare the relative weights of various melecules and atoms. 

According to Avogadro’s Law:

n: Number of moles, V: Volume, T: Temperature (Constant), P: Pressure (Constant)

Example: The reaction in which hydrogen and oxygen combine to form water can be displayed as the following.

Water Molecules Formation

Avogadro’s Constant
The number of molecules in one mole, that is the number of atoms in exactly 12 grams of carbon-12.  

Sources:
1. Molecules History
2. Avogadro’s Law
3. Introduction of the Concept of the Molecules

In the 18^th Century, the first scientific data on the atom were gathered by A. L. Lavoisier and others from quantitative measurements of chemical reactions. From the experiment, he suggested that there exist some elements which could not be disintegrated into any smaller composition by usual chemical method. He definied this as chemical element. 

From the results of Lavoisier experiments, John Dalton proposed the first systematic atomic theory. This theory of the atom compose of two basic chemcial laws: the law of constant proportions and the law of multiple proportions. 

Law Of Constant Proportions
“The composition of a pure chemical compound is independent of its method of preparation”

Example: Water is a compound of hydrogen and oxygen. The ratio of the weight of hydrogen to oxygen in water is fixed at the value 1:8, independent of how it is formed. 

Law Of Constant Proportions
“When two elements A and B combine to form more than one compound, the weights of B which combine with a fixed weight of A are in the proportion of small whole numbers (integers)”.

Example: Carbon and oxygen react to form CO or CO2 but not CO_1.1 or CO_1.2.

Carbon Dioxide and Monoxide

Sources:
1. The discovery of the Atom

- Maximum wavelength emitted by a black body radiator is infinite
- Blackbody emits a definite amount of energy at each wavelength for a particular temperature.

Fig 1. Black Body Radiation Plot

- According to Figure 1, the curve touches the x-axis at infinite wavelength. This shown that blackbody does radiate energy at every wavelength.
- At 5800K the peak wavelength (wavelength that emits most energy) is about 5e-7 m.

Fig 2. Black Body Radiation Curves

- Figure 2 shows how the blackbody radiation curves change at various temperatures.
- As the temperature increases, the peak wavelength emitted by the blackbody decreases.
- As the temperature increases, the total energy emitted increases, because the total area under the curve increases.

In the early 1900, four important events had revolutionized science and the way we think about the world around us. They are Planck’s solution to the blackbody radiation, Einstein’s explanation of the photoelectric effect, the Bohr model of the atom, and the de Broglie wavelength of material particles

In the early 19^th century, most phenomenons can be explained by classical (or Newtonian) mechanics. By the turn of the century, however, many anomalies instances spawned that can only be explained through Relativity and Quantum Mechanics. Relativity was discovered mainly to describe physics of very massive and very fast objects. Quantum Mechanics was developed to describe the physics of very small objects.

Both Quantum Mechanics and Relativity contradict Newtonian Mechanics which make it very difficult to comprehend. However, both produce results equivalence to Newtonian results when applied to our physical world.