Derivation of PV=nRT, The Equation of Ideal Gas

According to the kinetic theory of gas,

- Gases are composed of very small particles and their number of particles is very large.
- These particles 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, area of ends A. There is a single particle with speed v traveling left and right to the end of the box by colliding with the end walls.

Ideal Gas Demonstration

Ideal Gas Demonstration

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

1. t=\frac{2L}{v}

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

2. f=\frac{L}{t}=\frac{L}{2L/v}=\frac{v}{2}

Sources:
http://www.chem.ufl.edu/~itl/2045/kin_thr/kin_thr.html

Motion of Molecules

The idea that molecules are in constant motion was proposed by the kinentic theory of gases. The development of this theory in the 19th century are mostly based on the theory of atoms & molecules. Since there are no real experiments during that time, many leading physicists strongly opposed the idea. However, Brownian Motion, an observation done by botanist Robert Brown eliminated any opposition to the kinetic theory of gases.

Motion of Molecules

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:
http://en.wikipedia.org/wiki/Kinetic_theory
http://www.kutl.kyushu-u.ac.jp/seminar/MicroWorld1_E/Part1_E/P14_E/Motion_of_molecule_E.htm

Concept of the Molecule

Although the atomic theory proposed by John Dalton created a basic structure of the atom, the general idea of molecules was not cleared. In 1809, Frech chemist Joseph-Louis Gay-Lussac and others began doing numerous experiments with gases by measuring the amounts of gass that actually reacted. They found that two volumes of hydrogen reacted with one volume of oxygen to form two volumes of water, and that one volume of hydrogen gas reacted with one volume of chlorine gas to form two volumes of hydrogen chloride gas. 

 2H_{2}+O_{2}\rightarrow2H_{2}O

 H_{2}+{CL}_{2}\rightarrow2HCL

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:

\frac{V_1}{n_1}=\frac{V_2}{n_2}=Constant

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

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.  

Avogadro's\:Constant=N_A=6.0221367\times10^{23}\:mol^{-1} 

Sources:

http://science.jrank.org/pages/4410/Molecule-History.html
http://en.wikipedia.org/wiki/Avogadro’s_law
http://www.kutl.kyushu-u.ac.jp/seminar/MicroWorld1_E/Part1_E/P13_E/DiscoverMolecule_E.htm

Atomic Proposal

The Idea of the atom were first proposed by the Greek philosophers Democritus and Leucippus around 400 B.C. At that time, there is absolutely no real evidence that support this proposal. Even after 20 centuries later, no experiment was strong enough to verify the existence of the atom. 

In the 18th 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 CO1.1 or CO1.2.

Carbon Dioxide and Monoxide

Carbon Dioxide and Monoxide

Sources:
http://www.kutl.kyushu-u.ac.jp/seminar/MicroWorld1_E/Part1_E/P12_E/DiscoverAtom_E.htm

Introduction to Blackbody Radiation

- Blackbody is a theoretical object that absorbs 100% radiation.
- Carbon in graphite form absorbs all but about 3% of incoming radiation.
- At a particular temperature the black body would emit the maximum amount of energy possible for that temperature. This value is known as blackbody radiation.

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

Black Body Radiation Curves

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.

Black Body Radiation Curves

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.

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