The Gas laws

Equation of state of the gas

In 17th~18th centuries, several scientists has done some experiments on gas and found that volume, pressure and temperature of gas is all related. This is represented with few laws; Boyle's law, Charles' law and Gay-Lussac's law(law of pressures). 

Boyle's Law
Named after Robert Boyle (1627-1691)

Robert Boyle found with his experiment that when pressure on the gas is lowed, volume gets bigger. And he also found this can be plotted as straight line that go through origin. This is simplified as 

The volume of the gas is inversely proportional to its pressure
when the temperature of the body is constant.

Mathematically written as V ∝ 1/p or pV = Constant 
Therefore, for same sample of gas

p1V1 = p2V2

Charles' Law
Named after Jacques Charles (1746-1823)

Charles' Law indicates how temperature affects on the volume of a gas. Because gas liquefies when temperature is reduced, experiments couldn't obtain any results below liquefaction temperature. However, if we plot the graph and project it backwards, it was found that it intersects with temperature axis at about -273℃, aka absolute zero (0ºK).
When Celsius temperature are converted to thermodynamic temperatures, it can be said as 

With same mass of gas at constant pressure,
the volume of the gas is directly proportional to its thermodynamic temperature

Mathematically written as V ∝ T or V/T = Constant 

Therefore, for same sample of gas

V1 / T1 = V2 / T2

Gay-Lussac's Law / law of pressures
Named after Joseph Gay-Lussac (1778-1850)

With same mass of the gas at constant volume,
The pressure of the gas is directly proportional to its thermodynamic temperature.

Mathematically written as p ∝ T or pT = Constant 

Therefore, for same sample of gas

p1 / T1 = p2 / T2

Ideal Gas function

Combining these three laws, we can get

pV ∝ T

But as all of these laws is related to mass of the gas (m), we can derive it into 

pV ∝ mT

Mass of the gas can be find with number of moles of the gas

Moles(㏖) are defined as amount of atoms there are in 0.012kg of carbon-12. This is related with Avogardro Constant, Which is 6.02x10^23/㏖(meaning number of atoms per one mol) As atomic mass unit is 1.66x10^-27㎏, Multiplying these two, we can get approximately 1g (defined as relative molecular mass[1/12 of one more of carbon-12]).

for TL;DR, this is shorten to (where n is number of moles of the gas)

pV ∝ nT

This can be represented with another constant, R 

pV = nRT

or

pV = NkT

Where N is number of molecules (NOT moles!)
and k is called Boltzmann constant. R and k is related with Avogardro constant  R = kNA

However, In real life, these laws doesn't really work because its not really accurate with high air pressure. These is true only if the pressure of the sample is low enough, and gas is not under liquefaction temperature. Still, they can be define as an ideal gas.

An ideal gas, is one which obeys the equation of state
pV = nRT
at all pressures, volumes and temperature. 

List of variables/constants and units in this page

V = Volume = ㎥

p = Pressure = ㎩

T = Temperature = ºK
m = Mass = ㎏

n = Number of moles = ㏖

NA = Avogadro Contonstant = 6.02x10^23/㏖
Ar = Relative atomic mass = mass of 1/12 an atom Carbon-12
Mr = Relative molecular mass = mass of 1/12 of a mol of Carbon 12 (1g) (Ar*NA)
u = atomic mass unit = 1.66x10^-27㎏

R = molar gas constant/universal gas constant = 8.3J/K
k = Boltzmann constant = 1.38x10^-23J/K