Discuss the ideal gas laws.

Explanation

Boyle.s law: For a given gas at low pressure (density) kept in a container of volume V experiments revealed the following information.
When the gas is kept at constant temperature, the pressure of the gas is inversely proportional to the volume `P prop (1)/(V)`
Charles. law: When the gas is kept at constant pressure, the volume of the gas is direaly proportional to absolute temperature `V prop T`.
By combining these two equations we have
`PV = CT.` Here C is a positive constant.
We can infer that C is proportional to the number of particles in the gas container by considering the following argument, If we take two containers of same type of gas with same volume V, same pressure P and same temperature T, then the gas in each container obeys the above equation. `PV = CT`. If the two containers of gas is considered as a single system, then the pressure and temperature of this combined system will be same but volume will be twice and number of particles will also be double.

For this combined system, V becomes 2V, so C should also double to match with the ideal `(P(2V))/(T)=2C`. It implies that C must depend on the number of particles in the gas equation gas and also should have the dimension of `[(PV)/(T)]=JK^(-1)`, So we can write the constant C as k times the number of particles N.
Here k is the Boltzmann constant `(1.381 xx 10^(-23)JK^(-1))`and it is found to be a universal constant So the ideal gas law can be stated as follows
`PV = NkT" ...(1)"`
The equation (1) can also be expressed in terms of mole.
Suppose if a gas contains `mu` mole of particles then the total number of particles can be written as
`N = muN_(A)" ...(2)"`
where `N_(A)` is Avogadro number `(6.023 xx 10^(23)" mol"^(-1))`
Substituting for N from equation (2), the equation (1) becomes `PV = muN_(A) kT`. Here `N_(A)k=R` called universal gas constant and its value is `8.314 J//mol`. K
So the ideal gas law can be written for `mu` mole of gas as
`PV = muRT" ...(3)"`
This is called the equation of state for an ideal gas. It relates the pressure, volume and temperature of thermodynamie system at equilibrium.