The kinetic gas equation is given as:
PV = 1/3 mnc2
P = pressure
V = volume
M = mass of molecule
n = no. of molecules present in the given amount of gas
c = root mean square speed.
Kinetic energy for one mole of gas is given as c = c1 + c2 + … + cn /n
The average kinetic energy per molecule is = average K.E per mole / NA
\(\overline{K.E}\)= 3/2 RT/NA = 3/2 kT
Where k = R/NA and is known as the Boltzmann constant.
The total kinetic energy for 1 mol of the gas is
Etotal = NA.\(\overline{K.E}\)= 3/2 RT
Deduction of gas laws from kinetic gas equation:
The ideal gas laws can be derived from the kinetic theory of gases which is based on the following two important assumptions:
- The volume occupied by the molecules is negligible in comparison to the total volume of the gas
- The molecules exert no forces of attraction upon one another.
1. Deriving Boyle’s law:
From kinetic theory
PV = 1/3 mnc2
At constant temperature and at fixed amount of gas, c is constant.
Therefore, PV = constant … Boyle’s law
2. Deriving Charle’s law:
PV = 1/3 mnc2
At constant pressure and amount,
V ∝ c2
∵ c2 ∝ T
∴ V ∝ T … Charle’s law
3. Deriving Gay Lussac’s law:
PV = 1/3 mnc²
At constant volume and amount of gas
P ∝ c²
∵ c² ∝ T
∴ P ∝ T
4. Deriving Avogadro’s law:
Deriving Avogadro’s Maxwell showed that the average kinetic energies of molecules are equal at the same temperature, that is:
½ [m1c12] = kT = ½ [m2c22] and so [m1c12] = [m2c22]
But P1V1 = 1/3 [m1n1c12] and P2V2 = 1/3 [m2n2c22]
Now if P1 = P2 and V1 = V2, [m1n1c12] = [m2n2c22]
Therefore: n1 = n2 … Avogadro’s law
5. Deriving Dalton’s law of partial pressures:
For a mixture of gases:
PV = 1/3 ([m1n1c12] + 1/3 [m2n2c22] + …)
P = 1/3 ([m1n1c12]/V + 1/3 [m2n2c22]/V + …) = P1 + P2 + … Where P1, P2 … are the partial pressures of the gases, and this is Dalton’s law (the sum of the partial pressures of all the gases occupying a given volume is equal to the total pressure).
P = P1 + P2 + …