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What is Dalton’s Law of Partial Pressure?
We have so far developed the kinetic theory of gases to a certain stage. Now it is possible to see that the theoretical result of kinetic theory matches exactly with the thermodynamic gas laws obtained from experiments. This proves the success of the kinetic theory of gases.
1. Boyle’s law: According to the kinetic theory of gases, T ∝ E and E = \(\frac{3}{2}\)pV. So, pV ∝ T. If T = constant for a certain amount of gas, then pV = constant. This is Boyles law.
2. Charles’ law: According to the kinetic theory of gases, T ∝ E and E = \(\frac{3}{2}\)pV. So pV ∝ T. If p = constant for a certain amount of gas, then Voc T. This is Charles’ law.
3. Pressure law: According to the kinetic theory of gases. T ∝ E and E = \(\frac{3}{2}\)pV. So pV ∝ T. If V = constant for a certain amount of gas, then p ∝ T. This is Charles’ law of pressure.
4. Joule’s law: According to kinetic theory of gases, E = \(\frac{3}{2}\)RT. So E is a function of temperature only; it does not depend on the volume or pressure of the gas. This is Joule’s law or Mayer’s hypothesis, as discussed in the chapter First and Second Law of Thermodynamics.
5. Avogadro’s law: Let equal volumes of two gases be taken at the same pressure and temperature. The pressure, temperature and volume are p, T and V, respectively.
Now for the first gas,
N1 = total number of molecules in the container,
m1 = mass of each molecule,
c1 = rms speed of the molecules.
Then, n1 = \(\frac{N_1}{V}\) = number of molecules per unit volume.
According to kinetic theory, pressure of the gas,
p = \(\frac{1}{3}\)m1n1\(c_1^2\) = \(\frac{1}{3}\)m1\(\frac{N_1}{V} c_1^2\).
Pressure being the same, so we get similarly for the second gas,
Again, the temperature of the two gases is the same. So the average kinetic energy of a molecule is equal for the two gases. This means that
\(\frac{1}{2} m_1 c_1^2\) = \(\frac{1}{2} m_2 c_2^2\) or, \(m_1 c_1^2\) = \(m_1 c_2^2\) ……….. (2)
From relations (1) and (2), we get N1 = N2. So, equal volumes of different gases, at the same pressure and temperature, contain equal number of molecules. This is Avogadro’s law.
6. Dalton’s law of partial pressure: The pressure of a gas mixture on the walls of its container is equal to the sum of the partial pressures exerted by constituent gases separately, at same temperature as that of the mixture, provided that the gases do not react chemically with each other—this is Dalton’s law.
Let several gases be mixed in a closed container. Their densities are ρ1, ρ2, ρ3, … and the molecular rms speeds are c1, c2, c3, , respectively. Every gas molecule moves with its own kinetic energy which does not depend on the motion of the other molecules. So, the net pressure on the container will be the sum of the pressures exerted by all individual molecules. Then, the pressure of the gas mixture is
Here, p1, p2, p3. … are the partial pressures of the first, second, third, … gases on the walls of the container. So equation (3) expresses Dalton’s law of partial pressure.
7. Giaham’s law of diffusion: The rate of diffusion of a gas in a mixture is inversely proportional to the square root of the density of the gas—this is Graham’s law.
Let the densities of two gases be ρ1 and ρ2 and the rms velocities of the molecules be c1 and c2, respectively. The gases are allowed to diffuse with each other at the same temperature and same pressure. The diffusion is due to the motion of molecules; so the rate of diffusion is clearly proportional to the rms speed of the molecules.
8. Ideal gas equation: According to the kinetic theory of gases, T ∝ E and E = \(\frac{3}{2}\)pV. So, pV ∝ T. Then, pV = kT, where k is a constant. For 1 mol of an ideal gas, this constant k is called the universal gas constant, denoted by R.
So, pV = RT (1 mol of an ideal gas)
Equation (4) is known as the ideal gas equation.