Deductions from Kinetic Theory

 

Charles' Law

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At a given pressure, the volume of a given mass of a gas is proportional to its absolute temperature. This is Charles' law.

From (a), if p is constant in expression (a),

we get , which is Charles' law

Charles' Law of Pressure

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Regnault's or Gay-Lussac's Law

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At a given volume, the pressure of a given mass of a gas is proportional to its absolute temperature. This is Charle's law for pressure.

This is the definition of the absolute temperature, T. If one starts from

the fact that µ T and uses the fact that V is constant, one gets from (a),

Avogadro's Law

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At the same temperature and pressure, equal volumes of all gases contain equal number of molecules. This is Avogadro's law.

Consider equal volumes of two gases kept at the same pressure and temperature. Let,

m₁ = mass of a molecule of the first gas

m₂ = mass of a molecule of the second gas

N₁ = number of molecules of the first gas

N₂ = number of molecules of the second gas

p = common pressure of the two gases

V = common volume of the two gases

From equation (vi)

where v₁ and v₂ are rms speeds of the molecules of the first and the second gas respectively. Thus,

-----(b)

As the temperatures of the gases are the same, the average kinetic energy per molecule is same for the two gases (equation ix), i.e.,

From (b) and (c),

N₁ = N₂, which proves Avogadro's law.

Graham's Law of Diffusion

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When two gases at the same pressure and temperature diffuse into each other, the rate of diffusion of each gas is inversely proportional to the square root of the density of the gas. This is Graham's law of diffusion.

It is reasonable to assume that the rate of diffusion is proportional to the rms speed of the molecules of the gas. Then, if r₁ and r₂ be the rates of diffusion of the two gases,

-----(d)

From equation (iv),

If the pressure of the two gases are the same,

So that from (d)

Dalton's Law of Partial Pressure

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Dalton's law of partial pressure says that the pressure exerted by a mixture of several gases or vapors (which do not interact in any) equals the sum of the pressures exerted by each gas occupying the same volume as that of the mixture.

In Kinetic theory, we assume that the pressure exerted by a gas on the walls of a container is due to the collisions of the molecules with the walls. The total force on the wall is the sum of the forces exerted by the individual molecules. Suppose there are N₁ molecules of gas 1, N₂ molecules of gas 2, etc., in the mixture.

The force on a wall of surface area A is,

F = force exerted by N₁ molecules of gas 1

+  force exerted by N₂ molecules of gas 2

+…… = F₁ + F₂ + …….

Thus, the pressure is

If the first gas alone is in the container, its N₁ molecules exerts a force F₁ on the wall. If the pressure in this case is p₁,

It is the same for other gases.

Ideal Gas Equation

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Consider a sample of an ideal gas at pressure , volume V and temperature T. Let m be the mass of each molecule and v the rms speed of the molecules. Also, let v_tr be the rms speed of the gas at the triple point 273.16K.

Note : The international committee on weights and measures fixed triple point as a reference point for constructing a temperature scale according to which it that value of pressure and temperature at which water consists in equilibrium in all three states of matter.

---

From equation (vi),

-----(1)

And from equation (viii)

Substituting this expression for v² in (1),

Now is the average kinetic energy of a molecule at the triple point 273.16K. As the average kinetic energy of a molecule is the same for all gases at a fixed temperature (equation ix), is an universal constant. Accordingly, the quantity in the brackets, in equation (2), is also a universal constant. Writing this constant as k, equation (2) becomes,

The universal constant k is the Boltzmann constant and its value is

upto three significant digits. If the gas contains n moles, the number of molecules is

Using this, equation (x) becomes

where R = N_A k, is another universal constant known as the universal gas constant. Its value is

R = 8.314 J/mol-K

Equation (xi) is known as the equation of state of an ideal gas.