States of Matter

Dalton’s Law of Partial Pressure

“The total pressure exerted by the mixture of non-reactive gases is equal to the sum of partial pressures of individual gases.

p_total = `p_1+p_2+p_3+`…

Partial Pressure in Terms of Mole Fraction

Let us assume that at temperature T, three gases (enclosed in volume V) exert pressure p₁, p₂ and p₃ respectively. Then

`p_1=(n_1RT)/V`

`p_2=(n_2RT)/V`

`p_3=(n_3RT)/V`

Now, p_total = `p_1+p_2+p_3+`

`=p_1=(n_1RT)/V+p_2=(n_2RT)/V+p_3=(n_3RT)/V`

`=(n_1+n_2+n_3)(RT)/V`

On dividing p₁ by p_total we get

`(p_1)/(p_(t\o\ta\l))``=((n_1)/(n_1+n_2+n_3))(RTV)/(RTV)`

`=(n_1)/(n_1+n_2+n_3)=(n_1)/n=x_1`

Here, x₁ is called mole fraction of first gas.

So, p₁ = x₁ p_total

p₂ = x₂ p_total

p₃ = x₃ p_total

Kinetic Molecular Theory of Gases

Gases are composed of large number of identical particles (atoms or molecules). These particles are so small and so far apart that the actual volume of particles is negligible compared to the empty space between them. This assumption explains the great compressibility of gases.
There is no force of attraction between the particles of a gas at ordinary temperature and pressure.
Particles of gases are always in constant and random motion. Due to this, gases do not have fixed shape.
Particles of a gas move in all possible directions in straight lines. They collide with each other and with the walls of the container during their motion. Collision of particles with the walls of the container causes pressure exerted by the gas.
Collisions of gas molecules are perfectly elastic. This means that total energy of molecules remains the same before and after collision. There may be exchange of energy between colliding particles, individual energies may also change, but the sum of their energies remains constant. Had there been loss of kinetic energy, the motion of molecule would stop and gases would settle down. But it does not happen in real life.
At any particular time, different particles have different speeds and hence different kinetic energies.
The average kinetic energy of gas molecules is directly proportional to the absolute temperature. On heating the gas, kinetic energy of particles increases and they strike the walls of the container more frequently and thus exert more pressure.

Behavior of Real Gases

Deviation from Ideal Gas Behavior

When a graph of pV vs p is plotted for an ideal gas, we get a straight line that is parallel to x-axis. But same graph for real gases shows variation from ideal behavior. In case of H₂ and He, the value of pV increases with increase in pressure p. In case of methane and carbon monoxide, the value of pV reduces at first; followed by increase in its value with increase in pressure.

When a graph of pressure Vs volume is plotted, the graph for real gases show deviation from ideal gas. At high pressure, volume of real gas is more than that of ideal gas. But at low pressure, the measured and calculated volumes approach each other.

Deviation from ideal behavior happens because molecules interact with each other. Molecules of gases come very close to each other at high pressure. At this point, molecular interactions start operating. At this point, molecules do not strike the container with full impact because they are dragged back by other molecules due to attractive forces between molecules. Thus, the pressure exerted by a gas is lower than the pressure exerted by ideal gas.

p_total = p_real + `(an^2)/(V^2)`

Here, a is a constant.

At high pressure, repulsive forces also become significant because molecules are very close to each other. The repulsive forces cause the molecules to behave as small but impenetrable spheres. Now, the volume is restricted to V – nb instead of V, where nb is approximately the total volume occupied by the molecules themselves. Now, the above equation can be written as follows:

`(p+(an^2)/(V^2))(V-nb)=nR\T`

This equation is called van der Waals equation. Here, n is the number of moles. Constants a and b are called van der Waals constants and their value depends on the characteristic of the gas. Value of ‘a’ is the measure of magnitude of intermolecular attractive forces and is independent of temperature and pressure.

Intermolecular forces become significant at very low temperature as well.

The deviation from ideal behavior can be measured in terms of compressibility factor Z, which can be given by following equation.

`Z=(pV)/(nR\T)`

Z = 1 for ideal gas at all temperature and pressure.
At very low pressure, Z ∼ 1, for all gases.
Z > 1 at high pressure, for all gases.
At intermediate pressure, most gases have Z < 1

The temperature at which a real gas obeys ideal gas law over an appreciable range of pressure is called Boyle Temperature or Boyle Point. It depends on the nature of a gas. Above their Boyle point, real gases show positive deviation from ideality and Z values are greater than one. Below Boyle temperature, real gases first show decrease in Z value with increasing pressure, which reaches a minimum value. Thus, gases show ideal behavior at high temperature and low pressure.

`Z=(pV_(re\al))/(nR\T)`

If the gas shows ideal behavior then

V_ideal `=(nR\T)/p`

Substituting this value in previous equation, we get

`Z=(V_(re\al))/(V_(id\ea\l))`

It can be said that compressibility factor is the ratio of actual molar volume of a gas to the molar volume of it.