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.

ptotal = `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, ptotal = `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 ptotal 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 ptotal

p2 = x2 ptotal

p3 = x3 ptotal


Kinetic Molecular Theory of Gases


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.

Pressure Vs PV Graph of Gases Pressure Vs Volume Graph of Gases

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.

ptotal = preal + `(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)`

Deviation of Z value of Gases

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

Videal `=(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.



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