Diffusion- All about Graham’s Law

The natural process of spontaneous mixing of two or more gases irrespective of the force of gravity is diffusion.
eg. mixing of perfume in the air.

Effusion: The spontaneous escape of a gaseous molecule from a very tiny pinhole. eg. leakage of air from football.

Graham’s law of diffusion

It states that “Under the similar condition of temperature and pressure, the rate of diffusion of a gas is inversely proportional to the square root of the density of gas”. Let r be the rate of diffusion of gas and d is the density of the gas,

r\ \alpha\ \frac{1}{\sqrt{d}}

Let r1 and r2 are the rates of diffusion of two gases having densities d1 and d2 respectively, then

r_{1}\ \alpha\ \frac{1}{\sqrt{d_{1}}}, \;\; r_{2}\ \alpha\ \frac{1}{\sqrt{d_{2}}}\\ r_{1}= \frac{k}{\sqrt{d_{1}}}, \;\; r_{2}= \frac{k}{\sqrt{d_{2}}}\\ \frac{r_{1}}{r_{2}}= \frac{\sqrt{d_{2}}}{\sqrt{d_{1}}} ----(i)

We know, molecular mass (M) = 2 x vapour density (d)

d1=M1/2 and d2 = M2/2

Putting the value of d1 and d2 in equation (i),

\frac{r_{1}}{r_{2}}=\frac{\sqrt{M_{2}}}{\sqrt{M_{1}}}----(ii)

The rate of diffusion is the volume of gas diffused per unit time.

i.e. rate\ of\ diffusion(r)= \frac{volume\ of\ gas\ diffused(V)}{Time\ of\ diffusion(t)}\\ or,\ r_{1}=\frac{V_{1}}{t_{1}} \;\; and \;\; r_{2}=\frac{V_{2}}{t_{2}}\\ or,\ \frac{r_{1}}{r_{2}}= \frac{V_{1}t_{2}}{V_{2}t_{1}} ----(iii)

Combining equations (i), (ii), and (iii), we get,

\frac{r_{1}}{r_{2}}= \frac{\sqrt{d_{2}}}{\sqrt{d_{1}}}= \frac{\sqrt{M_{2}}}{\sqrt{M_{1}}}= \frac{V_{1}t_{2}}{V_{2}t_{1}}
Applications of diffusion
  • Measure the rate of diffusion of gas
  • Determine the vapour density and molecular mass of gas
  • Separate the isotopes of gas.


Note: The rate of diffusion can also be expressed in terms of the mass of gas diffused.

\frac{\sqrt{d_{2}}}{\sqrt{d_{1}}}= \frac{V_{1}t_{2}}{V_{2}t_{1}}= \frac{m_{1}d_{2}t_{2}}{m_{2}d_{1}t_{1}}\\ \frac{\sqrt{d_{2}}}{\sqrt{d_{1}}}=\frac{m_{2}t_{1}}{m_{1}t_{2}}

Postulates of the kinetic theory of gases
  1. All gases consist of a very large number of tiny particles called molecules. All gas molecules are identical in all respects.
  2. The molecules are small and the distance between them is large. So, the volume occupied by the individual gas molecule is negligible in comparison to the total volume of the gas.
  3. The average distance between gas molecules is large and hence the force of attraction between them is negligible.
  4. The gas molecules are always moving with high velocities in all direction. During their motion, they collide with each other and also in the walls of the container.
  5. All molecular collisions are elastic. There is no loss in kinetic energy in the molecule during the collision.
  6. There is no effect of gravity on the motion of the molecules.
  7. Pressure is exerted when the gas molecules collide with each other and in the walls of the container.
  8. The average kinetic energy of gas molecules is directly proportional to the absolute temperature. i. e. KE ∝ T

Explanation of gas law from kinetic theory of gas

1. Boyle’s law: According to kinetic theory, pressure is due to the collision of the gas molecule on the wall of the container. At a constant temperature, the average kinetic energy of the molecules is fixed and the average velocity is also fixed. For a fixed mass of gas, the total number of molecules is also fixed. When the volume of gas decreases, the number of molecular collision increases and the pressure decreases.

2. Charle’s law: When a fixed mass of gas in a container of fixed volume is heated, its pressure increases. If the volume of the container is not fixed and pressure is kept constant, then on heating the container, the gas expands and the expansion continues till the pressure decreases to its original value. Hence, the net result of increasing the temperature is the increase in the volume at constant pressure.


Differences between Real and ideal gas
Real gasIdeal gas
They do not follow the ideal gas equation under all condition of temperature and pressure.They follow the ideal gas equation under all condition of temperature and pressure.
They exist in nature.They have no existence.
The volume occupied by real gas is not negligible as compared to the total volume of gas.The volume occupied by real gas is negligible as compared to the total volume of gas.
The force of attraction between real gas is not negligible.The force of attraction between real gas is negligible.
It obeys all gas laws at high temperature and low pressure.It obeys all gas laws at all temperature and pressure.
Deviation of real gas from ideal gas behaviour

Real gas nearly obeys the ideal gas equation at high temperature and low pressure. On decreasing the temperature and increasing the pressure, there is a deviation of real gas from ideal gas behaviour due to the following reasons:

  1. The volume occupied by the individual gas molecule is negligible in comparison to the total volume of the gas.
  2. The force of attraction between gas molecules is negligible.
Vander Waal’s equation for real gas
\left ( P + \frac{an^{2}}{V^{2}} \right )\left ( V-nb \right )=nRT
Solved Numerical Examples

1. What are the relative diffusion rates of CH4 and SO2? If these two gases are simultaneously introduced into opposite ends of a 100 cm tube and allowed to diffuse each other, at what distance from SO2 will the molecules of two gases meet?
solution:

Molecular mass of CH4 = 16
Molecular mass of SO2 = 64
\frac{Rate\ of\ diffusion\ of\ CH_{4}}{Rate\ of\ diffusion\ of\ SO_{2}}= \frac{\sqrt{Mol.\ wt.\ of\ CH_{4}}}{\sqrt{Mol.\ wt.\ of\ SO_{2}}}\\ =\frac{\sqrt{64}}{\sqrt{16}}= \frac{2}{1}= 2:1
diffusion numerical image
Rate\ of\ diffusion = \frac{Distance\ covered}{Time\ taken}\\ Rate\ of\ diffusion\ of\ CH_{4}= \frac{(100-x)}{t}\\ Rate\ of\ diffusion\ of\ SO_{2}= \frac{x}{t}\\ or,\ \frac{2}{1}= \frac{(100-x)}{x}\\ or,\ x=33.33cm


2. An evacuated vessel weighs 50 gm when empty, 148 gm when filled with a liquid of density of 0.98 gm/cc and 50.5 gm when filled with an ideal gas at 760 mm Hg and at 300 K. Determine the molecular mass of gas.
solution:

Density of liquid (d) =0.98 gm/cc
Weight of liquid (m) =148-50 =98 gm
Weight of ideal gas (W) = 50.5-50 = 0.5 gm
Temperature (T) = 300K
Pressure (P) = 1 atm
Volume of liquid (V) =m/d = 100 cc = 0.1 lit

Volume of an ideal gas is the same as the volume of liquid
From ideal gas equation,

PV=nRT\\ 1\times 0.1= \frac{0.5\times 0.0821\times 300}{M}\; \left [ n=\frac{m}{M} \; and\ M= molecular\ mass\right ]\\ \therefore M=123.15
Some Important Questions
  1. 5gm of hydrogen diffused through a porous membrane in 30 sec. Find the time required to diffuse the same amount of SO2 gas at identical conditions. (5.3 min)
  2. 2 gm of hydrogen is diffused from a container in 10 minutes. How many grams of oxygen would diffuse through the same container at the same time under similar conditions? (8gm)
  3. A vessel of volume 100 ml contains 10% O2 and 90% unknown gas volume. The gases diffuse in 86 sec through a small hole in the vessel. If pure oxygen under the same condition is diffused in 75 sec. Find the molecular mass of unknown gas? (43.19)
  4. How long will it take 600 ml of hydrogen gas to diffuse through a porous partition if 300 ml of oxygen is diffused through it in 10 minutes under identical conditions? (5 min)
  5. A gas X is diffused 5 times as rapidly as another gas Y. Calculate the ratio of molecular mass of X to Y. (1:25)
  6. A mixture of ozone and oxygen-containing 20% by volume of ozone is diffused through a porous plug in 172 sec, while the same volume of pure oxygen took 164 seconds to diffuse through the same plug. Calculate the relative density of ozone. (24)
  7. The rate of diffusion of saturated hydrocarbon (CnH2n+2) gas is 1.206 times that of SO2 gas under identical conditions. Find the molecular mass and value of n for the gas. (44,3)
  8. What are the relative diffusion rates of CH4 and SO2? If these two gases are simultaneously introduced into opposite ends of a 100 cm tube and allowed to diffuse each other, at what distance from the SO2 will the molecules of the two gases meet? (33.3)

References:
Mishra, AD, et al. Pioneer Chemistry. Dreamland Publication.
Mishra, AD et al. Pioneer Practical Chemistry. Dreamland Publication
Wagley, P. et al. Comprehensive Chemistry. Heritage Publisher & Distributors Pvt. Ltd.

Sharing is Caring

Search Your Notes

Like our Page

Latest Notes

Subscribe To Our Newsletter

Get updates and learn from the best