# How the Rate of Effusion is Inversely Related to the Molar Mass of a Gas

Have you ever wondered why a balloon filled with helium gas deflates faster than a balloon filled with air? Or why a perfume bottle can spread its fragrance throughout a room? The answer lies in a phenomenon called effusion, which is the escape of gas molecules through a tiny hole into an evacuated space. Effusion is closely related to another phenomenon called diffusion, which is the gradual mixing of gas molecules due to their random motion. Both effusion and diffusion depend on the speed and mass of the gas molecules, and they are governed by a simple law known as Graham’s law.

### What is Graham’s Law?

Graham’s law was named after Thomas Graham, a Scottish chemist who studied the behavior of gases in the 19th century. He discovered that the rate of effusion or diffusion of a gas is inversely proportional to the square root of its molar mass. In other words, lighter gas molecules tend to move faster and escape or mix more quickly than heavier gas molecules. The mathematical expression of Graham’s law is:

��������=������������fracrA​rB​=sqrtfracMB​MA​

where ��rA​ and ��rB​ are the rates of effusion or diffusion of gases A and B, and ��MA​ and ��MB​ are their molar masses.

### How to Apply Graham’s Law?

Graham’s law can be used to compare the rates of effusion or diffusion of different gases under the same conditions of temperature and pressure. For example, if we want to compare the rates of effusion of helium (He) and ethylene oxide (C2H4O) through a small hole in a balloon, we can use Graham’s law as follows:

���������2�4�=����������2�4����fracrHe​rC2H4O​=sqrtfracMC2H4O​MHe​

The molar mass of helium is 4.00 g/mol, and the molar mass of ethylene oxide is 44.0 g/mol. Plugging these values into the equation, we get:

���������2�4�=��������44.04.00=����11������3.32fracrHe​rC2H4O​=sqrtfrac44.04.00=sqrt11approx3.32

This means that helium effuses through the hole about 3.32 times faster than ethylene oxide. Therefore, the balloon filled with helium will deflate much faster than the balloon filled with ethylene oxide.

Graham’s law can also be used to calculate the molar mass of an unknown gas if its rate of effusion or diffusion is known relative to a reference gas with a known molar mass. For example, if we want to find the molar mass of an unknown gas that diffuses 1.5 times slower than oxygen (O2) at the same temperature and pressure, we can use Graham’s law as follows:

��������������2=����������2��������fracrunknown​rO2​=sqrtfracMO2​Munknown​

The molar mass of oxygen is 32.0 g/mol. Solving for ��������Munknown​, we get:

��������=������2(��������������2)2=����32.0(����11.5)2=����32.00.444������72.1�����/���Munknown​=fracMO2​(fracrunknown​rO2​)2=frac32.0(frac11.5)2=frac32.00.444approx72.1textg/mol

Therefore, the molar mass of the unknown gas is about 72.1 g/mol.

### Why Does Graham’s Law Work?

Graham’s law is based on the kinetic molecular theory of gases, which assumes that gas molecules are tiny particles that move randomly in all directions with different speeds. The theory also assumes that gas molecules have negligible size and volume compared to the container they occupy, and that they do not interact with each other except during elastic collisions.

According to the kinetic molecular theory, the average kinetic energy of gas molecules is proportional to their absolute temperature in kelvins. This means that all gases at the same temperature have the same average kinetic energy, regardless of their molar mass or identity. The kinetic energy of a moving particle is given by:

��=����12��2KE​=frac12mv2

where �m is the mass and �v is the speed of the particle.

Since all gas molecules have the same average kinetic energy at a given temperature, we can equate the kinetic energies of two different gases A and B as follows:

����12����2=����12����2frac12mA​vA2​=frac12mB​vB2​

where ��mA​ and ��mB​ are the masses and ��vA​ and ��vB​ are the speeds of gas molecules A and B.

Solving for the ratio of the speeds, we get:

��������=������������fracvA​vB​=sqrtfracmB​mA​

This equation shows that the speed of a gas molecule is inversely proportional to the square root of its mass. Therefore, lighter gas molecules tend to move faster than heavier gas molecules at the same temperature.

The rate of effusion or diffusion of a gas is directly proportional to its speed, because faster gas molecules can escape or mix more quickly than slower gas molecules. Therefore, the rate of effusion or diffusion of a gas is inversely proportional to the square root of its mass, as stated by Graham’s law.

### What are the Applications and Limitations of Graham’s Law?

Graham’s law has many practical applications in science and engineering. For example, Graham’s law can be used to:

• Separate isotopes of an element by effusion through a porous membrane. This is how uranium-235 is enriched from natural uranium, which contains mostly uranium-238.
• Determine the molecular weight or identity of an unknown gas by comparing its rate of effusion or diffusion with a known gas.
• Estimate the rate of leakage of a gas from a container or a pipeline.
• Explain why some gases have stronger odors than others. For instance, ammonia (NH3) has a lower molar mass than hydrogen sulfide (H2S), so it diffuses faster and reaches our noses more quickly.

However, Graham’s law also has some limitations and assumptions that may not always be valid. For example, Graham’s law assumes that:

• The hole through which the gas effuses is very small compared to the mean free path of the gas molecules, which is the average distance they travel before colliding with each other or the walls of the container. If the hole is too large, the rate of effusion may not follow Graham’s law exactly.
• The gas molecules behave as ideal gases, which means that they have negligible size and volume, and that they do not interact with each other except during elastic collisions. If the gas molecules are too large or polar, they may experience intermolecular forces that affect their speed and rate of effusion or diffusion.
• The temperature and pressure of the gas are constant and uniform throughout the container. If there are temperature or pressure gradients in the container, they may cause convection currents that affect the rate of effusion or diffusion.

Therefore, Graham’s law is only an approximation that works well for most gases under normal conditions, but it may not be accurate for all gases under all circumstances.

### Conclusion

Graham’s law is a simple and useful law that relates the rate of effusion or diffusion of a gas to its molar mass. It states that lighter gas molecules tend to move faster and escape or mix more quickly than heavier gas molecules at the same temperature and pressure. Graham’s law can be derived from the kinetic molecular theory of gases, which assumes that all gases at the same temperature have the same average kinetic energy. Graham’s law has many applications in science and engineering, but it also has some limitations and assumptions that may not always be valid. Therefore, Graham’s law should be applied with caution and understanding.