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Diffusion and effusion are two related processes that allow us to understand the behavior of gases and matter in general at the molecular level. Effusion is governed quite exactly by Graham’s law, but it also allows an adequate (albeit approximate) description of the diffusion process, providing a model that explains why some gases diffuse more rapidly than others.
What is diffusion?
Diffusion is the movement of particles through space following their concentration gradient . That is to say, it is about the displacement of any type of particle, be it a gas or a solute in solution, from a region where its concentration is higher to another where its concentration is lower. Diffusion is a process of great importance in many scientific contexts, including chemistry, physics and biology.
What is the effusion?
Effusion is the process by which a gas passes from one compartment or container to another through a small hole or orifice . For the process to be considered an outpouring, the diameter of the hole must be considerably less than the mean free path of the gas particle. This mean path refers to the average distance that a particle can travel in a straight line without colliding with another particle under given conditions of temperature and pressure.
Effusion is the process by which, for example, a helium-filled balloon spontaneously deflates over time, or by which a sealed soft drink loses almost all of its carbon dioxide after a few years, despite being sealed. “hermetically”.
Graham’s law of effusion
The Scottish physicist Thomas Graham studied the effusion process in 1846 and experimentally determined that the rate of effusion of any gas is inversely proportional to the square root of the mass of its particles. This can be expressed as:
Where r represents the rate of effusion through a small hole or pore and MM corresponds to the molar mass of the gas (the letter r stands for rate in English, which is called rate ). This law of proportionality became known as Graham’s law or equation of effusion, although it is also often called Graham’s law or equation of diffusion because it also applies to this phenomenon.
The effusion rate ( r ) indicates the number of particles that pass through the pore or hole per unit of time. In the case of effusion through a porous surface, in which there are millions of tiny pores, the effusion rate may refer to the total number of particles (or mass of gas) that pass through the porous surface per unit area and per unit area. time unit. In the context of diffusion, r indicates the rate of diffusion and represents the amount of gas that diffuses per unit area and per unit time.
Ratio of the rates of effusion or diffusion of two gases
Graham’s formula can also be expressed in a different way to relate the effusion rates of two different gases under the same conditions. This makes it possible to compare, for example, which of the two gases escapes faster when both are contained in the same container with a porous surface. In this case, Graham’s law is written like this:
What this equation indicates is that, between two gases that are in the same conditions, the one with the lighter particles will escape more quickly. Furthermore, the ratio of the effusion rates varies as a function of the square root of the masses of the particles. That is, if a gas is 4 times heavier than another, then it will diffuse at half the rate.
Explanation of Graham’s law of diffusion and effusion
Graham’s law is an empirical law that was originally established based on experimental observations. In other words, it is the mathematical expression that relates the effusion rate to the mass of the particles. However, the development of the kinetic theory of gases allowed us to understand the origin of Graham’s formula, that is, this model explains why (ideal) gases comply with said equation.
Using a hard sphere model in which gases only collide through elastic collisions, it was determined that the effusion rate depends on the velocity of movement of the particles, and this, in turn, is inversely proportional to the square root. of its mass.
Applications of Graham’s Law of Diffusion and Effusion
Gas isotope enrichment
Graham’s law has two very important fields of application. On the one hand, it allowed the development of enrichment or purification systems based exclusively on the molecular weight of the gases. When passing a mixture of gases through a column with porous walls, all the gases in the mixture will tend to escape through the pores, but the lighter particles will do so faster than the heavier ones, so the gas mixture that escapes will be richer in these light particles.
This is the operating principle of the uranium-235 enrichment system that was used in the Manhattan Project for the manufacture of the first atomic bomb. To be usable in the bomb, uranium-235 must be enriched to a concentration much higher than the 0.7% that natural uranium contains.
To purify this isotope, all the uranium in a sample is transformed into the volatile compound uranium hexafluoride (UF 6 ), which is vaporized and the gaseous mixture is passed through a cascade of porous columns. Since 235 UF 6 is lighter than 238 UF 6 , the former diffuses faster than the latter (following Graham’s law) and the mixture ends up slightly enriched with uranium-235 after each pass through a column.
Determination of molecular weights
Another application of Graham’s equation is in the experimental determination of molecular weights or masses. If we have a mixture of a known and an unknown gas and we pass it through a porous column, the resulting mixture will be enriched in lighter gas. This enrichment is determined by the ratio between the effusion rates of the two gases. Since Graham’s formula relates these rates to the ratio of molar masses, knowing the molar mass of one of them can use Graham’s equation to calculate the molar mass of the unknown gas.
Examples of calculations with Graham’s law of diffusion and effusion
uranium enrichment.
Statement:
Knowing that the relative atomic mass of uranium-235 is 235.04 and that of uranium-238 is 238.05, and that the average atomic mass of fluorine is 18.998, determine the relationship between the effusion rates of 235 UF 6 and the 238 UF6 . _
Solution:
Since we are determining the relationship between two effusion rates, we will use Graham’s equation. To do this, we first need to calculate the molar masses of both gases.
Using these values, we can determine the relationship between the effusion rates:
This result indicates that each time a mixture of these two gases is passed through a porous column, the resulting gas mixture (the one that escapes through the pores) will contain a relative concentration 1.0043 times greater than it was before.
Determination of the molar mass of an unknown gas.
Statement:
Suppose we have an equimolar mixture of two gases. One is carbon dioxide (MM=44 g/mol) and the other is an unknown gas (MM=?). If carbon dioxide diffuses 3 times faster than the unknown gas, determine the molar mass of the unknown gas.
Solution:
In this case, we know the relationship between the two effusion rates, since by saying that carbon dioxide diffuses 3 times faster, what is meant is that its diffusion (or effusion) rate is:
Now, applying Graham’s law, we can determine the molar mass of the unknown gas:
Solving this equation, we get:
Therefore, the molar mass of the unknown gas is 76.21 g/mol.
References
Internet Academy. (2018, September 3). Graham’s Law, Gas Diffusion Law [Video]. Youtube. https://www.youtube.com/watch?v=Fd-a35TPfs0
Atkins, P., & dePaula, J. (2010). Atkins. Physical Chemistry (8th ed .). Panamerican Medical Editorial.
Diffusion . (2021, March 22). BYJUS. https://byjus.com/biology/diffusion/
Graham’s Laws of Diffusion and Effusion . (September 1, 2020). https://chem.libretexts.org/@go/page/41411
Lumen Learning. (nd). 8.4: Effusion and Diffusion of Gases | General College Chemistry I. Courses Lumenlearning. https://courses.lumenlearning.com/suny-mcc-chemistryformajors-1/chapter/effusion-and-diffusion-of-gases/
Graham’s Law | Effusion and Diffusion of Gases . Chemistry-Organic. Available at https://www.quimica-organica.com/ley-de-graham/ .
