Effusion is a physical process where a gas confined in a container escapes through a tiny opening, often called a pinhole, into a region of lower pressure, typically a vacuum. The rate at which the gas escapes is a measurable quantity tied directly to the properties of the gas molecules. This rate is influenced by how fast the molecules are moving and how frequently they collide with the container wall. Since all gas molecules at a given temperature have the same average kinetic energy, the speed of the gas is the primary determining factor for its effusion rate.
Understanding Graham’s Law of Effusion
The relationship between a gas’s properties and its escape rate is defined by Graham’s Law of Effusion. Thomas Graham, a Scottish chemist, discovered that the speed at which a gas effuses is systematically related to its molecular weight. This principle establishes that gases composed of lighter molecules will consistently effuse at a faster rate than gases made of heavier molecules, provided the temperature and pressure remain the same for both.
This behavior is rooted in the kinetic molecular theory, which explains that at a constant temperature, all gas molecules possess the same average kinetic energy. Since kinetic energy is calculated using both mass and velocity, a molecule with a smaller mass must have a greater average velocity to maintain the same energy level as a heavier molecule. The law quantifies this by stating that the rate of effusion is inversely proportional to the square root of the gas’s molar mass. Therefore, a gas four times heavier than another will only effuse half as fast, demonstrating a specific mathematical dependence on mass.
The Formula for Calculating Effusion Rate
To calculate the relative rates of effusion for two different gases, Gas A and Gas B, the Graham’s Law formula is used. This mathematical tool allows for a direct comparison of how much faster one gas will escape compared to the other under identical conditions. The relationship is expressed as a ratio of the two rates, which is set equal to the square root of the inverse ratio of their molar masses.
The formula is written as: \(\frac{\text{Rate}_A}{\text{Rate}_B} = \sqrt{\frac{Molar Mass_B}{Molar Mass_A}}\).
In this equation, \(\text{Rate}_A\) and \(\text{Rate}_B\) represent the effusion rates, typically measured in units like moles per second or volume per second. \(Molar Mass_A\) and \(Molar Mass_B\) are the respective molar masses of the gases, often expressed in grams per mole (\(\text{g/mol}\)). The inversion of the molar masses mathematically reflects the inverse relationship: the gas with the lower molar mass (Gas A) corresponds to the higher effusion rate. This calculation yields a numerical value representing the factor by which one gas effuses faster than the other.
Step-by-Step Calculation Example
A practical way to apply Graham’s Law is to compare the effusion rates of two common noble gases, Helium (He) and Neon (Ne). Helium is a light gas with a molar mass of approximately \(4.00 \text{ g/mol}\), while Neon is heavier, having a molar mass of about \(20.18 \text{ g/mol}\). We want to determine how much faster Helium will effuse compared to Neon when both are held at the same temperature.
The first step is to correctly assign the variables in the Graham’s Law formula. We will let Gas A be Helium and Gas B be Neon, because we expect the lighter Helium to have the faster rate. Substituting the known molar masses into the formula gives the setup for the calculation.
The equation becomes: \(\frac{\text{Rate}_{\text{He}}}{\text{Rate}_{\text{Ne}}} = \sqrt{\frac{Molar Mass_{\text{Ne}}}{Molar Mass_{\text{He}}}}\).
Next, we input the numerical values for the molar masses: \(\frac{\text{Rate}_{\text{He}}}{\text{Rate}_{\text{Ne}}} = \sqrt{\frac{20.18 \text{ g/mol}}{4.00 \text{ g/mol}}}\). Carrying out the division inside the square root yields a ratio of \(5.045\).
The final step is to calculate the square root of this mass ratio to find the rate ratio. Taking the square root of \(5.045\) gives a value of approximately \(2.25\). This result means that the rate of effusion for Helium is \(2.25\) times greater than the rate of effusion for Neon.
Effusion Versus Diffusion
While both effusion and diffusion describe the movement of gas molecules, they are distinct physical processes. Effusion is highly specific, involving the unhindered escape of gas molecules through a hole that is substantially smaller than the average distance a molecule travels before colliding with another molecule. This movement is essentially a stream of gas particles escaping into a vacuum or an area of very low pressure without colliding with each other.
Diffusion, in contrast, is the process of gases mixing together as a result of the random movement and constant collisions of the molecules. This mixing occurs throughout an entire volume, causing a net movement of particles from a region of high concentration to a region of low concentration until the mixture is uniform. An example of diffusion is the spreading of an aroma across a room, a process that is much slower than effusion because molecular collisions interfere with the path of the molecules.