Question

Graham's law states that the rate at which gas molecules escape through a small orifice (rate of effusion) is inversely proportional to the square root of the density of the gas. Derive Graham's law from the following assumptions: (a) temperature is directly proportional to the average kinetic energy of the molecules; (b) the rate of effusion is directly proportional to the root mean square speed of the molecules; (c) the density of a gas at constant temperature and pressure is directly proportional to the molecular mass.

Short answer

Graham's law states that the rate of effusion (R) of a gas is inversely proportional to the square root of its density (ρ). By connecting the given assumptions, we derived the relationship: \(R \propto \frac{1}{\sqrt{\rho}}\), confirming Graham's law. This relationship implies that gases with lower densities will effuse faster than those with higher densities.

Step-by-step solution

Relate temperature and average kinetic energy

According to the first assumption, temperature is directly proportional to the average kinetic energy of the molecules. This can be written as: \[T \propto \overline{KE}\] where T is the temperature and \(\overline{KE}\) is the average kinetic energy of the gas molecules.

Relate average kinetic energy and root mean square speed

The average kinetic energy can be related to the root mean square speed (v_rms) using the formula: \[\overline{KE} = \frac{1}{2}m(v_{rms})^2\] where m is the mass of the gas molecule.

Relate the rate of effusion and root mean square speed

The rate of effusion (R) is directly proportional to the root mean square speed, as stated in the second assumption: \[R \propto v_{rms}\]

Relate density and molecular mass

The third assumption states that the density (ρ) of a gas at constant temperature and pressure is directly proportional to the molecular mass (M). This can be represented as: \[\rho \propto M\]

Connect the relationships

We can connect the relationships from steps 1-4 to find the relationship between the rate of effusion and density. Using Step 3 and Step 2, we can write: \[R \propto v_{rms} \rightarrow R \propto \sqrt{\frac{2\overline{KE}}{m}}\] Now, combining the relationships from Step 1 and Step 4, we have: \[\rho \propto M \rightarrow \rho \propto \frac{m}{V} \rightarrow m \propto \rho V\] Substituting this in the proportionality found above, we get: \[ R \propto \sqrt{\frac{2\overline{KE}}{\rho V}}\] Since Graham's law refers to the same volume and temperature, the average kinetic energy and volume remain constant. Therefore, we can say: \[ R \propto \frac{1}{\sqrt{\rho}}\] This shows that the rate of effusion (R) is inversely proportional to the square root of the density (ρ), which is Graham's law.

Key concepts

Rate of Effusion

The rate of effusion is a term used to describe the speed at which gas particles pass through a tiny opening into a vacuum. It's an important concept in chemistry that helps explain how gases behave when they are given the opportunity to escape from their container. According to Graham's law, the rate of effusion is inversely proportional to the square root of the gas’s density. This means that lighter gases effuse more rapidly than heavier ones.
To understand this better, imagine you have two balloons, one filled with helium and the other with carbon dioxide. If both balloons have small holes in them, you'll notice that the helium balloon deflates quicker. This happens because helium has a lower density, so its rate of effusion is higher based on Graham’s law. This law helps predict how gases will effuse and is utilized in various practical applications such as separating isotopes or manufacturing nitrogen-filled tires.

Kinetic Molecular Theory

Kinetic Molecular Theory (KMT) describes the behavior of gases in terms of their molecular motion. The theory posits that gas particles are in constant, random motion and that they collide with each other and the walls of their container. These collisions are elastic, meaning that they do not result in a loss of energy. The kinetic energy of these particles, which is directly related to temperature, is what drives the movement of gas molecules.
From this perspective, the gas pressure is explained as the result of collisions against the container walls, and temperature is a measure of the average kinetic energy of the molecules. The KMT is essential for understanding concepts like the rate of effusion, as it directly connects temperature with molecular motion and ultimately with the speed at which gases can escape their container.

Root Mean Square Speed

The root mean square speed (\(v_{rms}\)) is a way to quantify the average velocity of gas molecules. This value takes into account the speeds of all the molecules in a sample and provides a useful measure for comparing the behavior of different gases. Root mean square speed is a component in the equation for kinetic energy, \(\overline{KE} = \frac{1}{2}m(v_{rms})^2\).
This indication is powerful because it ties together the mass of the gas particles and their temperature (which determines their kinetic energy). Gases with lower masses will have higher root mean square speeds at the same temperature than gases with larger masses. When we talk about the rate of effusion, understanding root mean square speed is crucial because, as per Graham's law, it's directly proportional to the rate at which a gas effuses.

Gas Density and Molecular Mass

Gas density is essentially a measure of how much mass of gas is present in a given volume. It's directly related to the molecular mass of a gas because, at constant temperature and pressure, a gas with heavier molecules will be denser than one with lighter molecules. This concept is expressed by the proportionality \(\rho \propto M\), where \(\rho\) represents density and \(M\) is molecular mass.
When it comes to effusion, understanding the relationship between gas density and molecular mass is crucial. Since the rate of effusion is inversely proportional to the square root of the gas's density, this means that a gas with a higher molecular mass (and thus, higher density) will effuse more slowly than a gas with a lower molecular mass. This is the foundational reason behind why different gases effuse at different rates—a core aspect of Graham’s law.