Problem 34 - Chemistry The Molecular Nature of Matter

Question

Suppose we have a mixture of helium and argon. On average, which atoms are moving faster at \(25^{\circ} \mathrm{C},\) and why?

Step-by-Step Solution

Answer

Helium atoms are moving faster on average at 25 degrees Celsius because they have a much smaller mass compared to argon atoms, thus needing to move faster to have the same kinetic energy at a given temperature.

Step 1: Understand the Concept of Average Speed of Gas Molecules

The average speed of gas molecules in a mixture at a given temperature can be estimated using the Kinetic Molecular Theory of Gases. This theory states that the average kinetic energy of gas molecules is directly proportional to the absolute temperature (in Kelvin) of the gas and is given by the equation \(\frac{1}{2}mv^2 = \frac{3}{2}kT\), where \(m\) is the mass of the molecule, \(v\) is the speed of the molecule, \(k\) is the Boltzmann constant, and \(T\) is the temperature in Kelvin.

Step 2: Convert Celsius to Kelvin

First, convert the temperature from Celsius to Kelvin by using the relation \( T(K) = T(\degree C) + 273.15 \). Thus, the temperature in Kelvin is \(25 + 273.15 = 298.15 K\).

Step 3: Relate Average Kinetic Energy to Mass and Speed

Since the kinetic energy is the same for all gas molecules at a given temperature, we can set the kinetic energies for helium and argon equal to each other to find a relationship between their masses and velocities: \(\frac{1}{2}m_{He}v_{He}^2 = \frac{1}{2}m_{Ar}v_{Ar}^2\).

Step 4: Identify the Masses of Helium and Argon Atoms

Identify the atomic masses of helium (He) and argon (Ar). The atomic mass of helium is approximately 4 u (unified atomic mass unit) and the atomic mass of argon is approximately 40 u.

Step 5: Use the Relationship Between Mass and Speed

Since helium has a much smaller mass compared to argon, and given that their kinetic energies are equal at the same temperature, helium atoms must be moving faster on average. This is because velocity must increase to compensate for the lower mass in order to maintain the same kinetic energy.

Key Concepts

Kinetic Molecular Theory of Gases

The Kinetic Molecular Theory of Gases provides us a framework to understand the behavior of gas molecules in terms of motion and energy. According to this theory, gases consist of tiny particles in constant, random motion. The pressure of a gas arises from collisions between the molecules and the walls of their container.

Furthermore, this theory highlights a few key points: gas particles are considered to be small, hard spheres with an insignificant volume compared to the container volume, and they are in constant motion, undergoing perfectly elastic collisions.

Importantly, at a given temperature, the theory also postulates that all gases, regardless of atomic mass, have the same average kinetic energy. This principle allows us to relate the kinetic energy of gas molecules with their speed and mass, providing insights into their behavior under different conditions, such as changes in temperature and volume.

Kinetic Energy of Gas Molecules

The kinetic energy of a gas molecule is directly related to its velocity and mass. Kinetic energy is the energy that an object possesses due to its motion, and for a molecule, it can be calculated using the formula \(\frac{1}{2}mv^2\), where \(m\) is the mass and \(v\) its velocity.

According to the kinetic theory, temperature is a measure of the average kinetic energy of the gas molecules. At the same temperature, different gases have the same average kinetic energy but may have different velocities. This is due to the fact that velocity is dependent on the mass of the molecules: lighter molecules will move faster than heavier ones to have the same kinetic energy. Recognizing this inverse relationship between mass and velocity is essential when predicting the behavior of gases, such as diffusion rates or effusion velocities.

Atomic Mass and Gas Velocity

The atomic mass of a gas molecule has an inverse relationship with its velocity according to the principles of kinetic theory. In our example with helium and argon, helium's lower atomic mass results in higher velocity at the same temperature when compared to argon.

Kinetic theory teaches us that at a constant temperature, lighter gas molecules (with lower atomic mass) will generally move faster to maintain energy equilibrium. Therefore, when we analyze helium (4u) versus argon (40u), we conclude that helium atoms move about ten times faster than argon atoms because their mass is about a tenth of argon's. This concept is crucial for understanding molecular speeds in various gases, and it explains why lighter gases diffuse more quickly than heavier ones. It's a key point in fields ranging from chemistry to environmental science when dealing with the movement and behavior of gases.