Question

Consider a 1.0-L sample of helium gas and a 1.0-L sample of argon gas, both at room temperature and atmospheric pressure. a. Do the atoms in the helium sample have the same average kinetic energy as the atoms in the argon sample? b. Do the atoms in the helium sample have the same average velocity as the atoms in the argon sample? c. Do the argon atoms, because they are more massive, exert a greater pressure on the walls of the container? Explain. d. Which gas sample has the faster rate of effusion?

Short answer

a. Yes, both gases have the same average kinetic energy at the same temperature. b. No, helium atoms have a higher average velocity than argon atoms. c. No, both gases exert the same pressure under the same conditions. d. Helium has a faster rate of effusion.

Step-by-step solution

Understanding Kinetic Energy and Temperature Relation

Recall that the average kinetic energy of gas particles is directly related to the temperature of the gas by the equation \(\frac{3}{2}kT\), where \(k\) is the Boltzmann constant and \(T\) is the absolute temperature in Kelvin. Since both gases are at room temperature, they have the same average kinetic energy.

Comparing Average Velocities

The average velocity of gas particles can be derived from the kinetic energy, where \(KE = \frac{1}{2}mv^2\) for a particle with mass \(m\) and velocity \(v\). Because helium atoms are lighter than argon atoms (i.e., have a lower molar mass), they have a higher average velocity at the same temperature.

Examining Pressure Exerted by Gases

According to the ideal gas law \(PV = nRT\), for a given volume and temperature, the pressure \(P\) is determined by the number of moles \(n\). Since both helium and argon gases are at the same temperature, pressure, and volume, they contain the same number of moles; therefore, the argon atoms do not exert a greater pressure on the container walls solely because of their greater mass.

Determining Rate of Effusion

The rate of effusion of a gas is inversely proportional to the square root of its molar mass according to Graham's law of effusion. Mathematically, \(\text{Rate of effusion} \propto \frac{1}{\sqrt{M}}\), where \(M\) is the molar mass. Since helium has a lower molar mass than argon, helium effuses faster.

Key concepts

Kinetic Energy and Temperature Relation

Understanding the kinetic energy and temperature relation in gases is fundamental in grasping the behaviours of different gases under similar conditions. At a fundamental level, the kinetic energy, associated with the motion of gas particles, is an indication of a gas's thermal energy. A pivotal equation to remember is \(KE = \frac{3}{2}kT\), where \(KE\) represents the kinetic energy, \(k\) is Boltzmann's constant, and \(T\) the temperature in Kelvin.

This tells us that for a gas at any given temperature, the kinetic energy of its particles remains constant. For example, helium and argon at room temperature, despite being different gases, possess the same average kinetic energy. This plays a critical role in predicting other properties such as velocity and rate of effusion, which are directly influenced by the kinetic energy of the gas particles.

Average Velocity of Gas Particles

Delving into the average velocity of gas particles provides insight into how different gases behave under the same energetic conditions. The average velocity (\(v\)) of gas particles can be calculated using the relationship \(KE = \frac{1}{2}mv^2\), with \(m\) being the mass of a particle. Lighter gas particles, such as those of helium, will have a higher average velocity compared to heavier gas particles like those of argon under the same temperature.

Understanding the inverse relationship between mass and velocity can explain why different gases diffuse at different rates. For identical conditions, we can expect a lighter gas to have particles moving faster on average than a heavier gas. This concept is frequently encountered in problems related to gas diffusion or effusion.

Ideal Gas Law

The ideal gas law is a cornerstone in understanding gas behaviour, represented by the equation \(PV = nRT\). This law relates pressure (\(P\)), volume (\(V\)), the number of moles (\(n\)), the gas constant (\(R\)), and temperature (\(T\)) in a beautifully simple way.

Crucially, the ideal gas law allows us to deduce that gases at the same pressure, volume, and temperature have the same number of moles irrespective of their mass. Therefore, contrary to what might seem intuitive, a gas with more massive particles does not exert a greater pressure than a gas with less massive particles if both are contained in the same volume and are at the same temperature. This is essential in understanding how different gases react and are measured in various scientific experiments and real-world applications.

Graham's Law of Effusion

Graham's law of effusion is yet another important concept in gas dynamics, which describes how the rate of effusion for a gas is connected to its molar mass. Given by the formula \(\text{Rate of effusion} \propto \frac{1}{\sqrt{M}}\), where \(M\) represents the molar mass, it shows us that gases with lower molar masses effuse more rapidly than those with higher molar masses.

Taking helium and argon as examples again, since helium has a substantially lower molar mass than argon, it would effuse faster through a small opening or porous membrane. This principle is used in various practical applications, such as the purification of gases and is also an important concept in understanding natural phenomena involving gases.