Question

Consider a \(1.0-\mathrm{L}\) sample of helium gas and a \(1.0-\mathrm{L}\) 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, the helium and argon atoms have the same average kinetic energy at room temperature. b. No, helium atoms have a higher average velocity due to their lower mass. c. No, both gases exert the same pressure when they are at the same temperature, volume and number of moles. d. Helium gas has the faster rate of effusion because of its lower molar mass.

Step-by-step solution

Understanding Kinetic Energy at the Same Temperature

The average kinetic energy of gas particles is given by the equation \( KE_{avg} = \frac{3}{2}kT \) where \( k \) is the Boltzmann's constant and \( T \) is the absolute temperature. At the same temperature, all gases have the same average kinetic energy, regardless of the type of gas or the mass of the particles.

Comparing Average Velocities Using the Kinetic Energy Equation

Knowing that the kinetic energy can also be expressed using the mass \( m \) and velocity \( v \) of the particles as \( KE_{avg} = \frac{1}{2}mv^2 \) and that helium has a lower molar mass than argon, we can deduce that the average velocity of helium atoms must be greater than that of argon atoms, in order to have the same kinetic energy.

Analyzing Pressure Exerted by Gas Particles

The pressure exerted by a gas on the walls of its container is dependent on the number of particles, their velocity, and their mass according to the ideal gas law, given as \( PV = nRT \) where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature. As both samples are at the same temperature and volume and contain the same number of moles, the pressure exerted by each gas is the same.

Determining Rate of Effusion

Graham's law of effusion states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass (molecular weight). Since helium has a lower molar mass than argon, it will effuse faster than argon.

Key concepts

Average Kinetic Energy of Gases

The concept of average kinetic energy is fundamental to understanding how gases behave. It reflects the energy that gas particles have due to their motion, which is directly related to the temperature of the gas. In a gas sample, particles are constantly moving and colliding, and each one has its own kinetic energy. However, when we talk about average kinetic energy (\( KE_{avg} \)), we refer to the energy possessed by the average particle in a gas sample.

The relationship between temperature and kinetic energy is given by the equation \( KE_{avg} = \frac{3}{2}kT \), where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature. This equation tells us that at a particular temperature, all gases have the same average kinetic energy, regardless of the type of gas. Whether we have a sample of helium or argon at room temperature, the average kinetic energy of their particles will be the same because the temperature determines it.

This understanding allows us to compare different gases and predict how they will behave under similar conditions. Despite the differences in mass or size of gas particles, the average kinetic energy remains a unifying factor determining gas behavior at a given temperature.

Average Velocity of Gas Particles

When students think about the movement of gas particles, they often ask about the speed at which these particles zip around. The average velocity of gas particles is an important aspect because it's linked to how quickly they spread out, their kinetic energy, and how they interact with their environment. It's different from the average speed because velocity includes both the speed and the direction of motion.

The kinetic energy of a gas molecule can also be expressed through its mass (\( m \) and velocity (\( v \) as \( KE_{avg} = \frac{1}{2}mv^2 \) - this immediately shows us that for a gas at a fixed temperature, lighter gas particles will have higher velocities to account for their lower mass and still have the same kinetic energy as heavier gas particles. Helium, for example, has a lower molar mass compared to argon, which means that individual helium atoms must move faster on average than argon atoms to maintain the same kinetic energy as dictated by the temperature. In essence, lighter gases 'hustle' more – they move faster to keep up with their heavier counterparts in terms of energy.

Ideal Gas Law

To tackle the behavior of gases in a variety of conditions, we turn to a powerful tool in chemistry: the ideal gas law. This mathematical equation serves as a cornerstone for understanding how changes in pressure, volume, and temperature affect a sample of gas. The ideal gas law is neatly encapsulated by the formula \( PV = nRT \), where \( P \) represents pressure, \( V \) indicates volume, \( n \) is the number of moles of gas, \( R \) is the universal gas constant, and \( T \) is the absolute temperature.

Let's consider the significance of this law. If we have a fixed amount of gas – say, 1.0 liter – at a constant temperature, any increase in pressure must be due to a decrease in volume and vice versa. It illustrates how the gas molecules interact with their container, exerting force (pressure) and spreading out (volume) while balancing the energy (temperature). This equation applies perfectly to ideal gases, which are hypothetical gases that perfectly follow the law without any attraction or repulsion between the particles. It becomes a useful approximation for real gases under many conditions, explaining why, in our original problem, both helium and argon exert the same pressure when measured at the same temperature, volume, and number of moles.

Graham's Law of Effusion

When gas particles escape through a tiny opening, this movement is known as effusion. Understanding which gas effuses faster under certain conditions can be rather useful, especially in industrial processes and in scientific research. Graham's law of effusion provides us with the means to compare the rates of effusion between different gases.

This law states that the rate of effusion for a gas is inversely proportional to the square root of its molar mass. Mathematically, we express this law as \( \frac{Rate_1}{Rate_2} = \sqrt{\frac{Molar\ Mass_2}{Molar\ Mass_1}} \).

With this equation, we can easily determine that lighter gases will effuse more rapidly than heavier gases. Therefore, among our two sample gases, helium, which has a significantly lower molar mass compared to argon, will effuse more quickly. Graham's law highlights the profound effect of molecular weight on the escape of gas particles and is essential for applications ranging from separating isotopes to the manufacture of semiconductors.