Problem 82 - Chemistry A Molecular Approach

Question

A flask at room temperature contains exactly equal amounts (in moles) of nitrogen and xenon. a. Which of the two gases exerts the greater partial pressure? b. The molecules or atoms of which gas have the greater average velocity? c. The molecules or atoms of which gas have the greater average kinetic energy? d. If a small hole were opened in the flask, which gas effuses more quickly?

Step-by-Step Solution

Answer

a. Both gases exert the same partial pressure. b. Nitrogen molecules have greater average velocity. c. Both gases have the same average kinetic energy. d. Nitrogen gas effuses more quickly.

Step 1: Understanding Partial Pressure

Partial pressure in a mixture of gases is determined by the number of moles, temperature, and volume according to the Ideal Gas Law: P = \(\frac{nRT}{V}\). Since the number of moles (n), temperature (T), and volume (V) are the same for both gases, and R is a constant, the partial pressures exerted by the nitrogen and xenon will be equal.

Step 2: Determining Average Molecular Velocity

Use the formula for average velocity (Root Mean Square Speed) for a gas from kinetic molecular theory: \(v_{rms} = \sqrt{\frac{3RT}{M}}\), where M is the molar mass of the gas. Since R and T are constants and the same for both gases, the gas with the lower molar mass will have a higher average velocity. Nitrogen (N2) has a lower molar mass than Xenon (Xe), thus nitrogen molecules will have a greater average velocity.

Step 3: Comparing Average Kinetic Energy

The average kinetic energy of a gas molecule can be derived from kinetic molecular theory and is given by the formula \(KE_{avg} = \frac{3}{2}kT\), where k is Boltzmann's constant and T is the temperature. Since the temperature is the same for both gases, the average kinetic energies of nitrogen and xenon molecules are the same.

Step 4: Analyzing Rates of Effusion

Graham's law of effusion states that the rate of effusion for a gas is inversely proportional to the square root of its molar mass: \(\text{Rate of Effusion} \propto \frac{1}{\sqrt{M}}\). Given that nitrogen has a lower molar mass compared to xenon, nitrogen will effuse more quickly through a small hole.

Key Concepts

Partial Pressure

Imagine you’re at a party where everyone is talking at the same time. The sound you hear is the combined noise of all the voices. In a similar way, partial pressure is the 'contribution' that each gas in a mixture makes to the total pressure. It’s as if each gas is speaking, or in this case, exerting pressure in the container all by itself.

Using our party analogy, if nitrogen and xenon are chatting at the same volume, since they have equal moles, their voices (partial pressures) contribute equally to the overall din (total pressure). This is because partial pressure depends on the mole fraction of the gas and the total pressure, as per Dalton's Law of Partial Pressures. When the flask contains equal amounts of nitrogen and xenon, neither yells louder; they exert the same partial pressure.

Average Molecular Velocity

Let’s switch gears to movement – think of gas molecules as runners in a race. The lighter the runner, the faster they can run. Average molecular velocity is like the average speed of these runners at any given moment. It's crucial in understanding how gases move and mix.

To compare our molecular racers, nitrogen is the lighter runner with a lower molecular mass. Following the equation for Root Mean Square Speed, that means nitrogen molecules sprint faster on average than the heavier xenon molecules. This concept helps to predict how quickly gases will diffuse and even determine rates of chemical reactions where gases are involved.

Kinetic Molecular Theory

Imagine millions of tiny particles zooming around, colliding with each other and the walls of their container – that's the gist of kinetic molecular theory. It’s a model that explains the behavior of gases, envisioning them as small, hard spheres all in constant, random motion.

Kinetic molecular theory not only accounts for the movement of particles but also provides insights into properties such as pressure and temperature. One of its fundamental assumptions is that the average kinetic energy of gas molecules is directly proportional to the temperature in kelvins. This theory also helps us understand why, in our textbook problem, the average kinetic energy of both nitrogen and xenon gases is the same when they're at the same temperature.

Graham's Law of Effusion

Picture two different types of perfume released in a room – one diffuses faster and soon you can smell it all over, while the other takes a bit longer to spread. This is like gas effusion, the process by which gases pass through a tiny hole into a vacuum or less pressurized space.

According to Graham's Law of Effusion, the rate at which a gas effuses is inversely proportional to the square root of its molar mass. This principle tells us that lighter molecules whizz through the hole more quickly than heavier ones. So in a race between nitrogen and xenon, nitrogen would be the first to seep out of our hypothetical flask with a small opening.