Question

Consider a container of gas with the pressure inside the container the same as the room pressure outside the container. If a tiny hole is punched in the side of the container, will the gas leak out? Explain your answer.

Short answer

In this scenario, the gas will not "leak out" in a conventional sense, as the pressure inside and outside of the container is equal (\(P_{inside} = P_{outside}\)). This means there is no force pushing gas particles through the hole. However, due to the random motion of gas particles, some may pass through the hole in either direction (effusion), causing the overall amount of gas in the container to remain relatively constant as the rate of gas molecules escaping and entering the container is equal.

Step-by-step solution

Understand Pressure Variables

The problem states that the pressure inside the container (P_inside) is equal to the room pressure outside the container (P_outside). So we can write the equation: \(P_{inside} = P_{outside}\)

No pressure difference

Since the pressure inside and outside of the container is equal, there is no pressure difference between the two: \(P_{inside} - P_{outside} = 0\) No pressure difference means that there is no force pushing the gas particles either in or out through the tiny hole.

Observe gas particle behavior

Gas particles are in constant motion and move randomly in all directions. Because of this, gas particles inside the container have chances to pass through the tiny hole to the outside and vice versa, gas particles from outside can enter the container.

Conclude the result

Since there is no pressure difference between the container and the room, the gas particles are not forced to move out through the hole or move in. However, due to their random motion, some particles may pass through the hole in either direction, in a process called effusion. Thus, the overall amount of gas in the container will remain relatively constant since the rate of gas molecules escaping and entering the container is equal. Therefore, the gas will not "leak out" in a conventional sense, because its overall amount in the container stays practically the same.

Key concepts

Pressure Equilibrium

When we talk about pressure equilibrium, we're referring to instances where the pressure inside a container is the same as the pressure outside. Here, let's imagine a container with gas inside it. If this gas container is surrounded by air at room pressure and there is no difference in pressure, we can say they are in equilibrium:

• No net force is pushing the gas particles either inward or outward.
• It's like a balanced seesaw; both sides are matched perfectly.
• In this balanced state, gas does not flow from one place to the other due to pressure differences.

This means that if you punch a tiny hole in our gas container, there's no driving force to make the gas leak out or rush in. Everything inside and outside is in balance.

Gas Particle Motion

Gas particles are in constant, speedy motion, bouncing around in all directions. This is due to their kinetic energy, which keeps them moving randomly. Imagine a bunch of tiny balls continuously bouncing off each other and the walls of their container:

• Each particle moves independently and changes direction upon colliding with another particle or a surface.
• They take up all available space they can reach, leading to uniform distribution over time.
• Their behavior is usually unpredictable due to their constant collisions and rapid motion.

Even with no pressure difference, these energetic particles can wander out and in through any opening, such as a tiny hole. This random motion is constant, but without a pressure difference, it doesn’t direct particles specifically in or out of the container.

Effusion

Effusion describes the process of gas particles passing through a tiny hole from one side of a barrier to the other without any significant interactions with other particles. Here’s a neat way to grasp this concept:

• Particles that reach the hole may slip out on one side or enter from the other side.
• As there’s no pressure difference, there is no preference for particles to exit or enter.
• This results in a stable exchange; for every particle that exits, typically one enters.

So, while some individual gas particles might depart through the hole, others will replace them by entering from the outside. The exchange rate ensures that the overall quantity of gas in the container remains fairly steady, preventing what we would typically consider leakage.