Problem 87 - Chemistry Matter and Change

Question

If two identical containers each hold the same gas at the same temperature but the pressure inside one container is exactly twice that of the other container, what must be true about the amount of gas inside each container?

Step-by-Step Solution

Answer

The second container has twice the amount of gas as the first.

Step 1: Understand the Relationship

We are given two identical containers holding the same gas at the same temperature. One container has a pressure that is twice that of the other. According to the ideal gas law, pressure is directly proportional to the amount of gas moles if volume and temperature remain constant.

Step 2: Apply the Ideal Gas Law Formula

The ideal gas law is given by \( PV = nRT \). Here, \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles of gas, \( R \) is the gas constant, and \( T \) is temperature. Since the volume and temperature are constant, we can simplify this to \( P \propto n \).

Step 3: Compare Moles in Each Container

Let the pressure in the first container be \( P \) and in the second container be \( 2P \). If \( n_1 \) is the amount of gas in the first container, then \( n_2 \), the amount of gas in the second container, must be such that \( n_2 = 2n_1 \). This means the second container has twice the amount of gas as the first.

Key Concepts

Pressure: The Force Exerted by Gas Molecules

Pressure in the context of gases refers to the force that gas molecules exert when they collide with the walls of a container. Think of pressure like a bunch of tiny invisible kicks on the container’s walls. The more kicks, the higher the pressure.

In physics and chemistry, pressure is usually measured in units such as atmospheres (atm), Pascals (Pa), or millimeters of mercury (mmHg). In the exercise, we observe that one container has a pressure twice that of another, indicating it has more gas molecules contributing to these impacts.

Key takeaway: If temperature and volume are unchanged, the pressure is directly affected by the number of gas moles. More gas moles result in more collisions and thus a higher pressure.

Volume: The Space Occupied by Gas

Volume is essentially the amount of space a gas occupies. Imagine a box that expands and contracts: when it expands, the volume increases; when it contracts, the volume decreases.

In our example, the volume remains constant, meaning the containers do not change in size. The significance of this is that any changes in pressure or the amount of gas will relate directly to changes in the other variables of the equation.

• This constant volume ensures that pressure variations are due to the amount of gas present only.
• Understanding that volume is fixed helps us focus on how other factors, like pressure and moles of gas, change.

Temperature: The Measure of Kinetic Energy

Temperature measures the kinetic energy of gas molecules—essentially, how fast they're moving around. When temperature increases, molecules move faster and hit the walls harder, thereby increasing pressure if volume is constant.

In the scenario we are studying, the temperature remains the same, meaning that any change in pressure is not due to temperature changes but other factors. Constant temperature helps isolate the effect of changing the number of gas moles on pressure.

• Constant temperature implies all changes in pressure are due to differences in the amount of gas present.
• This stability allows for a clearer understanding of how moles relate to pressure through the ideal gas law.

Moles of Gas: The Quantity of Gas Present

Moles represent the quantity of gas available in a container. In chemical terms, a mole is a fixed number of molecules or atoms—a standard unit in chemistry.

In the ideal gas law equation, the number of moles (n) plays a crucial role. In our exercise, the problem illustrates that if the pressure in one container is double the other, then the container with higher pressure must contain double the moles at constant volume and temperature.

• This is represented in the simplified relationship: Pressure (P) is directly proportional to the number of moles (n) when temperature (T) and volume (V) are constant.
• The container at twice the pressure simply has more molecules pushing against the walls.