Temperature & Pressure. Beginner Physics.

[Nachore]

Homework Statement

A rigid, perfectly insulated container has a membrane dividing its volume in half. One side contains a gas at an absolute temperature To and pressure Po, while the other half is completely empty. Suddenly a small hole develops in the membrane, allowing the gas to leak out into the other half until it eventually occupies twice its original volume. In terms of To and Po, what will be the new temperature and pressure of the gas when it is distributed equally in both halves of the container?

Homework Equations

None.

The Attempt at a Solution

I have no idea how to explain this answer. I don't know how to approach this. Please help.

 

[lfwake2wake]

The box has a definite volume of V and a definite amount of mass inside (n). If the gas starts off with a volume of V, Pressure of Po, Temp of To, and mass of n, use the Ideal Gas Law to solve for a change in volume and respective pressure/temperature changes.

PV=nRT where R is the ideal gas constant.

Case 1, Po*V=n*R*To
Case 2 P2*V2=n*R*T2

Po/To=(nR)/V
P2/T2=nR/V2

Since V2=2*V then P2/T2=0.5(nR)/V or the ratio of the pressure to the temperature of the final volume of gas will be half that of the original. P2/T2=.5*Po/To

 

[nlightner]

I can provide an explanation for this scenario using the ideal gas law, which states that the product of pressure and volume is directly proportional to the absolute temperature of a gas. In this case, we can use the equation P1V1/T1 = P2V2/T2, where P1 and V1 are the initial pressure and volume of the gas, and P2 and V2 are the final pressure and volume of the gas after it has expanded into both halves of the container.

Since the container is perfectly insulated, we can assume that there is no heat exchange between the gas and its surroundings, meaning the temperature remains constant. This allows us to simplify the equation to P1V1 = P2V2.

Initially, the gas is at a temperature To and pressure Po in one half of the container. When the membrane ruptures, the gas expands into the other half, resulting in a final volume of 2V1. Plugging this into the equation, we get P1V1 = P2(2V1), which simplifies to P1 = 2P2.

This means that the final pressure of the gas in both halves of the container will be half of its initial pressure, or P2 = Po/2. Since the temperature remains constant, the final temperature of the gas will also be To.

Therefore, the new temperature and pressure of the gas when it is distributed equally in both halves of the container will be To and Po/2, respectively.