# Thermo problem: Two containers separated by stopcock

• D_Tr
In summary, the solution provided by your professor uses the first law of thermodynamics to explain the transfer of energy between the two containers and the resulting increase in temperature in the low pressure container. The assumption of negligible gas molecule volume allows for simpler calculations and does not significantly affect the overall behavior of the gas.
D_Tr

## Homework Statement

Container 1 has a volume of 10 ft^3 and contains ideal gas at a pressure of a atm and temperature of 273 F.
Container 2 has a bolume of 1 ft^3 and contains the same quantity of ideal gas at a pressure of 0.1 atm and temperature of 273 F.
The two containers are separated by a stopcock. We then open the stopcock until pressure equilibrium is achieved. Find the final temperature of each container as well as the quantity of gas transferred between the two containers.

## Homework Equations

The state equation for ideal gases: P*V = n*R*T
The first law of thermodynamics for open systems (?)

## The Attempt at a Solution

I have a solution for the problem from the professor but I cannot make full sense of it. I can post the full solution if you think I sould. Basically, at the start, the equilibrium pressure is calculated using the law of partial pressures: Pf = (P1initial*V1 + P2initial*V2)/(V1+V2) = 2/11 atm.
At the end the temperatures calculated for each container are different. My question is why do we have different temperatures. Even if we close the stopcock as soon as pressures become equal witout caring about temperature, why should temperatures at the end be different? (352 F for container 2, -89 F for container 1). For this to happen the molecules in container 1 (high pressure) must lower their average kinetic energy right? When we open the stopcock molecules are only bouncing in the container walls and with other molecules of the same temperature. The solution seems to assume that the gas in the high pressure container acts as a piston that does work on the gas in the low pressure container, thus transfering energy to the low pressure gas and heating it. It uses the first law of thermodynamics for an open system (system is defined as the volume of the high pressure container) and arrives at: (T1f/T1i) = (Pf/P1i)^((γ-1)/γ). However, for the "piston made of gas molecules" assumption to hold we must assume that the molecules of the ideal gas have some volume otherwise each container's molecules would expand into the other container as if other molecules were not there. Have I correctly interpreted the solution to the problem? Sorry for the lengthy post and thanks for your time .

Hello, thank you for your post. It seems like you have a good understanding of the problem and the solution provided by your professor. Your interpretation of the solution is correct - the gas molecules in the high pressure container act as a "piston" and transfer energy to the gas molecules in the low pressure container, causing them to increase in temperature. This is due to the first law of thermodynamics, which states that energy cannot be created or destroyed, only transferred from one form to another. In this case, the energy is transferred from the high pressure gas to the low pressure gas, causing an increase in temperature in the low pressure container.

As for your question about the assumption of the gas molecules having some volume, this is a valid concern. In reality, gas molecules do have some volume, but it is usually very small compared to the volume of the container they are in. In the ideal gas law, we assume that the volume of the gas molecules is negligible compared to the volume of the container. This allows us to simplify the calculations and make assumptions about the behavior of the gas molecules. So, while the molecules do have some volume, it is not significant enough to affect the overall behavior of the gas in this problem.

I hope this helps clarify the solution for you. If you have any further questions, please feel free to ask.

## 1. How does the stopcock affect the temperature in the two containers?

The stopcock acts as a barrier between the two containers, preventing the transfer of heat between them. This means that the temperature in each container will remain independent of the other, unless the stopcock is opened.

## 2. Can the stopcock be used to control the temperature in the two containers?

Yes, by opening or closing the stopcock, you can control the flow of heat between the containers. This can be useful in experiments where you want to maintain a specific temperature in one container while allowing the other to reach a different temperature.

## 3. What factors can affect the rate of heat transfer between the two containers?

The rate of heat transfer is influenced by several factors, including the temperature difference between the two containers, the thermal conductivities of the materials used, and the surface area of the containers in contact with each other.

## 4. Is there a limit to how much heat can be transferred between the two containers?

Yes, there is a limit to the amount of heat that can be transferred between the two containers, which is determined by the thermal conductivities of the materials used and the temperature difference between the containers. Once the containers reach thermal equilibrium, no further heat transfer will occur.

## 5. How can the thermo problem with two containers be solved mathematically?

The thermo problem can be solved using the principles of thermodynamics, specifically the laws of heat transfer and conservation of energy. By applying these principles and using mathematical equations, the temperature change in each container can be determined at different time intervals as the heat is transferred between them.

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