Work done by an expanding gas with no moving boundary

[manuslum]

Hello,

I'm trying to find an expression for the work done by a gas that expands adiabatically from one chamber to another through a controlled opening and closing of a valve.

The setup is the following. There is spherical bulb divided inside into to chambers. One of the chambers (A) contains a gas that is pressurized at several atmospheres, while the other chamber (B) contains very little of the gas, with a pressure that is only 1% of the pressure in chamber A. The two chambers are connected trough a small hole that has a valve on the side of chamber A, keeping the gas from flowing into chamber B. Upon command, the valve will open in small and regular time intervals that would allow some gas to escape from A into B.

I need to find the work that the gas will do on [anything] while expanding. Note that there is no boundary moving, like a piston or anything similar. I know that if there were such a boundary, I could use dW = P dV, but I can't seem to find the change in volume in my scenario. First I thought that the work would be the product of the Force and the Distance the gas covers while moving into chamber B (the thickness of the dividing plate inside the sphere), but I am now too doubtful about this assumption.

Also, I'm taking into account the fact that the force that the gas exerts on the valve is:

F_Net = P_AA - P_BA = A(P_A-P_B), where P_A is the pressure in chamber A, P_B is the pressure in chamber B, and A is the cross-sectional area of the small hole connecting the two chambers (Pi R²)

I realize that this force is not constant as it depends on the relative pressures in A and B, which are changing during the duration of the entire process. I'm also trying to take this fact into account in my calculations.

What I ultimately intend to achieve is the cooling of chamber A, and to that end, I need to find a point (if it exists) where the temperature in A is a specific value. I believe that the work done by the gas while expanding will result in a decrease in its internal energy, which translates into a drop in temperature.

Please, any help is very appreciated. And please correct me if I am wrong about anything.

Thanks a lot

http://img36.imageshack.us/img36/9066/diagramj.png

 

[Legion81]

That would just be an adiabatic free expansion if I'm following your model right. So the work done would be zero since it is a free expansion. And if there is zero work and zero heat, then your internal energy is zero. That means the temperature is the same before and after.

 

[astrokreng]

Dear researcher,

Thank you for your inquiry. The work done by an expanding gas with no moving boundary can be calculated using the following formula: W = -∫PdV, where W is the work done, P is the pressure of the gas, and dV is the change in volume. In your scenario, since the gas is expanding from chamber A to chamber B, the volume change can be calculated as dV = Vb - Va, where Vb is the final volume of the gas in chamber B and Va is the initial volume of the gas in chamber A.

To calculate the final volume, you can use the ideal gas law: PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the gas constant, and T is the temperature. Since the process is adiabatic, the temperature remains constant, so you can rearrange the equation to solve for Vb: Vb = (nRT)/Pb, where Pb is the final pressure of the gas in chamber B.

Similarly, you can calculate the initial volume Va using the initial pressure and temperature in chamber A. Once you have both Va and Vb, you can calculate the change in volume and then use the formula for work to calculate the work done by the gas during the expansion process.

It is important to note that the work done by the gas will depend on the final pressure in chamber B. If you want to achieve a specific temperature in chamber A, you will need to adjust the final pressure in chamber B accordingly. This can be done by changing the size of the valve opening or the frequency of valve openings.

I hope this helps and please let me know if you have any further questions. Good luck with your research!