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If I had an apparatus such as the one below:

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where each of the four blue lines represents a valve, and then I was to open the three lower valves such that the pipes connecting the three containers would become pressurized (the red area on the picture below), would the total final pressure be 60 PSI or 180 PSI?

enter image description here

Thank you.

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I'll assume that the volume of the narrow part in negligible compared to the volume of the tanks. In that case, the pressure doesn't change when you open the valve.

A useful way to think about pressure in this case is something like $$ \frac{ \text{number of particles}} {\text{volume}}$$

that's actually density but for the purposes of this question it's good enough.

If you don't change the number of particles, and you don't (significantly) change the volume, then the pressure won't change.

Now open the second one. Again, nothing happens, because the pressure on both sides of the value are same. Nothing happens for the third valve either.

Finally, if you do want to factor in the volume of the narrow tube, then you must know the ratio of volumes to answer the question precisely. If the tank is much larger than the tube, then the pressure will decrease slightly upon each opening of a value. The pressure will never increase.

You may want to look into extrinsic and intrisic quanities, which change (don't change) when you increase the size of a system. Temperature doesn't increase when you increase the size, but mass does, for example.

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