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will the pressure inside a bottle remain low if I attach the bottle with a vaccum pump via a one way check valve like th following picture and then take the vacuum pump?and why?

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This is how brake boosters on cars work. The check valve will obviously leak slowly over time, but it can last several minutes, perhaps hours, and a better-designed system could likely last longer depending on your application.

There's no such thing as "vacuum" in some tangible sense. Vacuum is just a relatively lower pressure, and pressure tends to go from high pressure to low pressure.

So a vacuum pump doesn't create vacuum, it just creates a lower pressure than whatever is "not vacuum". I won't get into how a vacuum pump works, but it generally involves increasing the volume while not changing the amount of air. Ok, so now there's low pressure outside the bottle, and high pressure inside the bottle.

The check valve is essentially a door without a latch. So imagine you're standing in a room and your friend is standing outside the room, with a door that swings outward. Your friend pushes on the door with some force, say 10 lbs, which holds it closed. Now, you push outward. If you push with less than 10 lbs of force, nothing happens. But if you push with more force, say 15 lbs, the net force on the door is outwards, and the door opens.

Now, instead of people, it's air pressure. When the air pressure inside is (relatively) high, the door opens and air flows from inside to outside. When the air pressure inside is (relatively) low, the pressure outside forces the door closed and holds it there, keeping the outside air from getting in. The better the seal is, the less air inadvertently slips past and the longer the vacuum inside the bottle lasts.

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If the valve is totally effective, then yes, the pressure will remain low. The pressure of a gas is a measure of how much force its particles exert on the container it's in, over a given area. The pump removes many of the particles, so the force they collectively exert goes down; it stays down as long as no new particles enter.

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