We have already explained how decreasing the pressure in the straw allows the atmosphere to push the liquid up the straw.
What was not discussed, was how the pressure in the straw is decreased in the first place.
It is tempting to say that we reduce the pressure in the straw by inhaling, but that is not correct. Take a moment or two to prove this to yourself. If you have a straw available, use it to drink, but while you are bringing the liquid up the straw, notice that you can inhale and exhale through your nose while you are drinking (obviously not while you are swallowing, though).
If you don't have a straw, or if you have trouble breathing while drinking, try putting a single fingertip in your mouth and sucking on it like a straw. Then, with your cheeks “sucked” in, breathe through your nose. You should be able to breathe both in and out without affecting the “suction” on your finger at all.
Once you are convinced that the “suction” you create with your mouth is not about breathing (if you aren't convinced yet, go back and prove it to yourself – seriously, if you don't, you'll never really get the truth) pay attention to how you create that “suction.” Most people do it by lowering their jaw, although it can be done by lowering your tongue inside your mouth.
What matters most is that, no matter how you do it, you are making the volume of your mouth bigger. We know (Boyle's Law) that when volume goes up, the pressure goes down. So, as a result of dropping your jaw the pressure in your mouth is decreased.
Now, assuming that you have a straw in your mouth, rather than your finger, the pressure of the air in the straw will be more than the pressure of the air in your mouth, so air will be pushed into your mouth.
As a result of that air movement, there will be less air in the straw. We know (the Un-Named Law) that a decrease in amount means a decrease in pressure, so as the air leaves the straw and goes into your mouth, the pressure in the straw will decrease – allowing the atmosphere to push the water up into your mouth.
A last thought... if you are wondering how the width of the straws plays in here, it doesn't.
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