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Boyle's Law states that pressure is inversely proportional to volume. However, let's take an example of a balloon: as we fill air in it, its pressure increases, but its volume also increases. Can anyone explain?

(source: ucdavis.edu)

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Balloon is not an ideal system to study pressure-volume relationship. Because, on expansion, the elastic skin also expands and there needs to be an additional pressure build-up on the inner side to counter that force, very similar to the extra pressure in a spherical bubble. (where the surface tension acts exactly similar to the elastic balloon skin)

But in your arguments, you are forgetting an important point. Blowing a balloon also involves increasing the amount of air in it. Boyle's law holds for only a fixed volume of gas. If you change the number of moles, Boyle's law no longer stays valid.

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Let's start with the general gas equation:

$$P_1V_1 = nRT_1$$

Now to get to Boyle's law, we assume that both $T$ (temperature) and $n$ (the number of moles of the substance involved in our experiment) are both constant. Now we can rewrite the above equation as

$$P_1V_1 = \mathrm{constant}$$

Of course if we change $P$ or $V$ in our system their product must still produce the same constant, or

$$P_1V_1 = P_2V_2$$

So we can't blow additional air into our balloon (adding more moles of material and violating one of our initial assumptions) and still expect Boyle's law to hold. In a fixed system like the one we just described where Boyle's law holds, if we reduce the volume of our balloon $(V_2)$ by $1/2$ then the pressure $P_2$ must increase by a factor of $2$ in order to maintain the value of the constant.

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As Satwik explained, a balloon is not really a great container for ideal gases because of its elastic nature. It would be better to imagine a piston like in the picture below.

enter image description here

As this picture shows when the piston goes down (towards the base of the container), the volume decreases. At the same time, since the number of moles of gas stays the same (a constraint that Sathwik and Ron explained well), the pressure goes up since the ideal gas particles are now more likely to interact (collide) with the side of the container. Thus we can see that as the volume of a system decreases, the pressure increases (they are inversely proportional).

Image adapted from: https://opentextbc.ca/chemistry/chapter/9-5-the-kinetic-molecular-theory/

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