When air is added to an elastic system, like a balloon, the volume and pressure change. Would use Boyle's Law- $PV=nRT,$ but n does not remain constant in this situation.
Known: $V_1$ is at atmospheric pressure $(P_1)$. $V_2$ is known, but pressure is not $(P_2).$ What equation should be used to find the change in pressure?
As stated the relationship cannot be solved. You have one equation, $PV = nRT$, and at least two unknowns.
You might get away with treating balloon + lungs as a closed system undergoing an adiabatic transformation to a higher pressure and lower volume, but that would involve making some estimates about lung volume before and after.
Let's first assume the simple case where temperature is constant and the container you are introducing gas to is rigid and at some fixed volume $V_c$. Assume the gas in the container is initially at the same pressure as outside the container.
Now gas does have a springiness to it. In other words if you try to compress more molecules into a fixed volume, it stores the energy of compression and wants to 'push' back. If we further assume the ideal gas 'law' aka 'state equation', $PV=nRT$ is valid for our gas, then we can rearrange the equation as $$\frac{P}{n}=\frac{RT}{V}$$ If we let the volume in the denominator on the right side of the equation be the volume of the rigid container we can write $$\frac{P}{n}=\frac{RT}{V_c}$$ And the right side of the equation is a constant, and is known as the mass compliance. The compliance is not the compliance of the container, but rather the compliance of the volume within the container. What does this mean? Looking at the left side we see that this constant relationship of the container compliance is equal to pressure divided by volume. But we are interested in changes so we rather write $$\frac{\Delta P}{\Delta n}=\frac{RT}{V_c}$$ So if we force more molecules into the container, we raise the container pressure proportionally to the factor on the right side. But before we start forcing gas into the container we start with some molecules $n_o$ at the current atmospheric pressure (absolute) $P_o$.and so we can write $$\frac{\Delta P +P_o}{\Delta n + n_o}=\frac{RT}{V_c}$$ We further know that the mass is equal to the density of the gas times the volume of the gas $$n = \rho V$$ This can be substituted into the last equation getting $$\frac{\Delta P +P_o}{\rho (\Delta V + V_c)}=\frac{RT}{V_c}$$ finally we assume constant gas density, relative to the amount of compression getting $$\frac{\Delta P +P_o}{ \Delta V + V_c}=\rho \frac{RT}{V_c}=\frac{P_o}{V_c} = C$$
And this is the volume compliance. The volume compliance is linear, relating the change in pressure to the change of gas introduced into a fixed, rigid container. Furthermore the compliance can be simply determined just by knowing the [absolute] reference pressure and the container volume.
Outside these assumptions, you can change the temperature, volume etc. to arrive at a more generalized formulation. And for even more complex models, assume the wall the container itself is elastic and thus changes volume of the container as a function of internal pressure - as does the balloon. But this is a much more complex analysis.
