Elastic Balloon Filled with Ideal Gas

For a balloon that stretches ideally according to Hooke's law, I believe it resists with a constant pressure, meaning you would see the volume doubling.

That is, because the surface increases with the square of the radius, the stretching force also increases with the square of the radius.
But the resulting pressure, which is the quotient of the two, remains constant.

 

Say the volume of the balloon increased by √2, then the pressure inside of the balloon would have to increase by √2.

Look at it this way because the balloon is a closed system (xP)(yV) = nR(zT)

x · y = z

if you double z, then x and y have to change so that their product also doubles.

 

You have one equation, PV = nRT - or, the version I think with, P = NT - and three variables: P, V or N, and T. Specifying a change to one does not specify how the other two change, you need another equation. Something about the properties of the container (Hooke's law spring), something about heat flow, whatever - but without more information you cannot say what happens to V if you double T.