Ideal gas law and a bicycle pump

[LizzleBizzle]

Homework Statement

When you push down on the handle of a bicycle pump, a piston in the pump cylinder compresses the air inside the cylinder. When the pressure in the cylinder is greater than the pressure inside the inner tube to which the pump is attached, air begins to flow from the pump to the inner tube. As a biker slowly begins to push down the handle of a bicycle pump, the pressure inside the cylinder is 1.0E5 Pa, and the piston in the pump is 0.55 m above the bottom of the cylinder. The pressure inside the inner tube is 2.4E5 Pa. How far down must the biker push the handle before air begins to flow from the pump to the inner tube? Ignore the air in the hose connecting the pump to the inner tube, and assume that the temperature of the air in the pump cylinder does not change.

Homework Equations

PV = nRT
P1V1 = P2V2

The Attempt at a Solution

This should be straightforward, but maybe I'm over-thinking it. I used Boyle's law here.
P1 = 1.0E5 Pa
P2 = 2.4E5 Pa
V1 = 0.55 m (I am assuming I can use this as a number that is proportionate to volume.)
V2 = ?

(1.0E5)(0.55) = (2.4E5)V2
V2 = 0.23 m

Does that mean that when I push down the handle 0.23, the pressure in the cylinder has reached 2.4E5 Pa and will begin to flow? Or do I need to subtract 0.23 from 0.55 m?

Thanks for any help. :)
Liz

 

[collinsmark]

Hello Liz,

Does that mean that when I push down the handle 0.23, the pressure in the cylinder has reached 2.4E5 Pa and will begin to flow? Or do I need to subtract 0.23 from 0.55 m?

What do you think?

Perhaps it might make things easier to reformulate your equations just a little, for intuitive reasons. You've correctly determined that the volume in the cylinder is proportional to the cylinder's height. So the volume in the cylinder is V = Ah, where A is the cross sectional area of the cylinder; and is a constant. The variable h is the height of the piston above the bottom of the cylinder. As you've already indicated,

V₁P₁ = V₂P₂.​

So now we have,

(1.0 x 10⁵)A(0.55) = (2.4 x 10⁵)Ah₂

Now solve for h₂.

To answer your question, ask yourself, how did we define h? Is h₂ the change of piston height, or is h₂ the piston's height above the bottom of the cylinder? Is the problem statement asking for the the change of piston height or the piston's height of the bottom of the cylinder?