A manual bicycle tire pump has a cylinder 35 cm long, with an inner diameter of

Question

A manual bicycle tire pump has a cylinder 35 cm long, with an inner diameter of 2.5 cm. When the piston is extended, the valve to the tire closes, and air at atmospheric pressure enters into the cylinder through another valve in its back. When the piston is compressed, air in the cylinder is compressed. The valve to the tire opens when the pressure inside the tire is surpassed at the cylinder, and air is transferred to the tire during the rest of the stroke. The last 5% of the air remains in the cylinder and hose. If you are pumping on a tire that already contains air at 2.5 atm gauge pressure, how much air is transferred to the tire at this pressure? What is the volume of this air at atmospheric pressure? Sketch a p-V diagram of the process. Assume temperature is a constant.

Explanation / Answer

Hi,

In this case we can assume that when the piston is completely extended the whole volume of the cylinder is filled with air at atmospheric pressure. When the piston compresses the air at the beginning nothing will happen since the pressure inside the cylinder will be lower than that of the tire.

After the point when the pressures between the cylinder and the tire are equal the air will flow to the tire. This proccess continues until 95% of the air has been transferred. While the air is being transfered to the tire, the pressure inside the cylinder remains constant.

If we consider the air in this problem as an ideal gas, we have the following:

PV = nRT ; where P is the pressure, V is the volume, R is the ideal gas constant, T is the temperature and n are the moles

The volume of the cylinder (equal to the volume of air once the piston is completely extended) is:

Vo = D2h/4 = (2.5*10-2 m)235*10-2 m/4 = 1.72*10-4 m3 = 0.172 L

The volume of the air inside the cylinder once the pressure of 2.5 atm gauge (3.5 atm) has been reached is:

P1V1 = PoVo ::::::: V1 = Vo (Po/P1) = 0.172 L (1 atm/3.5 amt) = 0.049 L

Therefore, the quantyty of air that is transferred to the tire is:

V = (0.95)V1 = 0.047 L

The volume of said air at atmospheric pressure would be:

PV = PoVo' ::::: Vo' = V (P/Po) = 0.047 L (3.5 atm/1 atm) = 0.165 L

Unfortunately I cannot paste the diagram but I can describe its phases:

- First, an isothermal compression, where the pressure increases and the volume is reduced.

- Second, an isobaric compression (where the temperature remains constant, while the volume and the moles change).

- Third, an isothermal expansion, where the pressure drops and the volume increases (due to the incoming air from the outside).

I hope it helps.

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