# Ideal gas law and a bicycle pump

• LizzleBizzle
In summary, the pressure in the cylinder is proportional to the height of the piston above the bottom of the cylinder, and can be determined using the equation PV = nRT. To find the height of the piston at which air will begin to flow from the pump to the inner tube, one must use the equation P1V1 = P2V2 and solve for the desired variable. In this case, the height of the piston above the bottom of the cylinder can be calculated by dividing the product of the initial pressure and volume by the final pressure, and then multiplying by the initial height of the piston.
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.

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

Hello Liz,
LizzleBizzle said:
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?

## 1. What is the Ideal Gas Law?

The Ideal Gas Law is a mathematical equation that describes the relationship between the pressure, volume, temperature, and amount of gas in a closed system. It is often written as PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin.

## 2. How does the Ideal Gas Law relate to a bicycle pump?

The Ideal Gas Law can be used to explain the behavior of gases inside a bicycle pump. When the pump is compressed, the volume of air decreases, causing an increase in pressure. This increase in pressure forces the air out of the pump and into the tire, allowing it to inflate.

## 3. What is the ideal gas constant?

The ideal gas constant, denoted as R, is a constant value that relates the properties of an ideal gas. It has a value of 8.314 J/mol·K and is used in the Ideal Gas Law to convert between units of pressure, volume, temperature, and moles.

## 4. Can the Ideal Gas Law be applied to all gases?

The Ideal Gas Law is a theoretical equation that is most accurate for ideal gases, which are gases that have no intermolecular forces and occupy no volume. However, it can also be used to approximate the behavior of real gases under certain conditions, such as low pressure and high temperature.

## 5. How does temperature affect the pressure of a gas in a bicycle pump?

According to the Ideal Gas Law, as temperature increases, the pressure of a gas will also increase if the volume and amount of gas remain constant. This means that when using a bicycle pump, if the temperature of the air inside the pump increases, the pressure will also increase, making it easier to inflate the tire.

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