How Does Pumping Air Affect Bicycle Tire Pressure and Temperature?

[zezima1]

Homework Statement

A bicycle tire is to start with flat and has a pressure of one atmosphere. It is then pumped such that the pressure is 4atm, when the air inside the tire has the same temperature as before.
a) How much air has been pumped into the bicycle tire?
b) What is the temperature of the air inside the tire just after it has been pumped (assume no heat escapes during the compression).

Homework Equations

Ideal gas law
pV = nRT
Adiabatic compression:
V*T^(f/2) = constant

The Attempt at a Solution

For the first one I thought this:
Since T is the same before and after but pressure is 4 times a much n must be 4 times as much as before the pumping. Is this correct?

Is so, good. Next we have b). Here I want to use the equations for adiabatic compression but I'm unsure if I'm doing it well. I thought the pressure one would be hard to use since the 4atm is after heat has escaped. Instead I thought that the volume before must have filled up a space of 4 times as much as after, since the amount of molecules is quadropled. Then knowing this we can write an expression for Tfinal = Tf:
Vf*Tf^(5/2) = Vi*Ti^(5/2)
where I've used that normal air has on average about 5 DOF. IS THIS APPROACH CORRECT?

 

[anathema]

Hello,

Your approach for part a) is correct. Since the temperature is constant, the number of moles of gas in the tire must increase in order to increase the pressure. Therefore, the amount of air pumped into the tire must be four times the amount that was originally in the tire.

For part b), your approach is not quite correct. You are correct in using the adiabatic compression equation, but you don't need to consider the initial volume of the tire. Instead, you can use the ideal gas law to relate the pressure, volume, and temperature before and after the pumping. Since the only change is in pressure, we can write:

P1V1 = P2V2

where P1 = 1 atm, V1 is the initial volume of the tire, P2 = 4 atm, and V2 is the final volume of the tire. We can then rearrange this equation to solve for V2:

V2 = V1(P1/P2)

Now, we can use the adiabatic compression equation to relate the initial and final temperatures (T1 and T2) to the initial and final volumes (V1 and V2):

V1T1^(5/2) = V2T2^(5/2)

Substituting in the value of V2 from the first equation, we get:

V1T1^(5/2) = V1(P1/P2)T2^(5/2)

Simplifying, we get:

T2 = T1(P2/P1)^(2/5)

Substituting in the values of T1, P1, and P2, we get:

T2 = T1(4/1)^(2/5)

T2 = T1(4)^(2/5)

T2 = 1.319T1

Therefore, the temperature of the air inside the tire just after it has been pumped is 1.319 times the initial temperature. This means that the temperature has increased due to the compression of the gas.

I hope this helps! Let me know if you have any further questions.