[21385]

What volume of air at 1.0 atm and 22C is needed to fill a 0.98 L bicycle tire to a pressure of 5.0 atm at the same pressure? (Note that 5.0 atm is the gauge pressure, which is the difference between the pressure in the tire and the atmospheric pressure. Initially, the gauge pressure in the tire was 0 atm.)

What I don't really understand about this question is whether to use gauge pressure or total pressure for PV=PV. In the solutions manual, they used gauge pressure (ie. V(2)=(5.0atm)(0.98 L)/(1.0 atm)) but I do not think it is right because the actual pressure in the tire is 6.0 atm. Can someone help me out? Thanks

[Rabn]

Are you sure that you understand how the pressure gauge works?

[21385]

I'm not sure.

[Rabn]

The pressure that the gauge reads is usually the actual pressure inside the tire, unless the problem states the mechanism used by the gauge is unique you can assume that it reads the exact pressure in the tire. Do you understand how to derive the equation?

[billnotgatez]

According to this problem the tire pressure gage read 0 atm when allowed to read outside air pressure or did I read this wrong.

[Hunt]

The problem already gives the definition of gauge pressure.

Gauge Pressure inside the tire = Pressure inside the tire - atmospheric pressure

G = P - Patm , so P = G + Patm

Initially it is given that G = 0 , so pressure inside the tire is equal to atmospheric pressure = 1 atm. ( not 6 atm ). This makese sense as there mechanical equilibrium is established between the air inside the tire and that outside the tire.

This is why they used 1 atm. However, there seems to be something not logical in this problem. I'll post later.

[Hunt]

There should be more clearness in the given so that the reader would understand with less confusion. Anyway , this is how I believe the problem should be solved.

At the outset, the bicycle tire has some air in it , the pressure of the gas should be equal to the atmospheric pressure = 1 atm as stated earlier. ( Part of the given, can be computed from the gauge pressure ).

Imagine certain amount of air flows from the atmosphere to fill the tire. Eventually the tire becomes filled with air of gauge pressure = 5 atm. The initial volume of the gas is unknown , V1, but its final volume V2 should be equal to the volume of the tire ( ideal gas ). The gauge pressure is due to the total pressure of the air inside the tire.

I talk about "total" pressure of the gas inside because I assume there is the initial air present ( of partial pressure 1 atm ) and the new air that has passed in of partial pressure P.

Total pressure inside the tire , Pt = G + Patm =  5 atm + 1atm = 6 atm

But the tire already had certain amount of air of partial pressure 1 atm , so the partial of pressure of the air that has been transfered is P = Total pressure inside the tire - partial pressure of air already present = 6 - 1 = 5 atm

The process is isothermal and the number of particles of the moving is constant ( N is fixed ) , so the following holds : P1V1 = P2V2

P1 = 1 atm , the intial pressure of the gas ( atm pressure )
V1 = initial volume , unknown
P2= final partial pressure of the gas transfered = 5 atm
V2 = final volume occupied by the gas = 0.98 L

I hope this helps.

[Rabn]

I wonder if there is a typo at the end of the question, instead of reading "at the same pressure?" it should read "at the same temperature?" which would validate the isothermal expansion and constant "N"  assumptions that Vant_Hoff made.  If that is so I would say that Vant_Hoffs explanation is right on and should clarify it for you.

[Hunt]

Yes good point Rabn. Same pressure ? Seems inconsistent with the given , but I ignored that and solved the problem in a way that seems logical ( but not realistic ) to me. Ideal gases mix isothermally , and that's about it. 

Rabn , my assumption that N is constant holds true for all closed systems. The system here is the air itself entering the tire. Its amount is fixed however the temperature is varied. 

[21385]

Sorry about that, that is a typo.

Thanks for the explanation, I really appreciate it.