Feynman problem 39-2: Calculations for an adiabatic process

[aa_o]

Homework Statement

See attached

Relevant Equations

P*V / T= constant
P*V^(y) = constant (if process is adiabatic)

I'm not sure that this is an adiabatic process. As far as i can read, it is adiabatic if no HEAT or ENERGY is added. But pumping in molecules that are a non-zero temperature is an addition of energy, no?
Anyway - my solution with the assumption of an adiabatic process.
(skipping units for brevity):
P0 = 14.7
P1 = 50.0
T0 = 293
y = 1.40

We have:
P0 * V0^y = P1 *V1^y
So:
V1 / V0 = (P0 / P1)^(1/y)

And
P0*V0 / T0 = P1*V1 / T1
So:
P1 / P0 * V1 / V0 = T1 / T0
Inserting:
(P0 / P1)^(-1) * (P0 / P1)^(1/y) = T1 / T0
T1 = (P0 / P1)^(1/y - 1) * T0

Inserting the values we get

T1 = 415.7 K = 142.55 C

The answer in the book says 173 C.
Are my assumptions about adiabatic wrong? Or am i using the wrong equations?
Can i really use P*V / T = constant if we are adding air molecules with the pump?

Any suggestions would be appreciated.

 

[mjc123]

Can i really use P*V / T = constant if we are adding air molecules with the pump?

No. PV = nRT, and n is changing.

 

[aa_o]

Okay. So the pump has a start state of:
P0 * V0 = n0 * R * T0
And end state of:
P1 * V0 = n1 * R * T1

I thus have 2 unknowns i need to solve the problem T1, which is what i need in the end, and the ratio of the number of molecules between the 2 states. I just can't find that extra connection that gives me the information about the ratio!

I still haven't used the fact that y = 1.40. I just can't see where that fits in.

Any suggestions?

 

[mjc123]

Thinking about it again, if a constant amount of air is being compressed within the pump to 50 psig before being injected into the tyre (is this how the pump works?), then I think your original method is correct. But read carefully. The exit pressure is 50 psi gauge - what does that mean?

 

[aa_o]

Thanks, mjc123.

So n is constant for the whole system (tire + pump), but the volume then changes (compresses).

Ahh, i think that gauge was the missing piece. I didn't know about the meaning of gauge and simply skipped over it without paying much attention.

With that information (P1 = P0 + 50) i get an answer of 174.5 - close enough to the one in the book.

 

[mjc123]

So n is constant for the whole system (tire + pump), but the volume then changes (compresses).

No, n is constant for the air in the pump. As I read it, the pump compresses an amount of air from atmospheric to 50 psig, then injects it into the tyre (presumably the valve opens at that pressure).

 

[aa_o]

No, n is constant for the air in the pump. As I read it, the pump compresses an amount of air from atmospheric to 50 psig, then injects it into the tyre (presumably the valve opens at that pressure).

Yeah, that makes sense. I think there was a lot of assumptions that had to be made that wasn't explicitly stated in the problem.

But thanks a lot for the help!