(THERMO) - Adiabatic Expansion - known P1, P2, and T1

[Zomboy]

Question:

in a bicycle pump the preasure increases from [p1 = 10^5] to [p2 = 30^5]. If the process is adiabatic ant the air starts at [T1 = 293 K], find the maximum temperature of the air in the pump. (Assuming air can be treated as an ideal gas)

Attempt:

So using the 1st Law and given that its adiabatic (no heat input/output) I've written:

U = W

(C_v)dT = PdV

(C_v)dT = (nR) (1/v) dV

then integrated and re-arranged to obtain:

T2 / T1 = (V2/V1)^(nR/c)

then substituted in { C_v = nR5/2 } because its an adiabatic process, and { T2 P1 / P2 T1 } for V2/V1 using the equation for an ideal gas...

But then when I re-arranged to find T2 I found it to be 33K which is wrong of course...


Is this sort of the way to do it? Its just I can't think of another way seeing as it doesn't specify the initial volume of the tyre...

 

[nasu]

T2 / T1 = (V2/V1)^(nR/c)

This equation does not look right. If the volume increases (expansion) the temperature should decrease.
And there is no "n" in the exponent.
The change in internal energy is nCvΔT.

Anyway, what is the point in finding a T,V relationship when you know the pressures but not the volumes?
Use the gas law to substitute V in the first law and integrate to find a relationship between T and p. It's more straightforward.