Thermodynamics of open soda can

[helpmethermo2]

Homework Statement

C02 from a can of soda expands irreversibly against the atmospher. Assume the process is adiabatic and Pi = 3 bar. Assume C02 to be an ideal gas with Cp = 37 j/mol*K. Find the final temperature of CO2 after it has reached atmospheric pressure.

Thats all I get...

Homework Equations

Adiabatic implies dq = 0
constant Pext implies dw = -Pext dV
Ideal gas implies dU = CvdT
1st Law gives dU = -Pext dV
we go from here

The Attempt at a Solution

nCvdT = -Pext dV

integrating
nCv(T2-T1) = -Pext (V2-V1)

using PV=nRT the above equation reduces
nT2(Cv+R) = nT1(Cv+Pext*R/P1)

we can take Pext/T2 = P1/T1 ?? but this system does not yield a solution for T1, T2

here is were I am stuck seeing as how I don't know T1 or T2 (obviously n's cancel out)

If we take the can as the system
the first law energy balance gives

dU/dt = -dn_out/dt * H

this reduces to the standard adiabatic equation

(T2/T1)^Cp/R = Pext/P1

an attempt with the concervation of mass yields
dn/dt = -dn_out/dt we can sub in the ideal gas equation for n and get
d(p/T)/dt = -dn_out/dt * R/nV but I don't think we know anything about n or dn_out/dt

so I am stuck, help please!

Thanks

 

[Winzer]

I think it's before you integrate is your problem. Use ideal gas law to sub for pext.