Thermodynamics: polytropic processes

[silentwf]

Homework Statement

"During some actual expansion and compression processes in piston-cylinder devices, the gases have been observed to satisfy the relationship $$PV^n=c$$ where n and C are constants. Calculate the work done when a gas expands from 150kPa and .03 m^3 to a final volume of .2m^3 for the case of n = 1.3

Homework Equations

$$W = \int_{1}^{2} {P}dV$$

The Attempt at a Solution

$$PV^n=C \Rightarrow P=CV^{-n} \Rightarrow W = \int_{.03}^{.2} {150V^{-1.3}}dV = 621 kJ$$
Which...is wrong :(

The solution the book offers is:
$$P_{2} = P_{1}\frac{V_{1}}{V_{2}}^n = (150)\frac{.03}{.2}^{1.3} = 12.74 kPa \Rightarrow W = \int_{1}^{2} {P}dv = \frac{P_{2}V_{2} - P_{1}V_{1}}{1-n} =\frac{(12.74 \cdot .2 - 150 \cdot .03)}{1-1.3} = 6.51 kJ$$

Could someone explain why the way i did it is "unacceptable"?

 

[Mapes]

Because $C\neq150$; check your integrand.

 

[silentwf]

Oh...woops
I put in 150 because i was still thinking that $$W = P\Delta V$$ and I put in 150
then what would i put in? the question does not supply C though

 

[Borek]

You can calculate C from initial state.

--
methods

 

[silentwf]

oh lol!
ok, i got the answer, thanks :)