Adiabatic Equation Derivation Troubleshooting

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MotoPayton
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I have been struggling to find my mistake for over 2 hours now and I still haven't exactly matched my answers with the books solution. I feel that my method should give the same answer the solution manual obtained however they are slightly off.

I understand the book method completely.

Can someone tell me why my method doesn't work or find a way to make it work. I won't be able to sleep until I know.

Thanks

Book Question:

Capture-2.jpg


Book Solution:

solution.jpg


My solution:

DSCN8328.jpg


Derivation of adiabatic equation to avoid confusion:

DSCN8329.jpg

 
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