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(OP)
In an Adiabatic process (reversible) we have two different process:
• Expansion.
• Compression.
Let’s assume we have a cylinder and a piston with a gas inside when the gas expands the gas exert a work done in surroundings. By the other hand in compression work is done over the gas, by convention the work done over the system is positive and the work done by the gas in negative.
The first law of thermodynamics state the following:
ΔU=Q+W (1 expression)
We know that in an adiabatic process the heat supply is zero, so ΔU=W.
The ΔU=nCvdt (volume constant).
The work done W= pdv (pressure constant).
I want to find this expression TVᵞ⁻ᴵ (second expression)
The deduction I got it, but with this expression.
ΔU=Q-W; (Third expression) Q=0 then ΔU=-W
Here is my issued in a compression process the work is positive (1 expression), because the work is applied over the system.
The second expression is for both expansion and compression. Because if I do the calculations with the positive work the second expression is different, Please help with this.

### RE: Adiabatic process expression (expansion/compression)

I am assuming that you are studying thermodynamics at work and not in school. To answer your questions study the section on reversible polytropic process of an ideal gas with constant specific heat . Your first expression and the second constant volume expression are consistent with the material found in the reversible polytropic process. the second expression is derived from (P2/P1)/(v1/v2)^n where P is pressure and v=V/m is specific volume, V is volume and m is mass; and from P1v1/T1= P2v2/T2. where T is absolute temperature. For a constant pressure process, n=0 and p1=p2 so manipulate those two expressions in the third sentence and you should get P2/P1=(v1/v2)^n=(t1/t2)^n. n=0 for constant pressure process
See the attachment of a copy from my Eschbach engineering handbook for polytropic processes of ideal gases; on page 845, you'll note the second block for constant pressure process.

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