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What is the work done by the piston when the piston is compressed and there's no internal energy/heat lost from the gas(thermally insulated)? The reason I can't solve this problem is because I have trouble writing pressure as a function of volume/displacement.Here's how far I've done.

Subscript 0 indicates original piston position; 1 indicates piston position after compressed

x: height of piston

A: cross section area of piston

Ideal gas law PV=nRT

$W=\int_{x_0}^{x_1}F\left(x\right)dx$

$=A\int_{x_0}^{x_1}P\left(x\right)dx$

$=A\int_{x_0}^{x_1}\frac{nRT\left(x\right)}{V\left(x\right)}dx$

$=A\int_{x_0}^{x_1}\frac{nRT\left(x\right)}{Ax}dx$

$=nR\int_{x_0}^{x_1}\frac{T\left(x\right)}{x}dx$

I don't know how to express temperature as a function of height, since that would be related to the work done by the piston, which is something I'm calculating.

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A process in which there is no exchange of heat between the gas and the surroundings is called an 'adiabatic process' and it obeys the law $PV^γ =constant$

Now, dW = Fdx =PAdx

But, Adx = dV

Therefore dW = PdV

Now use PV = nRT and $PV^γ=c$ to get

$ dW = cdV/V^γ $

Integrate and substitute $ c=P_(initial)V_(initial)^γ $

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