(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

After integrating the pressure formula of an adiabatic system, I have to show how this is equal to the change in energy. I know that my integral is correct (it was very straight forward), but I'm having trouble showing that it is equal to [itex]\Delta[/itex]U.

2. Relevant equations

[itex]\Delta[/itex]U = W + Q (Q = 0)

PV = NkT

[itex]\Delta[/itex]U = 1/2 * N * k * f * [itex]\Delta[/itex]T

Where:

k = 1.381 * 10^{-23}

f = degrees of freedom

[itex]\Delta[/itex]T = change in temperature

3. The attempt at a solution

W = -P_{i}V_{i}(V_{f}^{1-[itex]\varphi[/itex]}- V_{i}^{1 - [itex]\varphi[/itex]}) / 1 - [itex]\varphi[/itex]

Given that, I need to somehow get that to be

[itex]\Delta[/itex]U = 1/2 * N * k * f * [itex]\Delta[/itex]T

I managed to reduce W to:

-NkT_{i}((V_{f}/ V_{i})^{1 - [itex]\varphi[/itex]}- 1) / 1 - [itex]\varphi[/itex]

But I'm stuck from there.

(Note: [itex]\varphi[/itex] = (f + 2) / f )

Any hints on what to do next would be very helpful.

Thanks

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# Thermaldynamics - Adiabatic system W = delta(U)

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