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I am considering a system with initial pressure $P_{A}$ and volume $V_{A}$. The internal energy $U_{A}$ should be zero. Then I have the following law for an ideal mono atomic gas $P^{3}V^{5}=const$. Now if the system is expanding to a final volume of $V_{B}$ and pressure $P_{B}$. Then I can use the following relation $P_{A}^{3}V_{A}^{5}=P^{3}V^{5}$ to obtain a relation for the pressure depending on V: $$P(V)=P_{A}V_{A}^{5/3}V^{-5/3}$$ and $$U_{B}-U_{A}=-\int_{V_{A}}^{V_{B}}P(V)*dV=\frac{3}{2}P_{A}V_{A}\left [\left ( \frac{V_{A}}{V_{B}} \right )^{3/2} -1 \right].$$ and by setting $U_{A}=0$,

$$U_{B}=-\int_{V_{A}}^{V_{B}}P(V)*dV=\frac{3}{2}P_{A}V_{A}\left [\left ( \frac{V_{A}}{V_{B}} \right )^{3/2} -1 \right]$$.

What is if I expand the system to the same volume $V_{B}$ but with different pressures $P_{B}$ and $P_{B}^{*}$ then the energy of the final state would be the same independent of the final pressure. What am I missing here??

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    $\begingroup$ Your derivation assumes that you are expanding or compressing the gas according to the adiabatic condition $P^3V^5=constant$. So the pressure at state B is fixed by the volume $V_B$ at state B. Trying to allow the pressure at state B to vary is in contradiction to the assumptions you made in deriving your final equation. You always have to keep in mind the assumptions and conditions you used in deriving any physics equation. $\endgroup$
    – user93237
    Commented Mar 13, 2018 at 18:37
  • $\begingroup$ As @SamuelWeir says, due to the adiabatic assumption, $P$ and $V$ are related to each other. you cannot choose $P_B$ freely if $V_B$ is fixed. If you choose to vary $P_B$, then $V_B$ will depend on $P_B$. $\endgroup$
    – Dlamini
    Commented Mar 14, 2018 at 4:49

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The comments are answering my question very nicely. Therefore to close this question which is not possible as far as I know without having an answer thread. I just want to summarize the comments of Samuel Weir and Dlamini.

Since $P^{3}V^{5}=constant$ one can not reach arbitrary points by an adiabatic expansion in a P-V diagram. The only allowed points to reach are those defined by the curve $P^{3}V^{5}=constant$. And therefore for every final pressure $P_{B}$ there exists a well defined volume $V_{B}$ and vice versa. To reach any point off this adiabatic curve one would have to use a combination of an adiabatic process and an isochoric process to alter the pressure to the desired value at constant volume.

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