Why do we use the formula Pdv for calculating work done in a irreversible process ? As per my knowledge , the term P indicates the pressure of the surrounding , which infinitesimally differs for the pressure of the system , thus P can be treated as a constant . Staying on topic,when is the formula Vdp used and why ?
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asked Aug 14, 2015 at 14:55
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1 Answer
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$p$ does not necessarily mean ‘the pressure of the surrounding’, its meaning depends on the problem. In an archetypical problem where a piston is moved down a cylinder filler with an Ideal Gas, an infinitesimal amount of work $dW=pdV$ is performed when the piston is moved an infinitesimal amount, with $p$ the pressure inside the cylinder.
To obtain a macroscopic, that is ‘practical’, amount of work this needs to be integrated between relevant boundaries:
$W=\int pdV$ and in such a case $p$ is unlikely to be constant. In fact $p$ must obey the Ideal Gas Law $pV=nRT$, so at a minimum $p$ is a function of $V$. The exact relation between $p$ and $V$ depends on the nature of the compression, e.g. isothermal or adiabatic.
$Vdp$ is a expression that can be useful in any number of contexts and can be obtained for example by partial derivation of relations between $p$ and $V$ to $p$ (i.e. $\frac{\partial}{\partial p}$). Expressions like $pdV$ and $Vdp$ are useful in all kinds of derivations with regards to calculating the parameters of an Ideal Gas.
