Reversible and Irreversible Expansion

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Reversible expansion occurs at equilibrium throughout the process, allowing for precise calculations of work done, typically represented by the integral of pressure with respect to volume. In contrast, irreversible expansion involves non-equilibrium conditions at some point along the path, resulting in an undefined or incalculable value for work. The distinction lies in the ability to accurately determine work in reversible processes due to their equilibrium nature, while irreversible processes lack this clarity. For exact formulas, the reversible work is expressed as W = ∫PdV, while irreversible work is often approximated and not easily defined.
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What's the difference between reversible and irreversible expansion? I know how they are in formulas, (Irreversible-PV, reversible- integral over P in terms of V), but why are they expressed like this? What's the difference? And I would also like to get an exact formula of these two as I can't really find a page on the web about this, if there is it would be nice if you can put a link to the site.
 
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Reversible: at equilibrium at every point on a defined path, giving you an exact calculation of work done.
Irreversible: non-equilibrium at at least one point somewhere on the path, leaving you with an undefined and incalculable value for work done.
 

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