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A reversible process means that it can be reversed by a very small change, while an irreversible process cannot be. For reversible expansion, it occurs when the internal and external pressure are equal (the system is at equilibrium)
The equations used to calculate work also vary depending on whether the process is reversible or irreversible. Reversible processes are isothermal, with the equation w=-nRT(ln(vf/vd)). Work for irreversible processes can be calculated using w=-Pext+Δv.
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