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# What is the relation between H and U?

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## Internal energy:The heat absorbed at constant volume is equal to the change in the internal energy."It is expressed as: $∆\mathrm{U}=\mathrm{qV}$This equation can also be written as $∆\mathrm{U}={\mathrm{q}}_{\mathrm{p}}–\mathrm{P}∆\mathrm{V}$ …………….(i) at constant pressure, Where ${\mathrm{q}}_{\mathrm{p}}$ is heat absorbed by the system and $–\mathrm{P}∆\mathrm{V}$ represent expansion work done by the system.Enthalpy:“The another thermodynamic function, the enthalpy H [Greek word enthalpien, to warm or heat content]”It is expressed as:$\mathrm{H}=\mathrm{U}+\mathrm{PV}$……………….(ii)Relation between H and U:From equation (i), it can be written as:${\mathrm{U}}_{2}–{\mathrm{U}}_{1}={\mathrm{q}}_{\mathrm{p}}–\mathrm{p}\left({\mathrm{V}}_{2}–{\mathrm{V}}_{1}\right)$ On rearranging, ${\mathrm{q}}_{\mathrm{p}}=\left({\mathrm{U}}_{2}+{\mathrm{PV}}_{2}\right)–\left({\mathrm{U}}_{1}+{\mathrm{PV}}_{1}\right)$Thus, this equation can become, ${\mathrm{q}}_{\mathrm{p}}={\mathrm{H}}_{2}–{\mathrm{H}}_{1}=∆\mathrm{H}$Finally, the change in enthalpy can be written as: $∆\mathrm{H}=∆\mathrm{U}+\mathrm{P}∆\mathrm{V}$ (since p is constant)

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