What is the relation between H and U?

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:
• ${\mathrm{U}}_2–{\mathrm{U}}_1={\mathrm{q}}_{\mathrm{p}}–\mathrm{p}({\mathrm{V}}_2–{\mathrm{V}}_1)$
• On rearranging, ${\mathrm{q}}_{\mathrm{p}}=({\mathrm{U}}_2+{\mathrm{PV}}_2)–({\mathrm{U}}_1+{\mathrm{PV}}_1)$
• 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)