Give the relation between `DeltaU and DeltaH`.

Explanation

When the system at constant pressure undergoes changes from an initial state `H_(1) ,U_(1) and V_(1)` to a final state with `H=U+PV`
In the initial state
`H_(1)=U_(1)+PV_(1)" "…(1)`
In the final sate
`H_(2)=U_(2)+PV_(2)" "…(2)`
Change in enthalpy is (2) -(1)
`(H_(2)-H_(1))=(U_(2)-U_(1))+P(V_(2)-V_(1))`
`DeltaH =DeltaU+PDeltaV" "....(3)`
As per first law of thermodynamics,
`DeltaU=q+w`
Equation (3) becomes
`DeltaH=q+w+PDeltaV`
`w=-PDeltaV`
`DeltaH=qp-PDeltaV+PDeltaV`
`DeltaH=pq" "...(4)`
`q_(p)` is the heat absorbed at constant pressure and is considered as heat content.
Consider a close system of gases which are chemically reacting to from gaseous products at constant temperature and pressure with `V_(i)and V_(f)` as the total volume of the reactant and product gases respectively, and `n_(i) and n_(f)` as the number of moles of gaseous reactants and product , then,
For reactants (initial state)
`PV_(i)=n_(i)RT" "...(5)`
For products (final state)
`PV_(f)=n_(i)RT" "...(6)`
(6)-(5)
`P(V_(f)-V_(i)) = (n_(f)-n_(i))RT`
`PDeltaV=Deltan_((g))RT" "....(7)`
Substituting in (7) in (3)
`PDeltaV=Deltan(g)RT" "...(8)`