Give the relation between `DeltaU and DeltaH`.

Explanation

(i)When the system at constant pressure undergoes changes from an initial state with `H_1,U_1,V_1` and P parameters to a final state with `H_2,U_2,V_2` and P parameters, the change in enthalpy `DeltaH`, is given by
`DeltaH=U+PV`
(ii)At initial state `H_1=U_1+PV_1`...(1)
(ii)At final state `H_2=U_2+PV_2` ...(2)
(2) -(1) `rArr (H_2-H_1) = (U_2-U_1) +P(V_2-V_1)`
`DeltaH=DeltaU+PDeltaV` ....(3)
Considering `DeltaH=q+w , w=-P DeltaV`
`DeltaH=q+w+PDeltaV`
`DeltaH=q_P - PDeltaV + PDeltaV`
`DeltaH=q_P` ....(4)
`q_P` is the heat absorbed at constant pressure and is considered as heat content.
(iii) Consider a closed system of gases which are chemically reacting to produce product gases at constant temperature and pressure with `V_i` and `V_f` as the total volumes of the reactant and product gases respectively, and `n_i` and `n_f` are the number of moles of gaseous reactants and products . Then ,
For reactants : `PV_i =n_i RT`
For products : `PV_f =n_f RT`
Then considering reactants as initial state and products as final state ,
`P(V_f-V_i)=(n-f-n_i)RT`
`PDeltaV=Deltan_g RT`
We know
`DeltaH=DeltaU+PDeltaV`
`therefore DeltaH=DeltaU +Deltan_g RT` ....(5)