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Delta H is just the difference between the final and initial values of H. Delta Hº is the value of the difference between the final and initial values of H when the products and reactions are at standard conditions.
deltaH^0 refers to the standard reaction enthalpy (the rxn enthalpy when all R and P are in their standard state). DeltaH is the general symbol for the amount of heat absorbed or released. There is also deltaHf^0 which refers to the Standard Enthalpy of Formation (the standard rxn enthalpy for the formation of 1 mol of a substance).
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