Relation between enthalpy change and internal energy of a gaseous reaction

Deduction of the Mathematical Relation between dH and dU for a Gaseous Chemical Reaction

To derive the mathematical relationship between the differential changes in enthalpy (dH) and internal energy (dU) for a gaseous reaction, we start with the fundamental thermodynamic definitions.

Enthalpy (H) is defined as:

H = U + PV

Differentiating both sides with respect to the reaction progress gives:

dH = dU + d(PV)

For an ideal gas, the product PV is related to the number of moles (n), gas constant (R), and temperature (T) by the equation:

PV = nRT

Assuming the reaction occurs at constant temperature (isothermal conditions), the differential of PV becomes:

d(PV) = d(nRT) = RT·dn + nR·dT

Since the temperature is constant (dT = 0), this simplifies to:

d(PV) = RT·dn

Substituting this back into the expression for dH:

dH = dU + RT·dn

In this expression, dn denotes the variation in the quantity of gas moles involved in the reaction, determined by:

\(dn = n_{products} - n_{reactants}\)

Hence, for a gaseous reaction occurring under constant temperature and pressure, the infinitesimal change in enthalpy can be expressed in terms of the change in internal energy as follows:

dH = dU + RT·dn

This equation shows that the change in enthalpy includes not only the internal energy change but also the work associated with the expansion or compression of gases as the number of moles changes during the reaction.