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*Subject - *Physical Chemistry, Thermodynamics

The differential reaction enthalpy ${\Delta}_{r}H$ is a partial differential quotient of the total differential of the state function $H=H(p,T,\xi )$. To demonstrate, this differential quotient is represented as a sum of the partial molar enthalpies of educts and products.

- $${\Delta}_{r}H={\left(\frac{\partial H}{\partial \xi}\right)}_{p,T}={\displaystyle \sum _{i}{\nu}_{i}{H}_{part,i}}\phantom{\rule{2em}{0ex}}\begin{array}{l}{\Delta}_{r}=\frac{\partial}{\partial \xi}=\text{differential operator}\\ \xi =\text{turnover variable}\\ p=\text{pressure}\\ T=\text{absolute temperature}\\ {\nu}_{i}=\text{stoichiometric number}\\ {H}_{part,i}=\text{partial molar enthalpies of substance i}\end{array}$$

The sum of partial molar enthalpies (partial molar size) can be written as a difference because the stoichiometric numbers of products are counted as positive while those of the educts enter the equation as negatives.

- $$\sum _{i}{\nu}_{i}{H}_{part,i}}={\displaystyle \sum _{i}{(|{\nu}_{i}|{H}_{part,i})}_{\mathrm{Pr}o}}-{\displaystyle \sum _{i}{(|{\nu}_{i}|{H}_{part,i})}_{Edu}$$

In case of an ideal reaction mixture, partial molar enthalpies can be substituted with molar enthalpies.

The integral reaction enthalpy $\Delta H$ is obtained by integration of the differential reaction enthalpies.

- $$\Delta H={\displaystyle \underset{{\xi}_{1}}{\overset{{\xi}_{2}}{\int}}{\Delta}_{r}H}d\xi =H({\xi}_{2})-H({\xi}_{1})\phantom{\rule{2em}{0ex}}\begin{array}{l}\Delta =\text{symbol for difference}\end{array}$$

Frequently, the integral reaction enthalpy $\Delta H\text{}$ is described as the difference of molar enthalpies ${H}_{\mathrm{m,i}}\text{}$ of the starting materials (products) and end products (educts) of a chemical reaction.

- $$\Delta H={\displaystyle \sum _{i}{({n}_{i}{H}_{m,i})}_{\mathrm{Pr}o}}-{\displaystyle \sum _{i}{({n}_{i}{H}_{m,i})}_{Edu}}$$

Molar instead of partial values can be used when assuming ideal conditions of the reaction mixture.

The integral reaction enthalpy is determined by measuring the heat of reaction of a reaction. Endothermic reactions show a positive enthalpy while reactions with negative enthalpy are called exothermic reactions.

If the pressure during the reaction remains constant, the integral reaction enthalpy equals the heat of reaction ${Q}_{p}$ being given or taken up by the reaction.

- $$\Delta H=\pm {Q}_{p}$$