Chemical Reactions vs. The First Law: Who Wins?

DOFPro Team

There are two principal methods for calculating enthalpy changes associated with reactions. The first is to calculate the heat of reaction and extent of reaction for each reaction. The reference states are the pure compounds at the reference temperatures and pressures. The First Law then becomes:

First Law for Steady-State Reactive Systems (no shaft work)

\[\sum\limits_{\substack{\mathrm{out} \\ \mathrm{streams}}}\left(\dot{n}_i + \sum\limits_k \nu_{ik} \dot{\xi}_k\right)\hat{H}_j\ - \sum\limits_{\substack{\mathrm{in} \\ \mathrm{streams}}} \dot{n}_i \hat{H}_i + \sum\limits_\mathrm{rxns} \dot{\xi}_k \left( \Delta \hat{H}^\circ_r\right)_k = \dot{Q}\]

Or implicitly accounting for the change in number of moles

\[\sum\limits_{\substack{\mathrm{out} \\ \mathrm{streams}}} \dot{n}_j \hat{H}_j\ - \sum\limits_{\substack{\mathrm{in} \\ \mathrm{streams}}} \dot{n}_i \hat{H}_i + \sum\limits_\mathrm{rxns} \dot{\xi}_k \left( \Delta \hat{H}^\circ_r\right)_k = \dot{Q}\]

First Law for Batch Systems at Constant Pressure


\[\sum\limits_{\mathrm{final}}\left(n_i + \sum\limits_k \nu_{ik} \xi_k\right)\hat{H}_j\ - \sum\limits_{\mathrm{initial}} n_i \hat{H}_i + \sum\limits_\mathrm{rxns} \xi_k \left( \Delta \hat{H}^\circ_r\right)_k = Q\]


Or implicitly accounting for the change in number of moles


\[\sum\limits_{\mathrm{final}} n_j \hat{H}_j\ - \sum\limits_{\mathrm{initial}} n_i \hat{H}_i + \sum\limits_\mathrm{rxns} \xi_k \left( \Delta \hat{H}^\circ_r\right)_k = Q\]

First Law for Batch Systems at Constant Volume


\[\sum\limits_{\mathrm{final}}\left(n_i + \sum\limits_k \nu_{ik} \xi_k\right)\hat{U}_j\ - \sum\limits_{\mathrm{initial}} n_i \hat{U}_i + \sum\limits_\mathrm{rxns} \xi_k \left( \Delta \hat{U}^\circ_r\right)_k = Q\]


Or implicitly accounting for the change in number of moles


\[\sum\limits_{\mathrm{final}} n_j \hat{U}_j\ - \sum\limits_{\mathrm{initial}} n_i \hat{U}_i + \sum\limits_\mathrm{rxns} \xi_k \left( \Delta \hat{U}^\circ_r\right)_k = Q\]


Remember: All of these absolute enthalpies and internal energies imply reference enthalpies at some standard temperature pressure and state of aggregation.

If enthalpy depends only on temperature:


\[\sum\limits_{\substack{\mathrm{out} \\ \mathrm{streams}}} \dot{n}_j \int\limits_{T_\mathrm{ref}}^{T_\mathrm{out}} C_{pj}\ dT - \sum\limits_{\substack{\mathrm{in} \\ \mathrm{streams}}} \dot{n}_i \int\limits_{T_\mathrm{ref}}^{T_\mathrm{in}} C_{pi}\ dT + \sum\limits_\mathrm{rxns} \dot{\xi}_k \left( \Delta \hat{H}^\circ_r\right)_k = \dot{Q}\]


where


\[\dot{n}_j = \dot{n}_i + \sum\limits_\mathrm{rxns} \nu_{ik} \dot{\xi}_k\]

The second method is to choose as reference states the pure elements at standard temperature and pressure. The enthalpy of a given species is then the enthalpy of formation plus any changes due to temperature (\(\int C_p dT\)), pressure, solution, or such. The 1st Law then becomes:

\[\sum\limits_{\substack{\mathrm{out} \\ \mathrm{streams}}} \dot{n}_j \hat{H}_j\ - \sum\limits_{\substack{\mathrm{in} \\ \mathrm{streams}}} \dot{n}_i \hat{H}_i = \dot{Q}\]

and its variants.

Don’t forget about latent heats and enthalpies of solution or mixing.

The Takeaways

  1. We presented the 1st Law for steady-state reactive systems.
  2. We presented the 1st Law for batch reactive systems at constant pressure.
  3. We presented the 1st Law for batch reactive systems at constant volume.
  4. We gave a second method that uses enthalpies of formation rather than enthalpies of reaction.






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The DOFPro Team