Standard Enthalpy and Gibbs free energy of reaction

Standard Enthalpy and Gibbs free energy of reaction 
From the foregoing discussion it may be apparent that a chemical reaction may be carried out in diverse ways by changing temperature, pressure, and feed composition. Each of the different conditions would involve different conversions and heat effects. Thus there is need to define a “standard” way of carrying out a reaction. If all reactions were carried out in the same standard manner, it becomes possible to compare them with respect to heat effects, and equilibrium conversion under the same conditions. In general all reactions are subject to heat effects, whether small or large. A reaction may either release heat (exothermic) or absorb heat (endothermic). However, it is expected that the heat effect will vary with temperature. Thus, there is a need to develop general relations that allow computation of the heat effect associated with a reaction at any temperature. 
Consider a reaction of the following form: 

The reactants (A1 and A2) and products (A3 and A4) may be gaseous, liquid or solid. The term is the stoichiometric coefficient corresponding to the chemical species Ai. For the purpose of development of the reaction equilibria relations it is convenient to designate the stoichiometric numbers of the reactants as negative, while those of the products as positive. This is to signify that reactants are depleted in proportion to their stoichiometric numbers, while the products are formed in proportion to their stoichiometric numbers. Consider, for example, the following gas-phase reaction:
The stoichiometric numbers are written as follows:
The standard enthalpy of reaction at say at any temperature is defined in the following manner: it is the change in enthalpy that occurs when  moles of A1 and  moles of B2 in their standard states at temperature T convert fully to form moles of A3 and  moles of A4 in their respective standard states at the same temperature T. The standard states commonly employed are as follows:
  • Gases: the pure substance in the ideal gas state at 1 bar
  • Liquids and Solids: the pure liquid or solid at 1 bar
    The conceptual schema of a standard reaction is depicted in fig. 8.1. All reactants enter and products leave the reactor in pure component form at the same temperature T, and at their respective standard states. In the literature, data on the standard enthalpy of reaction is typically reported.
temperature of 2980K. Using the sign convention adopted above, the standard enthalpy of reaction at any temperature T may be mathematically expressed as follows:

Where, is the standard state enthalpy of species ‘i’at the temperature T, and the summation is over all the reactants and products. 

If we further consider that each molecular species ‘i’ is formed from j elements each, an expression for the standard enthalpy of formation results:

Where, the summation is over all constituent elements that make up the ith molecule, is standard state enthalpy of formation of the ith molecule at T, and  the standard state enthalpy of the jth atomic species. If all  are arbitrarily set to zero as the basis of calculation then eqn. 8.3 simplifies to:

In such a case eqn. 1 becomes:

For simplicity in the subsequent equations we drop the subscript T, but implicitly all terms correspond to temperature T. Now writing  in a differential form:  

Where  is the specific heat of the ith species corresponding to its standard state. Note that since the standard state pressure for all substances is 1 bar, for gases , while for liquids and solids it is the actual value of the specific heat at 1 bar . Since the specific heat of liquids and solids are weakly dependent on pressure. The following summation may be applied on above equation to give:


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