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# Synthesis of Ethanol by Hydration of Ethylene

Consider the following gas-phase reaction to produce ethanol from ethylene: . Take the initial number of moles of ethylene to equal 1. The initial number of moles of water is determined by the selected steam-to-ethylene molar ratio. Initially, there is no ethanol in the reaction vessel. This Demonstration plots the extent of reaction versus pressure, , for user-set values of the temperature, . It follows from the stoichiometry of the reaction that the extent of the reaction is equal to the fractional conversion of ethylene at equilibrium.
The reaction gas mixture is described by a virial equation of state , where the second virial coefficient is given by .
The blue curve is the plot for a reacting gas mixture that is an ideal solution (the fugacity coefficients are computed using a virial equation of state that does not include the cross-coefficient terms: ). The magenta curve is the plot for a reacting mixture of real gases (the calculation of the fugacity coefficients includes the cross-coefficient terms in the virial equation of state). Finally, the brown curve is the plot for a reacting mixture that behaves as an ideal gas mixture: (all fugacity coefficients are set equal to unity). It is clear that the conversion approaches unity at low temperatures since the equilibrium constant is large for this exothermic reaction. Also, the conversion for all cases (ideal gas mixture, ideal solution, and real gas mixture) are identical at low pressures. Finally, since (i.e., the sum of the stoichiometric coefficients is negative), increasing the pressure will lead to higher conversions (Le Chatelier's principle). Present calculations are valid only for the gas-phase reaction, which means that the pressure, , cannot be higher than the approximate dew point pressure displayed in red in the extent of reaction versus plot.
Finally, the equilibrium compositions, computed for the real gas mixture case, are also displayed in a separate plot. The blue, magenta, and brown curves correspond to the mole fraction of ethanol, water, and ethylene at equilibrium, respectively.

### DETAILS

The equilibrium constant is equal to , where is the fugacity coefficient of species in solution, , , and .
1: For the ideal gas mixture, assume .
2: For the ideal solution mixture, assume , where is the fugacity coefficient of the pure species . From the virial EOS, , where is the pure-species second virial coefficient.
3: For the real gas mixture case: the second order cross virial coefficients must be used and we have , where .
Reference
[1] J. M. Smith, H. C. Van Ness, and M. M. Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., New York: McGraw-Hill, 2005.

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