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The number of moles at equilibrium is calculated for the Haber process, the reversible, exothermic reaction that synthesizes ammonia (NH3) from hydrogen (H2) and nitrogen (N2). The reaction is typically carried out at around 200 bar and 675–725 K. You can vary the pressure and temperature in this Demonstration. Gases are assumed to behave ideally, although realistically at the high pressures used in this reaction, there is significant deviation from ideal behavior. Four moles of reactants form two moles of product, so raising the pressure shifts equilibrium toward products. Initially, the system is filled with one mole of ammonia and allowed to reach equilibrium. At equilibrium, you can add additional nitrogen, hydrogen, and/or ammonia at constant pressure, and the effect on the equilibrium is observed. Le Chatelier's principle predicts that when nitrogen or hydrogen are added, the reaction goes to the right, whereas when ammonia is added, the reaction shifts to the left. However, when the nitrogen/hydrogen ratio is sufficiently high, adding nitrogen shifts reaction to the left (i.e. adding nitrogen decreases the amount of ammonia and increases the amount of hydrogen), contrary to what Le Chatelier's principle predicts. This happens because adding nitrogen decreases the mole fraction of hydrogen, and because the hydrogen mole fraction is cubed in the equilibrium expression, ammonia reacts to increase the number of moles of hydrogen and nitrogen.
The reaction is used in the Haber process. The moles of each component at equilibrium is:
,
where are the moles of component
added,
is the stoichiometric coefficient and
is extent of reaction (mol). Initially only 1 mol
is present.
The mole fraction at equilibrium is:
,
,
where is the total number of moles.
The extent of reaction is found by setting the equilibrium constant equal to the equilibrium rate constant
and solving for
:
,
,
where is Gibbs free energy (J/mol),
is the heat of reaction (J/mol),
is the entropy change of reaction (J/(mol K)),
is temperature (K),
is the ideal gas constant (J/(mol K)) and
is pressure (bar).
The screencast at [1] show solutions of an example problem on gas phase equilibrium.
Reference:
[1] Gas Phase Chemical Equilibrium [Video]. (Jan 20, 2017) www.youtube.com/watch?v=3ArBH_gbsNw.