Doing the calculations needed to analyze the stoichiometry of a chemical reaction can be a difficult task. This is especially true when the reaction is said to proceed from a given initial state to chemical equilibrium rather than proceeding all the way to completion. The assumed total consumption of the limiting reagent does simplify calculations, but for most reactions (including the most important types of reactions) this assumtion can be VERY wrong. Unit conversion is always an important part of solving any problem in reaction stoichiometry, but setting up unit conversions appropriate to a given problem can be quite difficult and clumsy at times. A group of college students once visited the campus of the university where I teach general chemistry to engineering majors. The visitors complained about having to do a "stoichiometry problem from hell" for the course they were taking. One of the host students, familiar with the arrow diagram method, took the problem from the complaining visitors and had it solved in a few minutes. Impressed, the visitors asked to photocopy some the host student's notes so they might use them to teach themselves problem solving with the arrow diagram method. As an example of a relatively difficult sort of problem (maybe a problem from hell for some students), consider the following:
Total consumption of the limiting reagent cannot be assumed in a problem like this. The measured fact given in the problem must be used as the basis for solving the problem:
Formulas for ideal gases are then used to convert the relationship for pressures into a relationship for moles of gas:
The gas constant, the temperature, and the volume all cancel out, leaving only moles of gas:
final mole N2= 1.30 ( final mole NH3+final mole NO )
This last equation is especially well suited to the use of the arrow diagram. In its general form, the preliminary arrow diagram for the given reaction would be set up as follows:
With all values in moles, the amount of each substance initially present is placed at the initial end of each arrow (Note: It is not always true that the initial amount of a given reaction product is zero moles.) Changes (in moles used for reactants and moles formed for products) are placed in the middle of each arrow. And moles present in the final reaction mixture (i.e., at chemical equilibrium) is placed at the terminal end of each arrow. The problem gives the grams of gas put in at the start of the reaction, with only the water vapor being not put in. The grams are converted to moles by conventional unit conversions, and the next set up for of the arrow diagram looks like the following:
The fundamental need to answer any question about a reaction is to calculate the value of "x" as a number in the arrow diagram, then use that value to answer the particular question asked. The relationship for final moles is given by the logical equation developed previously, so using terms from the arrow diagram:
( 0.8929 + 5x ) final moles N2 = 1.30 [ (1.294 - 4x) mole NH3 + (1.167 - 6x) mole NO ]
Solving this equation,x = 0.1281 with no unit, as the value of "x" applies to the reaction as a whole.
The value of "x", once known, should be used in the most direct way possible to answer the question asked in a particular problem. There is no real need to complete the arrow diagram with all numbers in place, but if done, the completed arrow diagram would look like the following:
Conventional unit conversions, using values from the numerically complete arrow diagram, would now be able to answer any given question about the reaction and the changes the reaction causes.
There are various ways to calculate the percent yield of the reaction to answer the question, but the simplest way would probably be to use the value of "x" calculated from the given fact about final partial pressures (this value being "xact" since it corresponds to the actual extent of the reaction) and the value represented as "x100" (this value representing what "x" would be to cause total consumption of the limiting reagent). The value of "x100" can usually be calculated by simply dividing the initial moles of each reactant by its stoichiometric coefficient in the reaction:
Then calculating the percent yield of the reaction:
For over twenty years, I have been teaching the arrow diagram method to Engineering Physics majors at Embry-Riddle Aeronautical University. Not all students can understand and learn the arrow diagram method well enough to use it effectively in problem solving, but most students manage to learn to use the method reasonably well. Ans the students who master it will probably never be confounded by a reaction stoichiometry problem, even one from hell.