# Limiting Reagent Problems

Limiting Reagent Problems
Dr. MJ Patterson

Most chemical reactions are examples of the limiting reagent (reactant) problem.  One of the reactants will run out before the others, and so it will limit how much product can be made.  When the reaction is over, the container (beaker, flask, drum...) will contain the products along with some of the other reactants that were present in excess.  It is very rare to run a reaction where exactly the right amounts of reactants are used so that absolutely none are left over.

There are usually very practical considerations for having a limiting reagent with others in excess.  One reagent might be very expensive, and so you want to use it in the smallest possible quantities and under conditions that favor it all being used.  Or, one reagent might be very cheap and readily available, so you throw in as much of it as you can.  There might also be safety considerations such as having an excess of one reagent to absorb the heat generated by the reaction.

Analogy

To illustrate the process of solving a limiting reagent problem, we can look at an analogy that is easier to visualize than a bunch of molecules reacting.  A florist might sell single roses packaged with a sprig of baby''s breath and 2 leafy green fronds.  We can represent the process of assembling the final package with an equation:

1 rose + 1 baby''s breath + 2 greens => 1 package

And, we can write a series of conversion factors based on the coefficients from the equation:

1 rose = 1 baby''s breath
1 rose = 2 greens
1 rose = 1 package
1 baby''s breath = 2 greens
1 baby''s breath = 1 package
2 greens = 1 package

If the florist has 24 roses, 32 sprigs of baby''s breath and 44 leafy green fronds, how many packages can be made?  What will be left over?

Let''s approach the problem by calculating how many packages can be prepped from each one of the starting materials, regardless of the others.

From the 24 roses, the florist can make

 (24 roses) x (1 package) = 24 packages (1) (1 rose)

From the 32 sprigs of baby''s breath, the florist can make

 (32 baby''s breath) x (1 package) = 32 packages (1) (1 baby''s breath)

From the 44 leafy green fronds, the florist can make

 (44 greens) x (1 package) = 22 packages (1) (2 greens)

So, which answer is correct?  24, 32 or 22?

The smallest of these answers is correct - 22.  After the florist has assembled 22 packages, all of the greens will have been used.  Even though there are enough roses and baby''s breath to make more, all of the greens are gone, so no more packages can be made.  The greens are the limiting reagent.  The roses and baby''s breath are the excess reagents.

How many of the roses and baby''s breath are left over?  There are several approaches to this part of the calculation.  I am going to demonstrate the most general of the approaches that I know.  If a different path makes more sense to you, follow it instead.

To calculate the left over reactants, I will first calculate how much of those reagents was actually used to make 22 packages.  Then, I can subtract the amount used from the amount that we started with to give the amount left over.

To calculate the number of roses used, start with the 22 packages created:

 (22 packages) x (1 rose) = 22 roses used (1) (1 package)

If 22 roses were used, then 24-22 = 2 roses are left over.

To calculate the number of sprigs of baby''s breath used, start with the 22 packages created:

 (22 packages) x (1 baby''s breath) = 22 baby''s breath (1) (1 package)

If 22 sprigs of baby''s breath were used, then 32-22 = 10 sprigs of baby''s breath are left over.

To wrap up, the florist started with 24 roses, 32 sprigs of baby''s breath and 44 greens, and then produced 22 packages with 2 roses, 10 sprigs of baby''s breath and 0 greens left over.

Limiting Reagents in Chemical Problems

The only difference between the example above and chemical limiting reagent problems is that the amounts given in chemical problems are typically in grams instead of numbers of moles.  Therefore, you have to convert from grams to moles with the molecular weight (molar mass) before using the coefficients to convert between compounds.  Since you are usually asked to find a mass in grams, you also must convert back to grams at the end of the problem with the molecular weight.

But, the basic idea remains the same:

• Calculate how much can be made from each reactant assuming an unlimited supply of the others.
• The smallest amount of product calculated in the first step is how much can actually be made, and it will indicate which is the limiting reagent.
• The amount of the limiting reagent left over is 0.
• To calculate how much of the excess reagents are left over, calculate how much was used, and then subtract that from how much you started with.