As plant manager at a chemical factory, your main responsibility is to ensure all production processes are working properly. One way to measure this is through the actual product amounts being obtained from the different chemical reactions, with respect to the amounts of reagents used.
A basic but very powerful concept used in this type of analysis is the theoretical yield. This value basically tells you the maximum amount of product you could expect from a given chemical reaction, based on the amount of reactants you use. This, in turn, can allow you to determine which type of reaction can be more cost-effective.
Through a very simple process, you can learn how to establish this important baseline to judge the performance of chemical reactors and any particular chemical reaction.
How to calculate the theoretical yield
To calculate a reaction’s theoretical yield follow these steps:
- Write down a balanced chemical equation for the reaction.
- Determine the limiting reagent of the reaction, meaning the one which would deplete entirely upon the completion of the chemical reaction.
- Calculate the ratio between the moles of the desired product and the limiting reagent through their stoichiometric coefficients in the balanced chemical equation.
- Calculate the moles of limiting reagent used in the reaction.
- Multiply the moles calculated in step 4 by the ratio obtained in step 3. The result is the theoretical yield of the product of interest in moles.
- Convert the theoretical yield to units of mass using the product’s molar mass.
What is the theoretical yield
Chemical equations are very useful not only because they describe how chemical compounds react, which products they form and, partly, how they do so; but also because they let us evaluate the performance of any real-life procedure. Imagine you want to produce potassium sulfate, a substance commonly used in fertilizers thanks to its content of both potassium and sulfur. A simple way to form potassium sulfate is by mixing potassium hydroxide, our source of K, with sulfuric acid, as a source of S. This typical acid-base reaction can be described by the following chemical equation:
Thanks to this equation, we know our reaction does not only form the desired product, potassium sulfate, but it also releases water. Furthermore, we can predict that if we mix 2 moles of potassium hydroxide with 1 mol of sulfuric acid, we will produce 1 mol of potassium sulfate and 2 moles of water. This is true, of course, in an ideal world. In reality, if you perform the reaction, you will probably notice these amounts do not necessarily match what you get.
This can occur due to your reagents not being entirely pure, or due to the presence of additional contaminants that get in the way of your reaction. Furthermore, there are fundamental conditions that can still cause a perfectly clean reaction not to proceed to completion, thus altering the expected amount of products. Let’s try to understand how.
Molecules tend to be stable. When two molecules react together, they most usually build new molecules that are more stable than their former versions. Nevertheless, since reactant molecules have a certain stability before forming a new molecule, some external energy might be needed to start the reaction.
Reaction mechanisms involve exchanging electrons or altering the way they behave in the atoms that compose each reacting molecule. Altering this electronic structure requires energy. So, in order for the reaction to take place, reactant molecules must first come close enough for their electronic clouds to interact with each other, exitate and initiate the transfer or relocation of electrons.
This happens usually through highly energetic collisions between reacting molecules. For example, when you heat up a recipient containing your reactants, you are providing thermal energy to them, which is converted into movement. Molecules then collide with enough energy for the reaction to start. Furthermore, many reactions also require an activation energy to be surpassed. This means the energy of the reaction, mostly associated with its temperature, must first overcome an energy barrier in order for it to start and proceed to completion.
All of the previous requirements can make a real-life experiment a bit different than what you expect from looking at a reaction’s chemical equation. On many occasions, some reactant molecules either do not come close enough to other reactant species or do not do so with enough energy for the reaction to take place. The result is that the amount of expected product is not fully reached, and some amounts of reagents might be left over once the reaction is finished.
Let’s go back to our potassium sulfate example. If you were to design your own reactor to produce potassium sulfate, you would probably want to design a device that creates a suitable environment for the reaction to take place. Additionally, you would want the reaction to take place efficiently, meaning, producing the highest possible amount of potassium sulfate. One way to evaluate this is by comparing the actual amount of product you get depending on the amount of reactants you use.
Thanks to the reaction’s chemical equation, you actually know how much product you can expect to get from a given amount of reagents. This value corresponds to the maximum amount of a certain product your reaction could yield and, since getting said amount would require all reagent molecules to comply with all of the requirements discussed above, it refers to an ideal scenario. This is why the maximum expected amount of a certain product of a chemical reaction is called its theoretical yield.
Procedure to calculate a reaction’s theoretical yield
The first thing we need to consider when evaluating a chemical reaction is the chemical equation. This expression represents what happens in reality. Since matter can’t be created or destroyed, the number of atoms of each species on both sides of the equation must be the same. Otherwise, it would mean some atoms appear or disappear during the chemical reaction, which is simply not the case.
This “correction” of a chemical equation is called balancing the chemical reaction. Since we can’t alter the subscripts of a chemical compound, which means we would alter the compound itself, we need to balance a chemical equation through its stoichiometric coefficients. Let’s say you are presented with the following chemical equation for the combustion of propane:
On the left hand side you have three carbon atoms, while on the right side you only have one. To solve this, you need to add a stoichiometric coefficient of 3 to carbon dioxide on the right hand side. This way, the three initial carbon atoms are preserved. If you do so, you end up with another discrepancy, though: you have a total of (3 x 2) + 1 = 7 oxygen atoms on the right side, while, on the left side, you have a total of 5 x 2 = 10 oxygen atoms. To solve this, you need to add a stoichiometric coefficient of 4 to water. This way, you end up with (3 x 2) + 4 = 10 oxygen atoms on the right side, and so mass if preserved. The balanced chemical equation is then:
Since the theoretical yield refers to the maximum amount produced of a certain product in a chemical reaction, we need to consider what limits its production. Let’s consider a very simple example: if you mix molecular oxygen with molecular hydrogen you can produce water, while releasing high amounts of energy. This reaction is actually used in very interesting applications, like last generation electric vehicles. Check out this video to learn how! This chemical reaction can be described as:
This equation tells us that 2 hydrogen molecules react with each oxygen molecule to build two water molecules. Let’s say you set four oxygen molecules and six hydrogen molecules to react together, as the following image shows. Each pair of hydrogen atoms will react with a single oxygen atom.
In this case, all of the hydrogen atoms will react. Nevertheless, only three of the four oxygen molecules will participate in the reaction, so only six water molecules are formed. This means, since hydrogen was fully consumed, there are no more molecules left to react with the remaining oxygen. In this case, hydrogen is the limiting reactant.
As you have probably already noticed, the amount of limiting reactant available in the reaction limits the amount of product that is formed. Since the theoretical yield is a measurement of the highest possible amount of the desired product that is expected from a reaction, the limiting reactant has to be considered when calculating it. This is why, to determine the theoretical yield, you need to determine which reactant is the limiting one, and how much is available to react.
If you are given a certain chemical reaction and the amounts used of the different reagents, first determine which will deplete first. This is easier if you convert the given amounts into moles, and then determine how many you will need of each reactant based on the most abundant one.
For example, imagine you have 20 moles of hydrogen and 8 moles of oxygen available to react and form water, according to equation 4. You can start by asking yourself how many moles of oxygen you will need to deplete those 20 moles of hydrogen. The answer is simple. You need 1 mol of O2 for every two moles of H2. Since you have 20 moles of hydrogen, you will need 10 moles of oxygen to deplete them all. Nevertheless, you only have 8 moles of oxygen available. In this case, oxygen is the limiting reactant.
The next step to determine the theoretical yield is to find the relationship between the limiting reagent and the desired product’s resulting moles according to the chemical equation. In our previous example, for every mol of oxygen that reacts, 2 moles of water are produced. We can then take the ratio between both stoichiometric coefficients:
Now, since you have 8 moles of oxygen available to react, and given that after those 8 moles are depleted no more water will be produced, we can calculate the maximum expected amount of water by multiplying the limiting reactant’s moles with the ratio we obtained in equation 5: 2 x 8 moles = 16 moles. Finally, using the molar mass of water (18 g/mol), we can calculate the theoretical yield in grams:
In our example, by using 20 moles of hydrogen and 8 moles of oxygen to produce water, the theoretical yield is 16 moles or 288 g of H2O.
Theoretical yield equation
The procedure discussed above to find the theoretical yield can be summarized like this:
- Determine if the chemical equation is balanced.
- Determine the limiting reagent and the amount used in the reaction.
- Find the ratio between the stoichiometric coefficients of the desired product and the limiting reagent.
- Convert the amount of limiting reagent you use for the reaction into moles.
- Multiply this amount by the ratio calculated in step 3.
- Convert the result into grams by means of the desired product’s molar mass.
This procedure can be expressed mathematically like this:
Where Yt is the theoretical yield of the product of interest; d and b are the stoichiometric coefficients of the product and the limiting reagent, respectively; xB and MB the mass (in grams) and the molar mass of the limiting reagent, respectively; and MD the molar mass of the desired product. This expression is very convenient, since it summarizes all the steps and directly provides any reaction’s theoretical yield in grams of product.
Exercise 1:
- Find the theoretical yield of potassium sulfate using the chemical reaction in equation 1, if 150 g of KOH and 100 g of H2SO4 are used.
- What is the limiting reagent in this case?
Answer A: 177,67 g
Answer B: the limiting reagent is sulfuric acid.
