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# Limiting reactant and reaction yields

AP.Chem:
SPQ‑4 (EU)
,
SPQ‑4.A (LO)
,
SPQ‑4.A.1 (EK)
,
SPQ‑4.A.2 (EK)
,
SPQ‑4.A.3 (EK)

## Limiting reactant and theoretical yield

It’s a classic conundrum: We have five hot dogs and four hot dog buns. How many complete hot dogs can we make?
Assuming that hot dogs and buns combine in a 1, colon, 1 ratio, we can make four complete hot dogs. Once we run out of buns, we'll have to stop making complete hot dogs. In other words, the hot dog buns limit the number of complete hot dogs we can produce.
A reaction with five hot dogs and four hot dog buns reacting to give four complete hot dogs and one leftover hot dog. The hot dog buns are the limiting reagent, and the leftover single hot dog is the excess reagent. The four complete hot dogs are the theoretical yield.
In much the same way, a reactant in a chemical reaction can limit the amounts of products formed by the reaction. When this happens, we refer to the reactant as the limiting reactant (or limiting reagent). The amount of a product that is formed when the limiting reactant is fully consumed in a reaction is known as the theoretical yield. In the case of our hot dog example, we already determined the theoretical yield (four complete hot dogs) based on the number of hot dogs buns we were working with.
Enough about hot dogs, though! In the next example, we'll see how to identify the limiting reactant and calculate the theoretical yield for an actual chemical reaction.

## Example 1: Using the limiting reactant to calculate theoretical yield

A 2, point, 80, space, g sample of A, l, left parenthesis, s, right parenthesis reacts with a 4, point, 15, space, g sample of $\ce{Cl2}(g)$ according to the equation shown below.
$\ce{2Al}(s) + \ce{3Cl2}(g) \rightarrow \ce{2AlCl3}(s)$
What is the theoretical yield of $\ce{AlCl3}$ in this reaction?
To solve this problem, we first need to determine which reactant, A, l or $\ce{Cl2}$, is limiting. We can do so by converting both reactant masses to moles and then using one or more mole ratios from the balanced equation to identify the limiting reactant. From there, we can use the amount of the limiting reactant to calculate the theoretical yield of $\ce{AlCl3}$.

### Step 1: Convert reactant masses to moles

Let's start by converting the masses of A, l and $\ce{Cl2}$ to moles using their molar masses:
\begin{aligned}2.80\; \cancel{\text{g Al}} &\times \dfrac{1\; \text{mol Al}}{26.98\; \cancel{\text{g Al}}} = 1.04 \times 10^{-1}\; \text{mol Al} \\\\ 4.15\; \cancel{\text{g Cl}_2} &\times \dfrac{1\; \text{mol Cl}_2}{70.90\; \cancel{\text{g Cl}_2}} = 5.85 \times 10^{-2}\; \text {mol Cl}_2\end{aligned}

### Step 2: Find the limiting reactant

Now that we know the quantities of A, l and $\ce{Cl2}$ in moles, we can determine which reactant is limiting. As you'll see below, there are multiple ways to do so, each of which uses the concept of the mole ratio. All of the methods give the same answer, though, so you can choose whichever approach you prefer!
Method 1: For the first method, we'll determine the limiting reactant by comparing the mole ratio between A, l and $\ce{Cl2}$ in the balanced equation to the mole ratio actually present. In this case, the mole ratio of A, l and $\ce{Cl2}$ required by balanced equation is
start fraction, start text, m, o, l, e, s, space, o, f, space, A, l, end text, divided by, start text, m, o, l, e, s, space, o, f, space, C, l, end text, start subscript, 2, end subscript, end fraction, start text, left parenthesis, r, e, q, u, i, r, e, d, right parenthesis, end text, equals, start fraction, 2, divided by, 3, end fraction, equals, 0, point, 6, start overline, 6, end overline
and the actual mole ratio is
start fraction, start text, m, o, l, e, s, space, o, f, space, A, l, end text, divided by, start text, m, o, l, e, s, space, o, f, space, C, l, end text, start subscript, 2, end subscript, end fraction, start text, left parenthesis, a, c, t, u, a, l, right parenthesis, end text, equals, start fraction, 1, point, 04, times, 10, start superscript, minus, 1, end superscript, divided by, 5, point, 85, times, 10, start superscript, minus, 2, end superscript, end fraction, equals, 1, point, 78
Since the actual ratio is greater than the required ratio, we have more start text, A, l, end text than is needed to completely react the $\ce{Cl2}$. This means that the start text, C, l, end text, start subscript, 2, end subscript must be the limiting reactant. If the actual ratio had been smaller than the required ratio, then we would have had excess start text, C, l, end text, start subscript, 2, end subscript, instead, and the start text, A, l, end text would be limiting.
Method 2: For the second method, we'll use the mole ratio between A, l and $\ce{Cl2}$ to determine how much $\ce{Cl2}$ we would need to fully consume 1, point, 04, times, 10, start superscript, minus, 1, end superscript moles of A, l. Then, we'll compare the answer to the amount of $\ce{Cl2}$ we actually have to see if $\ce{Cl2}$ is limiting or not. The number of moles of $\ce{Cl2}$ required to react with 1, point, 04, times, 10, start superscript, minus, 1, end superscript moles of A, l is
1, point, 04, times, 10, start superscript, minus, 1, end superscript, start cancel, start text, m, o, l, space, A, l, end text, end cancel, times, start fraction, 3, start text, m, o, l, space, C, l, end text, start subscript, 2, end subscript, divided by, 2, start cancel, start text, m, o, l, space, A, l, end text, end cancel, end fraction, equals, 1, point, 56, times, 10, start superscript, minus, 1, end superscript, start text, m, o, l, space, C, l, end text, start subscript, 2, end subscript
According to our earlier calculations, we have 5, point, 85, times, 10, start superscript, minus, 2, end superscript moles of $\ce{Cl2}$, which is less than 1, point, 56, times, 10, start superscript, minus, 1, end superscript moles. Again, this means that the $\ce{Cl2}$ is limiting. (Note that we could have done a similar analysis for A, l instead of $\ce{Cl2}$, and we would have arrived at the same conclusion.)
Method 3: For the third and final method, we'll use mole ratios from the balanced equation to calculate the amount of $\ce{AlCl3}$ that would be formed by complete consumption of A, l and $\ce{Cl2}$. The reactant that produces the smallest amount of $\ce{AlCl3}$ must be limiting. To start, let's calculate how much $\ce{AlCl3}$ would be formed if the A, l was completely consumed:
1, point, 04, times, 10, start superscript, minus, 1, end superscript, start cancel, start text, m, o, l, space, A, l, end text, end cancel, times, start fraction, 2, start text, m, o, l, space, A, l, C, l, end text, start subscript, 3, end subscript, divided by, 2, start cancel, start text, m, o, l, space, A, l, end text, end cancel, end fraction, equals, 1, point, 04, times, 10, start superscript, minus, 1, end superscript, start text, m, o, l, space, A, l, C, l, end text, start subscript, 3, end subscript
Then, let's calculate the amount of $\ce{AlCl3}$ that would be formed if the $\ce{Cl2}$ was completely consumed:
5, point, 85, times, 10, start superscript, minus, 2, end superscript, start cancel, start text, m, o, l, space, C, l, end text, start subscript, 2, end subscript, end cancel, times, start fraction, 2, start text, m, o, l, space, A, l, C, l, end text, start subscript, 3, end subscript, divided by, 3, start cancel, start text, m, o, l, space, C, l, end text, start subscript, 2, end subscript, end cancel, end fraction, equals, 3, point, 90, times, 10, start superscript, minus, 2, end superscript, start text, m, o, l, space, A, l, C, l, end text, start subscript, 3, end subscript
Since the $\ce{Cl2}$ produces a smaller amount of $\ce{AlCl3}$ than the A, l does, the $\ce{Cl2}$ must be the limiting reactant.

### Step 3: Calculate the theoretical yield

Our final step is to determine the theoretical yield of A, l, C, l, start subscript, 3, end subscript in the reaction. Remember that the theoretical yield is the amount of product that is produced when the limiting reactant is fully consumed. In this case, the limiting reactant is $\ce{Cl2}$, so the maximum amount of A, l, C, l, start subscript, 3, end subscript that can be formed is
5, point, 85, times, 10, start superscript, minus, 2, end superscript, start cancel, start text, m, o, l, space, C, l, end text, start subscript, 2, end subscript, end cancel, times, start fraction, 2, start text, m, o, l, space, A, l, C, l, end text, start subscript, 3, end subscript, divided by, 3, start cancel, start text, m, o, l, space, C, l, end text, start subscript, 2, end subscript, end cancel, end fraction, equals, 3, point, 90, times, 10, start superscript, minus, 2, end superscript, start text, m, o, l, space, A, l, C, l, end text, start subscript, 3, end subscript
Note that we had already calculated this value while working through Method 3! Since a theoretical yield is typically reported with units of mass, let's use the molar mass of $\ce{AlCl3}$ to convert from moles of $\ce{AlCl3}$ to grams:
3, point, 90, times, 10, start superscript, minus, 2, end superscript, start cancel, start text, m, o, l, space, A, l, C, l, end text, start subscript, 3, end subscript, end cancel, times, start fraction, 133, point, 33, start text, g, space, A, l, C, l, end text, start subscript, 3, end subscript, divided by, 1, start cancel, start text, m, o, l, space, A, l, C, l, end text, start subscript, 3, end subscript, end cancel, end fraction, equals, 5, point, 20, start text, g, space, A, l, C, l, end text, start subscript, 3, end subscript

## Percent yield

As we just learned, the theoretical yield is the maximum amount of product that can be formed in a chemical reaction based on the amount of limiting reactant. In practice, however, the actual yield of product—the amount of product that is actually obtained—is almost always lower than the theoretical yield. This can be due to a number of factors, including side reactions (secondary reactions that form undesired products) or purification steps that lower the amount of product isolated after the reaction.
Burglar cat with stripes and eye mask holding a stolen hot dog bun.
Oh no, a cat burglar stole one of our hot dog buns! This means that our actual yield is only three complete hot dogs. Given that the theoretical yield was four complete hot dogs, what is our percent yield?
The actual yield of a reaction is typically reported as a percent yield, or the percentage of the theoretical yield that was actually obtained. The percent yield is calculated as follows:
start text, P, e, r, c, e, n, t, space, y, i, e, l, d, end text, equals, start fraction, start text, a, c, t, u, a, l, space, y, i, e, l, d, end text, divided by, start text, t, h, e, o, r, e, t, i, c, a, l, space, y, i, e, l, d, end text, end fraction, times, 100, percent
Based on this definition, we would expect a percent yield to have a value between 0% and 100%. If our percent yield is greater than 100%, that means we probably calculated something incorrectly or made an experimental error. With all this in mind, let's try calculating the percent yield for a precipitation reaction in the following example.

## Example 2: Calculating percent yield

A students mixes 25, point, 0, space, m, L of 0, point, 314, M $\ce{BaCl2}$ with excess $\ce{AgNO3}$, causing A, g, C, l to precipitate. The balanced equation for the reaction is shown below.
$\ce{BaCl2}(aq) + \ce{2AgNO3}(aq) \rightarrow \ce{2AgCl}(s) + \ce{Ba(NO3)2}(aq)$
If the student isolates 1, point, 82, space, g of A, g, C, l, left parenthesis, s, right parenthesis, what is the percent yield of the reaction?
To solve this problem, we'll need to use the given information about the limiting reactant, $\ce{BaCl2}$, to calculate the theoretical yield of A, g, C, l for the reaction. Then, we can compare this value to the actual yield of A, g, C, l to determine the percent yield. Let's walk through the steps now:

### Step 1: Find moles of the limiting reactant

To determine the theoretical yield of A, g, C, l, we first need to know how many moles of $\ce{BaCl2}$ were consumed in the reaction. We're given the volume (0, point, 0250, space, L) and molarity (0, point, 314, M) of the $\ce{BaCl2}$ solution, so we can find the number of moles of $\ce{BaCl2}$ by multiplying these two values:
\begin{aligned}\text{Moles BaCl}_2 &= M_{\text{BaCl}_2} \times \text{liters soln} \\\\ &= \dfrac{0.314\; \text{mol BaCl}_2}{1\; \cancel{\text{L soln}}} \times .0250\; \cancel{\text{L soln}} \\\\ &= 7.85 \times 10^{-3}\; \text{mol BaCl}_2\end{aligned}

### Step 2: Determine the theoretical yield (in grams)

Now that we have the quantity of $\ce{BaCl2}$ in moles, we can find the theoretical yield of A, g, C, l in grams:
7, point, 85, times, 10, start superscript, minus, 3, end superscript, start cancel, start text, m, o, l, space, B, a, C, l, end text, start subscript, 2, end subscript, end cancel, times, start fraction, 2, start cancel, start text, m, o, l, space, A, g, C, l, end text, end cancel, divided by, 1, start cancel, start text, m, o, l, space, B, a, C, l, end text, start subscript, 2, end subscript, end cancel, end fraction, times, start fraction, 143, point, 32, start text, g, space, A, g, C, l, end text, divided by, 1, start cancel, start text, m, o, l, space, A, g, C, l, end text, end cancel, end fraction, equals, 2, point, 25, start text, g, space, A, g, C, l, end text

### Step 3: Calculate the percent yield

Finally, we can calculate the percent yield of A, g, C, l by dividing the actual yield of A, g, C, l (1, point, 82, space, g) by our theoretical yield from Step 2:
\begin{aligned} \%\;\text{yield} &= \dfrac{\text{actual yield}}{\text{theoretical yield}} \times 100\% \\\\ &= \dfrac{1.82\; \text{g AgCl}}{2.25\; \text{g AgCl}} \times 100\% \\\\ &= 80.9\% \end{aligned}

### Summary

The limiting reactant (or limiting reagent) is the reactant that gets consumed first in a chemical reaction and therefore limits how much product can be formed. As we saw in Example 1, there are many different ways to determine the limiting reactant, but they all involve using mole ratios from the balanced chemical equation.
The amount of product that can be formed based on the limiting reactant is called the theoretical yield. In reality, the amount of product actually collected, known as the actual yield, is almost always smaller than the theoretical yield. The actual yield is usually expressed as a percent yield, which specifies what percentage of the theoretical yield was obtained.