Percentages

Percentages express how large one quantity is relative to another quantity; so for example percentage yields express how large the yield of a reaction is compared to the theoretical yield. A percentage of a quantity is a way of expressing a number as a fraaction of 100. The symbol for percentage is %.

In general, if we have quantities \(x\) and \(y\) and we want to express how large \(x\) is compared to \(y\) (i.e., what percentage is \(x\) of \(y\)?), we can use the formula:

\[ \textrm{percentage} = \frac{x}{y} \times 100 \% \]

Example

What percentage is 8 of 160?

Solution:

\[ \frac{8}{160} \times 100\% = 5\% \]

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In Python, this looks like:

8 / 160 * 100

5.0

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Note

We may wish to convert between fractions and percentages:

1. To convert a fraction to a percentage, we multiply the fraction by 100%. For example, \(\frac{1}{2}\) as a percentage is \(\frac{1}{2} \times 100\% = 50\%\).

2. To convert a percentage to a fraction, we divide the percentage by 100% and simplify. For example, 25% as a fraction is \(\frac{25\%}{100\%} = \frac{1}{4}\).

We may also want to calculate the percentage of a quantity, for example, what is 28 % of 132? In general, we can use the formula:

\[ x = \frac{\textrm{percentage}}{100\%} \times y \]

Example

What is 28% of 132?

Solution:

\[ x = \frac{28\%}{100\%} \times 132 = 36.96 \]

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28 / 100 * 132

36.96

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Esterification Yield

In the esterification reaction:

CH₃OH + CH₃COOH → CH₃COOCH₃ + H₂O

The reactants combine in the following proportions based on their molar masses: 32 g of CH₃OH reacts with 60 g of CH₃COOH to form 74 g of the ester product.

In a particular reaction, 1.8 g of methanol is mixed with an excess of ethanoic acid. At the end of the experiment 2.6 g of the ester is extracted. What is the percentage yield?

\[ \textrm{percentage yield} = \frac{\textrm{actual yield}}{\textrm{theoretical maximum yield}} \times 100\% \]

Solution: First, we need to calculate the theoretical maximum yield:

From the question, we know that 32 g of CH₃OH yields 74 g of ester. Dividing through by 32 gives us that 1 g of CH₃OH yields \(\frac{74}{32}\) g of ester.
Multiplying by 1.8 gives use that 1.8 g of CH₃OH yields \(1.8 \times \frac{74}{32} = 4.2 \) g of ester (to on decimal point).

We know from the question that the actual yield is 2.6 g, so using the percentage yield formula, we have:

\[ \textrm{percengage yield} = \frac{2.6\;\textrm{g}}{4.2\;\textrm{g}} \times 100\% = 61.9\% \]

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In Python, we can store the theoretical maximum yield as a variable to be reused.

theoretical_maximum_yield = 1.8 * 74 / 32
percentage_yield = 2.6 / theoretical_maximum_yield * 100
percentage_yield

62.462462462462454

Notice, that by using Python variables, we can avoid carrying rounding errors through our calculation.

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