Question

You are making cookies and are missing a key ingredient-eggs. You have plenty of the other ingredients, except that you have only 1.33 cups of butter and no eggs. You note that the recipe calls for 2 cups of butter and 3 eggs (plus the other ingredients) to make 6 dozen cookies. You telephone a friend and have him bring you some eggs. a. How many eggs do you need? b. If you use all the butter (and get enough eggs), how many cookies can you make? Unfortunately, your friend hangs up before you tell him how many eggs you need. When he arrives, he has a surprise for you-to save time he has broken the eggs in a bowl for you. You ask him how many he brought, and he replies, "All of them, but I spilled some on the way over." You weigh the eggs and find that they weigh \(62.1 \mathrm{g} .\) Assuming that an average egg weighs \(34.21 \mathrm{g}\) c. How much butter is needed to react with all the eggs? d. How many cookies can you make? e. Which will you have left over, eggs or butter? f. How much is left over? g. Relate this question to the concepts of chemical stoichiometry.

Short answer

We can determine the amount needed of each ingredient and the number of cookies we can make using proportions. We need \(1.998 \) eggs with 1.33 cups of butter, and we can make about 3.995 dozen cookies. The friend brought about 1.814 eggs, requiring about 1.209 cups of butter. With those amounts, we can make about 3.629 dozen cookies. Since we have more eggs than needed for the 1.33 cups of butter, we'll have leftover eggs, with approximately 0.184 eggs leftover. This problem relates to chemical stoichiometry, as we are determining proportions between ingredients (reactants) and the amounts of final products (cookies) we can make with given amounts of ingredients.

Step-by-step solution

Determine the number of eggs we need given we have 1.33 cups of butter

To find the number of eggs needed when we have 1.33 cups of butter, we use the proportion of butter and eggs in the recipe. We have the proportion 2 cups butter : 3 eggs, and we can create an equation: \( \frac{1.33 \,\text{cups butter}}{x \,\text{eggs}} = \frac{2 \,\text{cups butter}}{3 \,\text{eggs}} \) Now we can solve for x (the number of eggs): \[ x = \frac{1.33 \,\text{cups butter} \times 3 \,\text{eggs}}{2 \,\text{cups butter}} \]

Calculate the number of cookies we can make with 1.33 cups of butter and enough eggs

Now that we know how many eggs are needed with 1.33 cups of butter, we can calculate the number of cookies using proportion again. The recipe says that 2 cups of butter and 3 eggs can make 6 dozen cookies. So, we can create an equation with the butter proportions: \[ \frac{1.33 \,\text{cups butter}}{2 \,\text{cups butter}} = \frac{x \,\text{dozen cookies}}{6 \,\text{dozen cookies}} \]

Find out how many eggs are needed based on their weight

According to the problem, our friend brought eggs weighing 62.1 g, and an average egg weighs 34.21 g. To find out how many eggs were brought, we will divide the weight of eggs by weight of one egg: \[ y = \frac{62.1 \,\text{g}}{34.21 \,\text{g/egg}} \]

Determine the amount of butter needed to react with the eggs our friend brought

Repeat the same as in step 1 with the new number of eggs (y) that our friend brought: \[ \frac{x \,\text{cups butter}}{y \,\text{eggs}} = \frac{2 \,\text{cups butter}}{3 \,\text{eggs}} \]

Calculate the number of cookies we can make with the new egg and butter amounts

Use the butter proportion as before to create an equation: \[ \frac{x \,\text{cups butter}}{2 \,\text{cups butter}} = \frac{z \,\text{dozen cookies}}{6 \,\text{dozen cookies}} \]

Determine which will be left over, butter or eggs, and how much

Compare the amount of butter calculated in step 4 with 1.33 cups of butter available. The lesser amount will decide the leftover ingredient.

Relate the question to chemical stoichiometry

The proportion between butter and eggs is similar to how the stoichiometry of a chemical reaction determines the proportions between reactants and products. This analogy helps us understand the relationship between proportions of ingredients (reactants) and the amounts of final products (cookies) we can make with given reactants.

Key concepts

Stoichiometric Calculations

Stoichiometric calculations are crucial in chemistry as they allow us to predict the quantities of reactants and products involved in chemical reactions. A stoichiometric calculation is essentially a method for quantifying the exact amounts of substances needed to react fully without any excess. Using the recipe analogy provided in the exercise, let's imagine that you’re baking cookies.

Understanding stoichiometry is much like following a cookie recipe. If the recipe requires 2 cups of butter for 3 eggs to make 6 dozen cookies, and you only have 1.33 cups of butter, calculating the corresponding amount of eggs (the reactants in this case) relies on stoichiometry.

The recipe calls for a specific ratio of butter to eggs. By setting up a proportion, we can use simple math to scale down the eggs to match the quantity of butter we have. This problem-solving approach is directly analogous to determining the amounts of reactants in chemical reactions, where maintaining the correct proportions ensures that all reactants will be used up with no waste.

Proportional Relationships

Proportional relationships are at the heart of both chemistry and cooking. In our cookie recipe, the proportions of ingredients determine the outcome—too much or too little of one ingredient and the cookies might not turn out as expected.

Applying this to our example, the original recipe calls for a ratio of 2 cups of butter to 3 eggs. It's a direct relationship – as the amount of butter increases or decreases, the number of eggs must be adjusted correspondingly to keep the cookies delicious.

For chemical reactions, it’s the same: reactants combine in set ratios, defined by their stoichiometry. Diverging from these ratios may leave you with too much of one reactant (excess) and too little of another (limiting reactant), which influences the yield of the product, analogous to the number of cookies one can make with given amounts of ingredients.

Chemical Reaction Analogy

The process of combining recipe ingredients can serve as an excellent analogy for understanding chemical reactions. Think of ingredients as reactants and the finished cookies as the product of a chemical reaction. Just as varying the amount of butter or eggs in your recipe will affect the final batch of cookies, altering the quantities of reactants in a chemical reaction will impact the amount of the product formed.

In the example given, eggs can be viewed as one reactant and butter as another. When your friend brings an unknown quantity of eggs, it's like having an imprecise measurement of a reactant in a lab - it's essential to figure out exactly how much of this reactant you have to 'react' properly with the butter. Thus, by weighing the eggs, we establish the quantity of one reactant, much like a chemist would do in a laboratory to ensure a successful reaction.

Converting Mass to Moles

One of the fundamental skills in chemistry is converting mass to moles. Moles provide a way to count atoms and molecules in a standardized format. To convert mass to moles, you use the molar mass of the substance in question. In the recipe analogy, weighing the mixed eggs and converting that weight to the number of eggs is akin to determining the number of moles from a given mass.

Assuming an average egg weighs 34.21 g, and you have 62.1 g of eggs, you can determine the number of eggs by dividing the total mass by the mass of one egg - effectively converting mass into 'egg-moles'.

In an actual chemical reaction, you would do similarly: measure the mass of a substance, then convert that mass into moles using the substance's molar mass (analogous to the 'average egg weight'). This allows chemists to work with proportional relationships and stoichiometric calculations to predict and measure the outcomes of chemical reactions.