Problem 109 - Chemistry The Science in Context

Question

A recipe for 1 cup of hollandaise sauce calls for \(\frac{1}{2}\) cup of butter, \(\frac{1}{4}\) cup of hot water, 4 egg yolks, and the juice of a medium-sized lemon. How many cups of this sauce can be made from a pound (2 cups) of butter, a dozen eggs, 4 medium lemons, and an unlimited supply of hot water?

Step-by-Step Solution

Answer

Answer: 3 cups.

Step 1: Determine the number of cups that can be made from each ingredient

We will find out how many cups of hollandaise sauce can be made from each ingredient. To do this, we will divide the quantity of each available ingredient by the quantity needed for one cup of hollandaise sauce.

Step 2: Find the number of cups that can be made from the butter

We are given 2 cups of butter, and we need 1/2 cup of butter to make one cup of hollandaise sauce. Therefore, the number of cups that can be made from the butter is: \(\frac{2 \text{ cups}}{\frac{1}{2} \text{ cup}} = 2 * 2 = 4\) cups

Step 3: Find the number of cups that can be made from the eggs

We are given a dozen (12) eggs, and we need 4 egg yolks to make one cup of hollandaise sauce. Therefore, the number of cups that can be made from the eggs is: \(\frac{12 \text{ eggs}}{4 \text{ eggs}} = 3\) cups

Step 4: Find the number of cups that can be made from the lemons

We are given 4 medium lemons, and we need the juice of 1 medium lemon to make one cup of hollandaise sauce. Therefore, the number of cups that can be made from the lemons is: \(\frac{4 \text{ lemons}}{1 \text{ lemon}} = 4\) cups

Step 5: Find the minimum number of cups that can be made from all the ingredients

To find the total number of cups that can be made from all the ingredients, we will find the minimum value among the cups from butter, eggs, and lemons: Minimum (4 cups from butter, 3 cups from eggs, 4 cups from lemons) = 3 cups

Step 6: Answer the question

With the given ingredients, we can make a total of 3 cups of hollandaise sauce.

Key Concepts

Stoichiometry

Understanding stoichiometry is akin to becoming a recipe master in cooking; both require precise measurements and a thorough understanding of how ingredients combine to create the desired outcome. In chemistry, stoichiometry refers to the calculation of reactants and products in chemical reactions. In the context of our cooking scenario, we want to create hollandaise sauce, and stoichiometry helps us figure out how to adjust our recipe based on the ingredients we have on hand. Imagine each item—butter, water, eggs, and lemon juice—like a reactant in a chemical equation, where the product is the final, delicious sauce. The initial recipe represents the balanced chemical equation, and the 'yield' is the final amount of sauce produced. By leveraging stoichiometry, we can make efficient use of our ingredients, much like a chemist ensuring none of the reactants go to waste.

Mole Concept

While we won't find moles in the kitchen, the mole concept is a fundamental cornerstone of chemistry, helping scientists count particles in a substance using the unit 'mole', which is equivalent to Avogadro's number (\(6.022 \times 10^{23}\) particles). For a cook, the concept of the 'mole' can be thought of in terms of units like 'cups' or 'dozen' for eggs. In our exercise, the 'dozen' eggs is a counting unit, similar to a mole in chemistry. The mole allows chemists to convert between atoms/molecules and grams, just as we convert a dozen eggs to the number of cups of hollandaise sauce they can produce. Without a firm understanding of the mole concept, chemists would struggle to scale reactions or interpret recipes—much like trying to cook without knowing how many teaspoons are in a tablespoon.

Chemical Calculation

Chemical calculations are the numerical side of chemistry, encompassing the quantitative relationships within a chemical reaction. For instance, in the preparation of hollandaise sauce, we performed various calculations to determine the maximum amount of sauce possible from the given ingredients. Much like chemical calculations provide the proportions of reactants needed for a reaction without excess, our kitchen calculation helped us use our resources without excess or waste. We need to remember that the 'limiting reactant,' or in our case the ingredient in shortest supply, determines the maximum amount of product formed—in our exercise, it was the eggs that limited the production to 3 cups of sauce.

Unit Conversion

Unit conversion is an essential skill both in the lab and the kitchen, enabling us to transition between different measurement systems. For cooks, this might mean converting between cups, tablespoons, and milliliters, while chemists convert among moles, liters, and grams. Our hollandaise sauce problem required converting cups of butter into the number of possible sauce cups, and eggs from a 'dozen' count to individual units needed per cup of sauce. Mastery of unit conversion is crucial for both chefs wanting their recipes to come out perfectly and chemists needing accurate measurements to avoid a failed reaction.