Problem 110 - Chemistry The Science in Context

Question

A factory making toy wagons has 13,466 wheels, 3360 handles, and 2400 wagon beds in stock. What is the maximum number of wagons the factory can make?

Step-by-Step Solution

Answer

Answer: 2,400 wagons.

Step 1: Determine the number of wagons that can be made from each component

To find how many wagons can be made from each component, we will divide the number of components by the number of components needed for each wagon. Each wagon needs 4 wheels, 1 handle, and 1 wagon bed. Wheels: Wagons_wheels = \(\frac{13466}{4}\) Handles: Wagons_handles = \(\frac{3360}{1}\) Wagon Beds: Wagons_beds = \(\frac{2400}{1}\)

Step 2: Evaluate the division and find the smallest number

Now, we will evaluate the division from Step 1 to find the number of wagons we can make from each component. Wagons_wheels = \(\frac{13466}{4}\) = 3366.5 Since we cannot have a fraction of a wagon, we will round down to the nearest whole number: Wagons_wheels = 3366 Wagons_handles = \(\frac{3360}{1}\) = 3360 Wagons_beds = \(\frac{2400}{1}\) = 2400 Now, we will compare the number of wagons we can make from each component and find the smallest number: Wagons_wheels = 3366 Wagons_handles = 3360 Wagons_beds = 2400 The smallest number is 2400.

Step 3: Conclusion

The maximum number of wagons the factory can make is 2400, since that is the number of wagons we can make from the component with the smallest number of available parts (wagon beds).

Key Concepts

Mathematical Calculations in Chemistry

Stoichiometry, an essential concept in chemistry, often involves mathematical calculations similar to the ones used in determining the number of toy wagons a factory can produce. In a chemistry context, these calculations are vital for predicting yields of chemical reactions, just as we predict the yield of wagons based on available parts.
In a chemical reaction, knowing the amount of reactants available can help us calculate the quantity of products formed. These calculations are based on the balanced chemical equation, which acts much like our list of needed components for wagons. For instance, if a chemical reaction requires 2 moles of hydrogen gas to react with 1 mole of oxygen gas to produce 2 moles of water, we can use the mole ratios to determine how many moles of water will form, given certain amounts of hydrogen and oxygen.
Key Steps in Stoichiometric Calculations:

• Write the balanced chemical equation.
• Convert all quantities to moles.
• Use mole-to-mole ratios to find the ratio between reactants and products.
• Convert moles back to desired units (grams, liters, etc.).

Through this process, chemists can ensure that the right proportions of reactants are used to optimize the yield of products and minimize waste, echoing our toy wagon factory's need to use resources efficiently.

Limiting Reactant Concept

The concept of the limiting reactant in chemistry is akin to the bottleneck in our toy wagon production scenario. In chemistry, reactions occur when reactants come together. However, not all reactants are present in the perfect stoichiometric ratio, and often, one reactant will run out first, halting the reaction.
This reactant, present in the smallest stoichiometric amount, is called the 'limiting reactant'. Identifying it is crucial because it determines the maximum amount of product that can be formed. The excess reactants remain unused, similar to the surplus of wheels and handles in the factory example.
Identifying the Limiting Reactant:

• Calculate the mole ratio of reactants from the balanced chemical equation.
• Determine the moles of each reactant available.
• Compute how much product each reactant can produce.
• The reactant that produces the least amount of product is the limiting reactant.

In the toy wagon problem, wagon beds were the limiting 'reactant'. An understanding of this concept helps students address real-world problems by translating the abstract into tangible examples.

Mole-to-Mole Ratios

Central to both stoichiometry problems in chemistry and our wagon assembly problem is the concept of mole-to-mole ratios. A mole is a unit that signifies a very large quantity (6.022 x 10²³ entities) of very small entities, such as atoms, molecules, or, in our analogy, wagon parts.
In the toy wagon scenario, we divide the available parts by the parts needed per wagon to find how many complete wagons can be assembled. Similarly, in chemistry, the balanced equation provides mole-to-mole ratios between reactants and products, which are crucial for predicting the amounts of substances involved in the reaction.
Using Mole-to-Mole Ratios:

• Identify the ratio from the balanced chemical equation.
• Use the ratio to calculate the moles of one substance from the moles of another.
• Apply the concept to real-life situations, such as calculating the efficiency of a chemical process or the yield of a reaction.

These ratios serve as a bridge between the reactants and products, elucidating how a change in the quantity of one affects the other. For students tackling stoichiometry, mastering these ratios is like ensuring they have the right blueprint for efficient production in any manufacturing scenario.