Question

A bottling plant has 126,515 bottles with a capacity of \(355 \mathrm{~mL}\), 108,500 caps, and 48,775 L of beverage. (a) How many bottles can be filled and capped? (b) How much of each item is left over? (c) Which component limits the production?

Short answer

(a) 108,500 bottles can be filled and capped. (b) There are 18,015 leftover bottles, 0 leftover caps, and 10,257.5 L of leftover beverage. (c) The limiting component is the number of caps.

Step-by-step solution

Calculate the number of bottles that can be filled with the given beverage volume

First, we need to find out how many bottles can be filled with the given volume of beverage. The volume of beverage is 48,775 L (or 48,775,000 mL) and the capacity of one bottle is 355 mL. To find the number of bottles that can be filled, we need to divide the total volume of the beverage by the capacity of each bottle: Number of bottles that can be filled = \( \frac{48\,775\,000}{355}\) Number of bottles that can be filled = 137,394 (approximately)

Calculate the number of bottles that can be capped

We are given that we have 108,500 caps. As one cap is required for one bottle, we can cap 108,500 bottles.

Find the minimum number of filled and capped bottles

In order to find out how many bottles can be filled and capped, we need to find the minimum between the number of bottles we can fill and the number of bottles we can cap. Number of filled and capped bottles = min(137,394, 126,515, 108,500) Number of filled and capped bottles = 108,500

Calculate leftover items

Now, we will find out the leftover items by subtracting the used items from the given items. Leftover bottles = Total bottles - Filled and capped bottles Leftover bottles = 126,515 - 108,500 Leftover bottles = 18,015 Leftover caps = Total caps - Used caps Leftover caps = 108,500 - 108,500 Leftover caps = 0 To find the leftover beverage, we will first find the used beverage volume: Used beverage volume = Filled and capped bottles * Capacity of one bottle Used beverage volume = 108,500 * 355 mL Used beverage volume = 38,517,500 mL (38,517.5 L) Leftover beverage = Total beverage - Used beverage Leftover beverage = 48,775 - 38,517.5 Leftover beverage = 10,257.5 L

Determine the limiting component

As we have used all the available caps and still have leftover bottles and beverage, the limiting component in the production process is the number of caps. In summary, (a) we can fill and cap 108,500 bottles, (b) there are 18,015 leftover bottles, 0 leftover caps, and 10,257.5 L of leftover beverage, and (c) the limiting component is the number of caps.

Key concepts

Limiting Reactant

The concept of a limiting reactant in chemistry is similar to the item that runs out first during a production process, preventing more of the final product from being produced. It's like when you're making sandwiches and you run out of bread; no matter how much cheese or ham you have left, you can't make more sandwiches without more bread.

In the context of the bottling plant exercise, the limiting 'reactant' is the number of caps. Since we only have 108,500 caps, we can only produce 108,500 bottles, regardless of the amount of beverage or the number of bottles available. Think of it as each bottle needing one 'reactant' (cap) to become a 'product' (a capped bottle). Once the caps are gone, the production process halts, no matter how much of the other 'reactants' (beverage and bottles) are left.

To determine which item is the limiting one, we compare the amount of each item required for one bottle to the total amount available. The one that runs out first, in this case, the caps, is the limiting reactant.

Mole Concept

In stoichiometry, the mole concept is a fundamental principle that helps chemists quantify substances. A mole is just like a 'dozen' represents twelve things, but in this case, it represents approximately 6.022 x 10^23 particles of a substance, be it atoms, molecules, or ions. This is Avogadro's number, and it creates a bridge between the atomic world and the macroscopic world we can measure.

Although the bottling plant exercise does not directly involve moles, the underlying principle of counting and conversion is quite relevant. Understanding the mole concept allows chemists to convert between masses, volumes, and particle counts in a predictable way, similar to how the plant needs to convert between the volume of beverage and the number of bottles that volume can fill.

Chemical Calculations

Chemical calculations are pivotal in chemistry for quantitatively analyzing reactions and processes. These calculations often involve converting between units, reacting quantities, and determining yields.

For the bottling plant, the exercise involves calculations to determine the number of bottles that could be filled with the given volume of liquid. Dividing the total volume of the beverage by the capacity of a single bottle gives us the total possible number of filled bottles. This is akin to predicting how much product will result from a certain amount of reactant in chemistry. Chemical calculations also involve identifying excess and limiting reactants (or components), as well as calculating percentages and conversion factors, which in the industrial setting translates to efficient production and cost savings.

Chemistry Problem-Solving

Chemistry problem-solving is a systematic approach to understanding and working through chemical problems. This often requires a step-by-step method, identifying what information you have, what you need to find, and which principles or formulas to apply.

In our exercise, we systematically determined the number of bottles we could fill and cap, calculated the leftover resources, and identified the limiting component. Just as a chemist follows a procedure to tackle complex reactions, we applied logical steps to our bottling problem. Problem-solving skills in chemistry not only help in theoretical exercises but also have practical applications in labs and industries, such as this bottling plant scenario, emphasizing the importance of understanding the underlying chemical principles in various contexts.