Question

By using estimation techniques, determine which of the following is the heaviest and which is the lightest: a 5-lb bag of potatoes, a \(5-\mathrm{kg}\) bag of sugar, or 1 gal of water \((\) density \(=1.0 \mathrm{~g} / \mathrm{mL})\)

Short answer

The heaviest item is the 5-kilogram bag of sugar weighing 5000 g, and the lightest item is the 5-pound bag of potatoes weighing 2267.96 g.

Step-by-step solution

Convert all weights to grams

To do this, we can use the following conversion factors: 1 pound (lb) = 453.592 grams (g) 1 kilogram (kg) = 1000 grams (g) 1 gallon (gal) = 3.78541 liters (L) Since 1 mL of water weighs 1.0 g, we can convert the volume of water to weight directly.

Calculate the weight of each item in grams

\(\textbf{Potatoes} :\) Weight of a 5-pound bag of potatoes in grams: \(Weight_{potatoes} = 5~\mathrm{lb} \times \frac{453.592~g}{1~\mathrm{lb}} = 2267.96~g\) \(\textbf{Sugar} :\) Weight of a 5-kilogram bag of sugar in grams: \(Weight_{sugar} = 5~\mathrm{kg} \times \frac{1000~g}{1~\mathrm{kg}} = 5000~g\) \(\textbf{Water} :\) Weight of 1 gallon of water in grams: \(Weight_{water} = 1~\mathrm{gal} \times \frac{3.78541~L}{1~\mathrm{gal}} \times \frac{1000~mL}{1~L} \times \frac{1.0~g}{1~\mathrm{mL}} = 3785.41~g\)

Compare the weights and determine the heaviest and lightest items

Now we have the weights in grams: Potatoes: 2267.96 g Sugar: 5000 g Water: 3785.41 g Comparing the weights, the 5-kilogram bag of sugar is the heaviest, and the 5-pound bag of potatoes is the lightest. So, the heaviest item is the 5-kilogram bag of sugar and the lightest item is the 5-pound bag of potatoes.

Key concepts

Estimation Techniques

When faced with a problem that requires a quick assessment of quantity and comparison, estimation techniques can prove invaluable, especially in fields like chemistry where precision matters. Estimation allows us to quickly derive a close approximation without the need for detailed calculations, and it's essential for making educated guesses when exact data is unavailable.

Estimation is not just about taking a wild guess; it involves using known information to come to a logical conclusion. In our exercise, we need to determine which of three items is the heaviest and which is the lightest. We are comparing items in different units: pounds for potatoes, kilograms for sugar, and gallons for water. Before we can estimate, we need to understand the relationship between these units.

For example, knowing that 1 kilogram is approximately 2.2 pounds helps us to quickly estimate that 5 kilograms of sugar would be roughly equivalent to 11 pounds, which seems more substantial than 5 pounds of potatoes. Similarly, with the knowledge that 1 gallon of water weighs about 8.34 pounds, we can infer that the water's weight is likely between that of the potatoes and the sugar.

Quick Conversion References

Common approximations such as 'a kilogram is a bit more than double a pound' or 'a gallon of water weighs about eight pounds', guide our initial assessment before we delve into exact conversions and calculations. Estimation is a skill that becomes sharper with practice and familiarity with the units involved.

Weight Conversion

Weight conversion is crucial in many scientific calculations, including chemistry, where the conversion between different units of mass is often needed. To compare weights accurately, we must convert all quantities to the same unit. In our scenario, we are converting pounds (lb), kilograms (kg), and gallons (gal) all into grams (g), as grams are a standard unit for mass in the metric system, which is commonly used in science.

Firstly, it's important to remember some key conversion factors: 1 lb is equivalent to 453.592 g and 1 kg equals 1000 g. With these factors, the conversion becomes a simple multiplication. For example, converting 5 lb to grams involves multiplying 5 by 453.592. It's crucial to use accurate conversion factors, as the right numbers lead to precise and reliable results.

Using Conversion Factors

The calculated weight of each item in grams allows us to make a direct comparison. Ensuring the correct use of these conversion factors is the foundation for the accuracy of the final results in any weight-related problems. Often, creating a conversion chart or memorizing common conversions can speed up the process for frequent use.

Density and Volume

Density and volume are interrelated concepts in chemistry critical for determining the mass of a substance from its volume. Density is defined as the mass per unit volume of a substance and is expressed in grams per cubic centimeter (g/cm³) or grams per milliliter (g/mL) in the metric system.

For substances like water, with a density of 1 g/mL, this relationship simplifies the conversion process. The volume of water in liters can be easily converted to mass in grams since 1 mL of water weighs 1 gram. Therefore, understanding the density of a substance allows us to directly convert its volume to weight, which is particularly useful when dealing with liquids.

Applying Density in Calculations

In our example, the volume of water in gallons is converted to liters, then to milliliters, and eventually to grams, thanks to water's density. With solids of uniform composition, or with liquids like our example, knowing the density eliminates the need for measuring mass directly, thereby streamlining the problem-solving process in many chemistry-related tasks.