Problem 77 - Chemistry The Central Science In Si Units

Question

By using estimation technique, determine which of the following is the heaviest and which is the lightest: a \(10-\mathrm{lb}\) bag of fertilizer, a \(10-\mathrm{kg}\) bag of rice, or 2 gal of olive oil \(\left(\right.\) density \(\left.=0.918 \mathrm{~g} / \mathrm{cm}^{3}\right)\).

Step-by-Step Solution

Answer

The heaviest is the 10 kg bag of rice, and the lightest is the 10 lb bag of fertilizer.

Step 1: Convert all weights to the same units

To compare the weights, we need them all in the same unit. We'll convert everything to kilograms.- The 10 lb bag of fertilizer: Using the conversion 1 lb = 0.453592 kg, we convert 10 lb to kg: \[ 10\, \text{lb} \times 0.453592\, \frac{\text{kg}}{\text{lb}} = 4.53592\, \text{kg} \]- The 10 kg bag of rice is already in kilograms.- For 2 gallons of olive oil: Use the conversion 1 gal = 3.78541 liters. Olive oil density is 0.918 g/cm³ or 0.918 kg/L, so: \[ 2\, \text{gal} \times 3.78541\, \frac{\text{L}}{\text{gal}} \times 0.918\, \frac{\text{kg}}{\text{L}} = 6.9474\, \text{kg} \]

Step 2: Compare the converted weights

Now that all items are converted to kilograms, compare their weights: - Bag of fertilizer: 4.53592 kg - Bag of rice: 10 kg - Olive oil: 6.9474 kg From this comparison, we can see which is the heaviest and which is the lightest.

Step 3: Identify the heaviest and lightest item

Based on the comparisons: - The heaviest is the 10 kg bag of rice. - The lightest is the 4.53592 kg bag of fertilizer.

Key Concepts

Weight Comparison

When faced with the task of determining which item is the heaviest or lightest, your first step should be to ensure that all measurements are in the same unit. This makes it possible to accurately compare their weights.
For example, to compare a bag weighing 10 pounds with another weighing 10 kilograms and a certain volume of liquid, they must all be converted to a common unit, like kilograms. This step is critical since direct comparison of differing measurement units, such as pounds and kilograms, can lead to errors.
After conversion, simply order the weights to determine the lightest and heaviest items. In our case:

• The 10 kg bag of rice is the heaviest.
• The 4.53592 kg bag of fertilizer is the lightest.

This technique is straightforward but requires careful attention to unit conversions.

Density Calculation

Density is a concept that links mass and volume. It essentially tells us how much mass is contained in a particular volume. In simpler terms, density can be visualized as how tightly packed the substance's molecules are.
To find the mass of a liquid, you need its density and the volume it occupies. This can be calculated using the formula: \[ \text{Mass} = \text{Volume} \times \text{Density} \]In the case of olive oil, given the volume in gallons and density in grams per cubic centimeter (which is the same as kilograms per liter), we first convert the volume from gallons to liters.
Then by multiplying this volume by the density, we obtain the weight in kilograms. For olive oil, this results in approximately 6.9474 kg.
A clear understanding of density allows you to perform these calculations accurately, especially in different scientific and real-world applications.

Metric System

The metric system is a decimal-based system of measurement used worldwide, especially for scientific purposes.
It simplifies conversions significantly, typically involving simple multiplication or division by powers of ten. Key units include meters for length, liters for volume, and kilograms for mass.
To convert from imperial units like pounds or gallons to metric units like kilograms or liters, use fixed conversion rates. For example, knowing that 1 pound equals 0.453592 kilograms or that 1 gallon equals 3.78541 liters makes these conversions straightforward.
Understanding the metric system not only simplifies internal calculations but also ensures that you can interact with global data more effectively. In the exercise, we saw it applied in converting weights and volumes to a common, workable unit for comparison purposes.