Problem 45 - Chemistry: The Central Science

Question

Using your knowledge of metric units, English units, and the information on the back inside cover, write down the conversion factors needed to convert (a) \(\mathrm{mm}\) to \(\mathrm{nm}\), (b) \(\mathrm{mg}\) to \(\mathrm{kg}\), (c) \(\mathrm{km}\) to \(\mathrm{ft}\), (d) in. \({ }^{3}\) to \(\mathrm{cm}^{3}\).

Step-by-Step Solution

Answer

To convert the units, use the following conversion factors: a) 1 mm = \(1\,\text{mm} \times 1000\,\text{μm/mm} \times 1000\,\text{nm/μm}\) = 1,000,000 nm b) 1 mg = \(1\,\text{mg} \times \frac{1\,\text{g}}{1000\,\text{mg}} \times \frac{1\,\text{kg}}{1000\,\text{g}}\) = 0.000001 kg c) 1 km = \(1\,\text{km} \times 1000\,\text{m/km} \times 3.28084\,\text{ft/m}\) = 3,280.84 ft d) 1 in³ = \(1\,\text{in}^3 \times (2.54\,\text{cm/in})^3\) = 16.3871 cm³

Step 1: a) Millimeters (mm) to Nanometers (nm)

We are given that 1 millimeter is equal to 1000 micrometers, and 1 micrometer is equal to 1000 nanometers. To convert mm to nm, we just need to multiply the given millimeters by these conversion factors: \(1\,\text{mm} \times 1000\,\text{μm/mm} \times 1000\,\text{nm/μm}\)

Step 2: b) Milligrams (mg) to Kilograms (kg)

We are given that 1 gram is equal to 1000 milligrams, and 1 kilogram is equal to 1000 grams. To convert mg to kg, we just need to multiply the given milligrams by these conversion factors: \(1\,\text{mg} \times \frac{1\,\text{g}}{1000\,\text{mg}} \times \frac{1\,\text{kg}}{1000\,\text{g}}\)

Step 3: c) Kilometers (km) to Feet (ft)

We are given that 1 kilometer is equal to 1000 meters, and 1 meter is equal to 3.28084 feet. To convert km to ft, we just need to multiply the given kilometers by these conversion factors: \(1\,\text{km} \times 1000\,\text{m/km} \times 3.28084\,\text{ft/m}\)

Step 4: d) Cubic inches (in³) to Cubic centimeters (cm³)

We are given that 1 inch is equal to 2.54 centimeters. To convert cubic inches (in³) to cubic centimeters (cm³), we need to use the conversion factor for inches to centimeters and then raise it to the power of 3: \(1\,\text{in}^3 \times (2.54\,\text{cm/in})^3\) Calculate the number to find the conversion factor for cubic inches to cubic centimeters.

Key Concepts

Metric Units

Metric units are an essential part of the metric system used worldwide to measure various quantities like length, mass, and volume. They are based on powers of ten, which makes calculations and conversions straightforward and standardized.
The basic units you'll encounter include:

• Meter (m) for length
• Gram (g) for mass
• Liter (L) for volume

For example, to convert between units of length, such as millimeters (mm) to nanometers (nm), one simply uses conversion factors. A millimeter is 1,000 times larger than a micrometer, which is 1,000 times larger than a nanometer. These conversions are as simple as multiplying by the appropriate power of ten. Understanding metric units allows for seamless navigation between different magnitudes of measurement, making them highly convenient.

English Units

Unlike metric units, English units, also known as imperial units, are primarily used in the United States. These measurements can feel less intuitive because they don't follow a base-10 system, leading to more complex conversion factors.
Common English units include:

• Inches and feet for length
• Pounds and ounces for weight
• Gallons and quarts for volume

When converting between English units and metric units, additional steps and conversion factors are necessary. For instance, converting inches to centimeters requires understanding that 1 inch equals 2.54 centimeters. For volumetric conversions, you must cube the conversion factor when dealing with cubic inches to convert accurately to cubic centimeters.

Conversion Factors

Conversion factors are crucial in changing units from one system to another. They act as bridges that relate different units, allowing you to convert back and forth between them seamlessly.
Here’s how you can use them effectively:

• Identify the unit you're converting from and the unit you're converting to.
• Use the conversion factor that equates the two units.
• Multiply your given value by these factors to switch units.

For example, converting kilometers to feet involves two conversion factors: 1 kilometer equals 1,000 meters, and 1 meter equals 3.28084 feet. By combining these factors, you can easily transform any given number of kilometers entirely into feet. Practicing with different units will solidify understanding and make complex conversions intuitive.

Chemistry Education

In chemistry education, understanding units and unit conversions is fundamental. Chemistry often involves precise measurements and calculations, requiring students to be adept at switching between different units.
Knowing how to use conversion factors fluently aids in solving chemical equations, understanding the properties of substances, and accurately describing reactions in a universal scientific language.
This skill is not only limited to classroom exercises. It extends to lab work, where measurements must be precise to ensure safe and accurate experimentation. Being bilingual in both metric and English units allows students, especially those in regions where English units are still prevalent, to thrive in both educational settings and future scientific endeavors.