Problem 43 - 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) \(\mathrm{in} .{ }^{3}\) to \(\mathrm{cm}^{3}\).

Step-by-Step Solution

Answer

To convert the given units, use the following conversion factors: 1. mm to nm: \(10^6\) 2. mg to kg: \(10^{-6}\) 3. km to ft: approximately 3,280.84 4. in^3 to cm^3: approximately 16.387

Step 1: 1. Millimeters to nanometers

To convert millimeters to nanometers, we need to know the relationship between the two units. Both are metric units, and we know that 1 mm equals 1,000,000 nm or \(1 mm = 10^6 nm\). So, the conversion factor is \(10^6\).

Step 2: 2. Milligrams to kilograms

For converting milligrams to kilograms, we need to understand the relationship between these metric units of mass. We know that 1 kg equals 1,000,000 mg, or \(1 kg = 10^6 mg\). Therefore, the conversion factor is \(\frac{1}{10^6}\) or \(10^{-6}\).

Step 3: 3. Kilometers to feet

To convert kilometers to feet, we need to know the relationship between both units. It's necessary to go through meters first because kilometers and meters are metric units, while feet are an English unit. We know that 1 km equals 1,000 m or \(1 km = 10^3 m\), and then 1 m equals 3.28084 ft or \(1 m = 3.28084 ft\). Therefore, to convert from km to ft, we first convert km to m, then m to ft. The conversion factor is \(10^3\times3.28084\), which is approximately equal to 3,280.84.

Step 4: 4. Cubic inches to cubic centimeters

To convert cubic inches to cubic centimeters, we need to understand the relationship between inches and centimeters first. We know that 1 in equals 2.54 cm, or \(1 in = 2.54 cm\). To convert cubic inches to cubic centimeters, we need to cube the linear relationship of inches to cm: \((1 in^3 = 2.54^3 cm^3 = 16.387 cm^3)\). So, the conversion factor is approximately 16.387.

Key Concepts

Metric Units

Metric units are part of a global standard for measurement used worldwide in various disciplines, including science, engineering, and everyday life. These units are decimal-based, meaning they scale by powers of ten, which makes calculations and conversions straightforward. For example, millimeters (mm), centimeters (cm), and kilometers (km) are metric units of length. Similarly, milligrams (mg) and kilograms (kg) are metric units of mass.

A useful feature of the metric system is its consistent use of prefixes such as "milli-," "centi-," and "kilo-," which indicate the relative size of the unit. For instance:

• "Milli-" denotes a thousandth (\(10^{-3}\)) of a unit, such as millimeter or milligram.
• "Kilo-" signifies a thousand (\(10^{3}\)) times a unit, such as kilogram or kilometer.

This system makes it easy to understand and convert between different magnitudes of the same unit, aligning with our step by step examples, like converting milligrams to kilograms or kilometers to meters.

English Units

English units, also known as Imperial or customary units, are uses commonly in the United States and a few other countries. Unlike the metric system, English units do not have a uniform base, and conversions between units often require memorization of specific conversion factors.

Examples include inches, feet, and yards for measuring length, and ounces, pounds, and tons for mass. The disparity between different measurement systems can lead to complexities, such as converting kilometers to feet as in our task, where an intermediary step through meters (a metric unit) is needed because the conversion between English and metric units is not straightforward.

Understanding both English and metric units is crucial for comprehending and performing unit conversions in a global context.

Conversion Factors

Conversion factors are values used to express a quantity in different units without changing the amount. They are crucial in converting measurements from one system of units to another, whether within the same system (like metric to metric) or between different systems (like metric to English). These factors are mathematical ratios derived from the relationship between two units and can be used to perform accurate and consistent conversions.

For example, converting millimeters to nanometers involves using the conversion factor \(10^{6}\), since we know that 1 millimeter equals 1,000,000 nanometers. Similarly, when converting between cubic inches and cubic centimeters, the conversion factor \(16.387\) applies because 1 cubic inch equals approximately 16.387 cubic centimeters.

Employing conversion factors successfully requires understanding the relationship between units, using precise mathematical operations, and often integrating additional steps when systems don't align directly, as seen when converting km to ft via an intermediate step through meters.