Problem 46 - 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) \(\mu \mathrm{m}\) to \(\mathrm{mm}\), (b) \(\mathrm{ms}\) to \(\mathrm{ns}\), (c) \(\mathrm{mi}\) to \(\mathrm{km},(\mathrm{d}) \mathrm{ft}^{3}\) to \(\mathrm{L}\)

Step-by-Step Solution

Answer

(a) 1 μm = \( \frac{1}{1000} \) mm (b) 1 ms = 1,000,000 ns (c) 1 mi ≈ 1.60934 km (d) 1 ft³ ≈ 28.3168 L

Step 1: (a) Conversion factor from micrometers to millimeters

To convert micrometers to millimeters, we should know the relative sizes of the two units. Due to the following definition: 1 millimeter (mm) = 1000 micrometers (μm) We can directly state that the conversion factor is: 1 μm = \( \frac{1}{1000} \) mm

Step 2: (b) Conversion factor from milliseconds to nanoseconds

To convert milliseconds to nanoseconds, we should also know the relationship between the two units: 1 millisecond (ms) = 1,000,000 nanoseconds (ns) Thus, the conversion factor is: 1 ms = 1,000,000 ns

Step 3: (c) Conversion factor from miles to kilometers

To convert miles to kilometers, we will use the relationship between English and metric units for length: 1 mile (mi) ≈ 1.60934 kilometers (km) So, the conversion factor is approximately: 1 mi ≈ 1.60934 km

Step 4: (d) Conversion factor from cubic feet to liters

Finally, to convert cubic feet to liters, we need to use the relationship between English and metric units for volume: 1 cubic foot (ft³) ≈ 28.3168 liters (L) Therefore, the conversion factor is approximately: 1 ft³ ≈ 28.3168 L

Key Concepts

Metric Units

Metric units are part of the International System of Units (SI), which is widely used around the world for scientific and everyday purposes. The metric system makes use of units like meters for length, liters for volume, and grams for mass. These units are especially convenient because they are based on powers of ten. This means converting between units is often as simple as moving a decimal point.

For instance:

• 1 millimeter (mm) is equal to 1000 micrometers (μm)
• 1 meter (m) is equal to 1000 millimeters (mm)
• 1 liter (L) is equal to 1000 milliliters (mL)

This system simplifies calculations and conversions, such as converting micrometers to millimeters or milliseconds to nanoseconds, which often involves multiplying or dividing by powers of ten.
Understanding the basic metric units and their prefixes, like kilo- (thousand), centi- (hundredth), and milli- (thousandth), helps in making accurate conversions easily.

English Units

English units, also known as Imperial units, are primarily used in the United States and some other countries for everyday measurements. This system includes units like inches, feet, and miles for length, or gallons for volume. Unlike the metric system, English units do not follow a base-10 format and can be a bit trickier for conversion.

For example:

• 1 mile is usually converted to feet as 5280 feet
• 1 inch is equivalent to 2.54 centimeters
• 1 gallon consists of 4 quarts

For tasks involving the conversion of English units to metric (or vice versa), like converting miles to kilometers, it's essential to know the specific conversion factors, since they don't follow a simple decimal or linear scale. English units can vary greatly depending on the unit being converted, so having a reliable reference or list of conversions can be incredibly helpful for accurate measurements.

Unit Conversions

Unit conversions involve changing one unit of measure to another, allowing for common understanding and comparison. This is a critical skill in many scientific, engineering, and everyday applications. It requires knowing the correct conversion factors, which are ratios that describe how much of one unit equals a certain amount of another unit.

Here are some common examples:

• To convert micrometers (μm) to millimeters (mm), we use the conversion factor: 1 μm = \( \frac{1}{1000} \) mm
• Milliseconds (ms) to nanoseconds (ns) involves: 1 ms = 1,000,000 ns
• Miles (mi) to kilometers (km) uses: 1 mi ≈ 1.60934 km
• Cubic feet (ft³) to liters (L) requires: 1 ft³ ≈ 28.3168 L

Each conversion requires care to ensure calculations are accurate, often using multiplication or division by the conversion factor.
Unround numbers conservatively to maintain accuracy and solve real-world problems where precision is essential. Whether you're converting distances for a road trip or changing recipe quantities, knowing how to convert units is an invaluable skill.