Problem 93 - Chemistry Matter and Change

Question

Convert the following mass measurements as indicated. (Chapter 2\()\) \(\begin{array}{ll}{\text { a. } 1.1 \mathrm{cm} \text { to meters }} & {\text { c. } 11 \mathrm{mg} \text { to kilograms }} \\ {\text { b. } 76.2 \mathrm{pm} \text { to millimeters }} & {\mathrm{d} .7 .23 \mu \mathrm{g} \text { to kilograms }}\end{array}\)

Step-by-Step Solution

Answer

1.1 cm = 0.011 m, 76.2 pm = 7.62 × 10⁻⁸ mm, 11 mg = 1.1 × 10⁻⁵ kg, 7.23 µg = 7.23 × 10⁻⁹ kg.

Step 1: Convert Centimeters to Meters

To convert 1.1 cm to meters, use the conversion factor that 1 meter = 100 centimeters. Divide 1.1 cm by 100 to get the length in meters:\[\text{Meters} = \frac{1.1 \text{ cm}}{100} = 0.011 \text{ m}\].Hence, 1.1 cm is equal to 0.011 meters.

Step 2: Convert Picometers to Millimeters

To convert 76.2 pm to millimeters, we first convert picometers to meters, knowing that 1 meter = \(10^{12}\) picometers. Then, convert meters to millimeters:1. Convert picometers to meters:\[\text{Meters} = \frac{76.2 \text{ pm}}{10^{12}} = 7.62 \times 10^{-11} \text{ m}\].2. Convert meters to millimeters (1 millimeter = 0.001 meters):\[\text{Millimeters} = \frac{7.62 \times 10^{-11} \text{ m}}{0.001} = 7.62 \times 10^{-8} \text{ mm}.\]Thus, 76.2 pm is equal to \(7.62 \times 10^{-8}\) millimeters.

Step 3: Convert Milligrams to Kilograms

To convert 11 mg to kilograms, use the conversion that 1 kilogram = 1,000,000 milligrams. Divide 11 mg by 1,000,000:\[\text{Kilograms} = \frac{11 \text{ mg}}{1,000,000} = 1.1 \times 10^{-5} \text{ kg}\].Therefore, 11 mg is \(1.1 \times 10^{-5}\) kilograms.

Step 4: Convert Micrograms to Kilograms

To convert 7.23 µg to kilograms, use the conversion factor 1 kilogram = \(10^9\) micrograms. Divide 7.23 µg by \(10^9\):\[\text{Kilograms} = \frac{7.23 \text{ µg}}{10^9} = 7.23 \times 10^{-9} \text{ kg}\].Thus, 7.23 µg is \(7.23 \times 10^{-9}\) kilograms.

Key Concepts

Mass Conversion

Mass conversion is the process of converting one unit of mass to another unit. It's important in science and everyday life when you need to switch between different units.

For example, in the exercise, we converted milligrams (mg) and micrograms (µg) to kilograms (kg).

• To convert from milligrams to kilograms, remember that 1 kilogram equals 1,000,000 milligrams. This means you need to divide the number of milligrams by 1,000,000 to get the result in kilograms.
• Similarly, for micrograms to kilograms, since 1 kilogram is 1,000,000,000 micrograms, divide the micrograms by 1,000,000,000.

Using simple division can help you get the appropriate conversion. It's essential to use the correct conversion factors to accurately switch between units. Understanding these basic relationships is crucial for mastering mass conversions.

Metric System

The metric system is a decimal-based system of measurement used by scientists and most of the world due to its simplicity and scalability. It is built on powers of ten, making conversions much more straightforward.

The basic units in the metric system cover different quantities such as meters for length, kilograms for mass, and liters for volume.

• A meter is the base unit for length.
• A kilogram is the base unit for mass in the metric system.
• Volume is often measured in liters or cubic meters.

The metric system's standardization and use of prefixes make it easy to convert between different magnitudes of measurement. Memorizing these basic units, along with their respective prefixes, helps in understanding and performing unit conversions efficiently.

Prefix Conversions

Prefix conversions in the metric system revolve around changing the scale of measurement using standard prefixes. These prefixes indicate a specific multiple of the base unit. Here are some common metric prefixes:

• Kilo- (k), which means a thousand, or 10 extsuperscript{3}.
• Centi- (c), which is one hundredth, or 10 extsuperscript{-2}.
• Milli- (m), meaning one thousandth, or 10 extsuperscript{-3}.
• Micro- (µ), meaning one-millionth, or 10 extsuperscript{-6}.
• Pico- (p), meaning one trillionth, or 10 extsuperscript{-12}.

To convert between units with different prefixes, you can either multiply or divide by powers of ten.

For instance, to convert picometers (pm) to millimeters (mm), convert picometers to meters first and then meters to millimeters using the appropriate power of ten based on their prefixes.

Scientific Notation

Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It involves representing the number as a product of a number between 1 and 10 and a power of ten.

This notation is particularly useful in scientific and mathematical work, where dealing with very large or small numbers is frequent.

• An example from the exercise includes converting lengths such as 76.2 picometers to meters and millimeters. By expressing these in scientific notation, you simplify the calculation and representation: 76.2 picometers is written as 7.62 x 10 extsuperscript{-11} meters.
• Similarly, micrograms to kilograms conversion shows 7.23 micrograms as 7.23 x 10 extsuperscript{-9} kilograms.

This method makes it easier to perform calculations and ensure precision across wide-ranging scales.