Problem 53 - Chemistry: The Central Science in SI Units, Global Edition

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) in. to \(\mathrm{cm}(\mathbf{b}) \mathrm{lb}\) to \(\mathrm{g}\) (c) \(\mu g\) to \(g\) (d) \(\mathrm{ft}^{2}\) to \(\mathrm{cm}^{2}\).

Step-by-Step Solution

Answer

The conversion factors for the given units are: (a) \(1\ \mathrm{in} = 2.54\ \mathrm{cm}\) (b) \(1\ \mathrm{lb} = 453.592\ \mathrm{g}\) (c) \(1\ \mu\mathrm{g} = 10^{-6}\ \mathrm{g}\) (d) \(1\ \mathrm{ft}^2 = 929.0304\ \mathrm{cm}^2\)

Step 1: 1. Conversion factor for inches (in) to centimeters (cm)

Remember that 1 inch is equal to 2.54 centimeters. Therefore, the conversion factor for inches to centimeters is: \(1\ \mathrm{in} = 2.54\ \mathrm{cm}\)

Step 2: 2. Conversion factor for pounds (lb) to grams (g)

Recall that 1 pound is equal to 453.592 grams. Therefore, the conversion factor for pounds to grams is: \(1\ \mathrm{lb} = 453.592\ \mathrm{g}\)

Step 3: 3. Conversion factor for micrograms (μg) to grams (g)

Remember that 1 gram is equal to 1,000,000 micrograms. Therefore, the conversion factor for micrograms to grams is: \(1\ \mathrm{g} = 1,000,000\ \mu\mathrm{g}\) To convert from micrograms to grams, we can find the reciprocal of this relationship: \(1\ \mu\mathrm{g} = 10^{-6}\ \mathrm{g}\)

Step 4: 4. Conversion factor for square feet (ft²) to square centimeters (cm²)

First, let's convert square feet to square inches. We know that 1 foot is equal to 12 inches. Therefore, to find the conversion factor for square feet to square inches, we can square this relationship: \(1\ \mathrm{ft}^2 = (12\ \mathrm{in})^2 \) \(1\ \mathrm{ft}^2 = 144\ \mathrm{in}^2 \) Now, we've already found the conversion factor for inches to centimeters in step 1. To find the conversion factor for square inches to square centimeters, we can square the conversion factor from inches to centimeters: \(1\ \mathrm{in}^2 = (2.54\ \mathrm{cm})^2\) \(1\ \mathrm{in}^2 = 6.4516\ \mathrm{cm}^2\) Finally, we can combine both conversion factors from square feet to square inches and square inches to square centimeters. Multiply both conversion factors: \(1\ \mathrm{ft}^2 = 144\ \mathrm{in}^2 \times 6.4516\ \mathrm{cm}^2\) \(1\ \mathrm{ft}^2 = 929.0304\ \mathrm{cm}^2\)

Key Concepts

Metric System

The metric system is a universal system of measurement used by scientists and most of the world. It is based on a series of units that are all multiples of ten.
Because of its base ten nature, it simplifies calculations and conversions. The international standard for most scientific measurements is the metric system.

• Base units: These include meters for length, kilograms for mass, and seconds for time.
• Derived units: These are formed by combining the base units, for example, area in square meters or volume in cubic meters.

By using prefixes like kilo (1000 times), centi (1/100th), and milli (1/1000th), it becomes easier to express and convert values. For instance, length in the metric system can easily switch between kilometers, meters, centimeters, and millimeters simply by moving the decimal point.

Measurement Units

Measurement units are essential for quantifying and comparing different quantities. Whether using metric or English units, it's crucial to understand their relationships.

In scientific work, metric units are often preferred due to their consistency and simplicity. To convert between metric and English units, conversion factors are used. These are fixed numerical values that relate one unit of measurement to another.
For example, the conversion factor for converting inches to centimeters is 2.54, since 1 inch equals 2.54 cm. When dealing with areas, like converting square feet to square centimeters, remember that both length dimensions must be changed. This is done by squaring the linear conversion factor (1 ft = 12 inches becomes 1 ft² = 144 in²). Then, you multiply by the conversion factor for inches squared to centimeters squared (1 in² = 6.4516 cm²).

Scientific Calculations

Scientific calculations are used to make precise and accurate measurements in experiments and studies, requiring conversions to ensure all measurements are in the same units. Without proper conversion, results may not make sense or could be inaccurate.

When performing scientific calculations involving converting measurement units, attention to detail is crucial.

• Steps in Conversion:
• Identify the unit to convert from and the unit to convert to.
• Find the appropriate conversion factor.
• Apply the conversion factor by multiplying (or dividing) the given value.

Precision in these calculations ensures data validity. For example, correctly converting micrograms (µg) to grams (g) requires understanding that 1 µg is equivalent to \(10^{-6}\) g. Similarly, converting pounds to grams means recognizing the factor of 453.592. Such calculations help maintain consistency in scientific measurements.