Problem 58 - Chemistry: The Central Science in SI Units, Global Edition

Question

Carry out the following conversions: (a) 0.105 in. to \(\mathrm{mm}\), (b) \(0.650 \mathrm{qt}\) to \(\mathrm{mL}\), (c) \(8.75 \mu \mathrm{m} / \mathrm{s}\) to \(\mathrm{km} / \mathrm{hr}\) (d) \(1.955 \mathrm{~m}^{3}\) to \(\mathrm{yd}^{3}(\mathbf{e}) \$ 3.99 / \mathrm{lb}\) to dollars per \(\mathrm{kg}\), (f) \(8.75 \mathrm{lb} / \mathrm{ft}^{3}\) to \(\mathrm{g} / \mathrm{mL}\).

Step-by-Step Solution

Answer

\(a) 2.667\mathrm{~mm}, b) 615.130\mathrm{~mL}, c) 0.0315\mathrm{~km/hr}, d) 2.555\mathrm{~yd}^{3}, e) 8.800\mathrm{~dollars/kg}, f) 140.162\mathrm{~g/mL}\)

Step 1: Identify the conversion factor

We know 1 inch (in) = 25.4 millimeters (mm).

Step 2: Apply the conversion factor

Multiply 0.105 in by 25.4 mm/in. \(0.105 \times 25.4 = 2.667\mathrm{~mm}\) b) Conversion of 0.650 qt to mL

Step 1: Identify the conversion factor

We know 1 quart (qt) = 946.353 milliliters (mL).

Step 2: Apply the conversion factor

Multiply 0.650 qt by 946.353 mL/qt. \(0.650 \times 946.353 = 615.130\mathrm{~mL}\) c) Conversion of \(8.75 \mu \mathrm{m} / \mathrm{s}\) to \(\mathrm{km} / \mathrm{hr}\)

Step 1: Identify the conversion factors

We know 1 micrometer (µm) = 1e-6 meters (m), 1 kilometer (km) = 1000 meters (m), and 1 hour (hr) = 3600 seconds (s).

Step 2: Apply the conversion factors

Convert micrometers to kilometers and seconds to hours. \((8.75 \times 10^{-6})\mathrm{~m/s} \times \frac{1\mathrm{~km}}{1000\mathrm{~m}} \times \frac{3600\mathrm{~s}}{1\mathrm{~hr}} = 0.0315\mathrm{~km/hr}\) d) Conversion of \(1.955 \mathrm{~m}^{3}\) to \(\mathrm{yd}^{3}\)

Step 1: Identify the conversion factor

We know 1 cubic meter (m³) = 1.3080 cubic yards (yd³).

Step 2: Apply the conversion factor

Multiply 1.955 m³ by 1.3080 yd³/m³. \(1.955 \times 1.3080 = 2.555\mathrm{~yd}^{3}\) e) Conversion of \(3.99 / \mathrm{lb}\) to dollars per \(\mathrm{kg}\)

Step 1: Identify the conversion factor

We know 1 pound (lb) = 0.453592 kilograms (kg).

Step 2: Apply the conversion factor

Divide 3.99 \(/\mathrm{lb}\) by 0.453592 kg/lb. \(\frac{3.99}{0.453592} = 8.800\mathrm{~dollars/kg}\) f) Conversion of \(8.75 \mathrm{lb} / \mathrm{ft}^{3}\) to \(\mathrm{g} / \mathrm{mL}\)

Step 1: Identify the conversion factor

We know 1 pound per cubic foot (lb/ft³) = 16.0185 grams per milliliter (g/mL).

Step 2: Apply the conversion factor

Multiply 8.75 lb/ft³ by 16.0185 g/mL/lb/ft³. \(8.75 \times 16.0185 = 140.162\mathrm{~g/mL}\)

Key Concepts

Conversion Factors

Conversion factors are essential tools in unit conversion. They act as a bridge, allowing us to switch from one unit of measurement to another seamlessly. A conversion factor is essentially a ratio or fraction that expresses how many of one unit is equivalent to another. For example:

• 1 inch = 25.4 millimeters: This means for every inch, there are exactly 25.4 millimeters.
• 1 quart = 946.353 milliliters: Likewise, one quart can be expressed as 946.353 milliliters.

The beauty of using conversion factors is their ability to simplify unit conversion tasks. To convert a measurement, you multiply by the conversion factor, ensuring the unwanted units cancel out, leaving you with the desired units. For example, converting 0.650 quarts to milliliters involves multiplying by 946.353, as shown by the formula \(0.650 \times 946.353 = 615.130\, \text{mL}\). This clear-cut process underscores why conversion factors are vital in making measurement conversions straightforward and error-free.

Metric System

The metric system is a globally recognized measurement system that is easy to use. It helps in standardizing measurements across different regions and fields, primarily because it is based on powers of ten. This system includes units like meters for length, kilograms for mass, and liters for volume.

• The simplicity of the metric system lies in its base units and the ease of converting between them by moving the decimal point. For example, to convert from micrometers to meters, you simply multiply or divide by powers of ten.
• Additionally, because the metric system is universally used in scientific and many everyday contexts, understanding and using it can greatly help in international communication and technology.

Joining the ease of the metric system with conversion factors can significantly simplify even complex conversions, making its grasp crucial for students and professionals alike.

SI Units

SI units, or the International System of Units, are a comprehensive and consistent set of units adopted globally for scientific and technical applications. They form a subset of the metric system, designed to maintain uniformity in measurements worldwide.

• Key SI units include: meter (m) for length, kilogram (kg) for mass, and second (s) for time.
• The SI system facilitates conversions, such as converting micrometers per second to kilometers per hour, by standardizing the units used in various fields.

Using SI units ensures that your measurements are clear and easily understood globally. Whether you're engaging in science, engineering, or everyday tasks, SI units provide the precision and reliability necessary for consistent results. Thus, a robust understanding of SI units is essential for anybody dealing with measurements.

Measurement Conversions

Measurement conversions leverage conversion factors, the metric system, and SI units to seamlessly transition from one unit to another. A good grasp of measurement conversions involves understanding both the calculation process and the context in which these conversions occur.

• Common conversions include: linear (inches to millimeters), volumetric (quarts to milliliters), and currency (dollars per pound to dollars per kilogram).
• For example, converting cubic meters to cubic yards involves a conversion factor of 1.3080, dictating that \(1.955 \times 1.3080 = 2.555 \, \text{yd}^3\).

Successfully performing conversions requires practice and familiarity with the units involved. As you gain confidence, these conversions become second nature, empowering you to tackle a wide array of scientific and practical tasks with ease. With practice, the process of converting units becomes intuitive, allowing more focus on analyzing and interpreting results.