Problem 52 - Chemistry: The Central Science

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 \(\mathrm{yd}^{3}\), (e) \(\$ 3.99 / \mathrm{lb}\) to dollars per kg, (f) \(8.75 \mathrm{lb} / \mathrm{ft}^{3}\) to \(\mathrm{g} / \mathrm{mL}\).

Step-by-Step Solution

Answer

\(2.67 \ \text{mm}\), \(615.13 \ \text{mL}\), \(0.0000315 \ \mathrm{km/hr}\), \(2.558 \ \mathrm{yd^3}\), \(\$ 8.80 \ \mathrm{per \ kg}\), \(0.139 \frac{\mathrm{g}}{\mathrm{mL}}\)

Step 1: Find the conversion factor between inches and millimeters.

The conversion factor between inches and millimeters is \(1 \ \text{in.} = 25.4 \ \text{mm}\).

Step 2: Convert 0.105 inches to millimeters.

To convert 0.105 inches to millimeters, we multiply by the conversion factor derived in Step 1. \(0.105 \ \text{in.} \times 25.4 \frac{\text{mm}}{\text{in.}} = 2.67 \ \text{mm}\) #b) Convert 0.650 quarts to milliliters#

Step 1: Find the conversion factor between quarts and milliliters.

The conversion factor between quarts and milliliters is \(1 \ \text{qt} = 946.353 \ \text{mL}\).

Step 2: Convert 0.650 quarts to milliliters.

To convert 0.650 quarts to milliliters, we multiply by the conversion factor derived in Step 1. \(0.650 \ \text{qt} \times 946.353 \frac{\text{mL}}{\text{qt}} = 615.13 \ \text{mL}\) #c) Convert 8.75 μm/s to km/hr#

Step 1: Find the conversion factor between μm/s and km/hr.

The conversion factor between micrometers per second (μm/s) and kilometers per hour (km/hr) is \(1 \ \mathrm{\mu m/s} = 0.0000036 \ \mathrm{km/hr}\).

Step 2: Convert 8.75 μm/s to km/hr.

To convert 8.75 μm/s to km/hr, we multiply by the conversion factor derived in Step 1. \(8.75 \ \mathrm{\mu m/s} \times 0.0000036 \frac{\mathrm{km/hr}}{\mathrm{\mu m/s}} = 0.0000315 \ \mathrm{km/hr}\) #d) Convert 1.955 m³ to yd³#

Step 1: Find the conversion factor between m³ and yd³.

The conversion factor between cubic meters (m³) and cubic yards (yd³) is \(1 \ \mathrm{m^3} = 1.30795 \ \mathrm{yd^3}\).

Step 2: Convert 1.955 m³ to yd³.

To convert 1.955 m³ to yd³, we multiply by the conversion factor derived in Step 1. \(1.955 \ \mathrm{m^3} \times 1.30795 \frac{\mathrm{yd^3}}{\mathrm{m^3}} = 2.558 \ \mathrm{yd^3}\) #e) Convert $3.99 per lb to dollars per kg#

Step 1: Find the conversion factor between pounds and kilograms.

The conversion factor between pounds (lb) and kilograms (kg) is \(1 \ \mathrm{lb} = 0.453592 \ \mathrm{kg}\).

Step 2: Convert $3.99 per lb to dollars per kg.

To convert $3.99 per lb to dollars per kg, we divide the price by the conversion factor derived in Step 1. \(\frac{\$ 3.99} {\mathrm{lb}} \times \frac{1 \ \mathrm{kg}}{0.453592 \ \mathrm{lb}} = \$ 8.80 \ \mathrm{per \ kg}\) #f) Convert 8.75 lb/ft³ to g/mL#

Step 1: Find the conversion factors between lb/ft³ and g/mL.

The conversion factors between pounds per cubic foot (lb/ft³) and grams per milliliter (g/mL) are: 1. \(1 \ \mathrm{lb} = 453.592 \ \mathrm{g}\) 2. \(1 \ \mathrm{ft^3} = 28,316.8 \ \mathrm{mL}\)

Step 2: Convert 8.75 lb/ft³ to g/mL.

To convert 8.75 lb/ft³ to g/mL, we multiply by the respective conversion factors (Step 1). \(8.75 \frac{\mathrm{lb}} {\mathrm{ft^3}} \times \frac {453.592 \ \mathrm{g}}{1 \ \mathrm{lb}} \times \frac {1 \ \mathrm{ft^3}}{28,316.8 \ \mathrm{mL}} = 0.139 \frac{\mathrm{g}}{\mathrm{mL}}\)

Key Concepts

conversion factors

Conversion factors serve as a bridge between differing measurement units. They allow us to translate quantities from one unit to another, ensuring accuracy in calculations. For instance, to convert inches to millimeters, we use the conversion factor of 1 inch equals 25.4 millimeters. This is expressed as:

• 1 in. = 25.4 mm

With this, any measurement in inches can be converted into millimeters simply by multiplying the inches by 25.4. Conversion factors are essential tools in science and everyday calculations, permitting us to work interchangeably with different units without losing precision. Knowing the right conversion factor is the first step in unit conversion problems.

measurement units

Measurement units are standardized quantities used to express dimensions like length, volume, and mass. They create a shared language for quantifiable observations. Common units include inches and millimeters for length, quarts and milliliters for volume, and pounds and kilograms for mass.
For any measurement task, selecting the appropriate unit is crucial. Making conversions between units, like from quarts to milliliters, ensures consistency and compatibility of measurements across different systems.
Understanding different units, such as how many milliliters are in a quart (946.353 mL), is key to performing precise calculations and switching smoothly between familiar and international systems.

dimensional analysis

Dimensional analysis is a systematic approach widely used in converting units by using conversion factors. It helps in ensuring that our calculations only involve units we are interested in, while others cancel out. By "unit cancellation," we can neatly convert between quantities.Let's consider converting cubic meters (\( ext{m}^3\)) to cubic yards (\( ext{yd}^3\)). The conversion factor is 1 m³ = 1.30795 yd³, and the calculation involves multiplying the two quantities:

• \[1.955 ext{ m}^3 imes 1.30795 rac{ ext{yd}^3}{ ext{m}^3} = 2.558 ext{ yd}^3\]

The units of cubic meters cancel each other out, leaving us with the final desired unit of cubic yards. Dimensional analysis is immensely powerful because it ensures that no step is forgotten, leading to highly accurate results.

metric system conversions

Metric system conversions, such as converting meters to kilometers or grams to kilograms, are often simpler because of the decimal-based nature of the system. Each unit is categorically related by factors of ten. This allows for straightforward conversions using multiplication or division.To convert micrometers per second to kilometers per hour, we need to consider both distance and time unit conversion:

• 1 μm/s = 0.0000036 km/hr

By multiplying the original speed by this conversion factor, we effortlessly convert it to the desired unit:

• \[8.75 ext{ μm/s} imes 0.0000036 rac{ ext{km/hr}}{ ext{μm/s}} = 0.0000315 ext{ km/hr}\]

The metric system's coherency and clarity make unit conversion straightforward, efficient, and consistent across scientific and everyday use.