Question

Perform the following conversions: (a) \(0.076 \mathrm{~L}\) to \(\mathrm{mL}\), (b) \(5.0 \times 10^{-8} \mathrm{~m}\) to \(\mathrm{nm}\), (c) \(6.88 \times 10^{5} \mathrm{~ns}\) to \(\mathrm{s}\), (d) \(0.50 \mathrm{lb}\) to \(\mathrm{g}\), (e) \(1.55 \mathrm{~kg} / \mathrm{m}^{3}\) to \(\mathrm{g} / \mathrm{L}\), (f) \(5.850 \mathrm{gal} / \mathrm{hr}\) to \(\mathrm{L} / \mathrm{s}\).

Short answer

(a) 76 mL, (b) 50 nm, (c) \(6.88 \times 10^{-4}\) s, (d) 226.796 g, (e) 1.55 g/L, (f) \(4.550 \times 10^{-3}\) L/s.

Step-by-step solution

a) Convert 0.076 L to mL

To convert liters to milliliters, use the following relationship: 1 L = 1000 mL The conversion can be done in one step: 0.076 L × \( \frac{1000 \mathrm{~mL}}{1 \mathrm{~L}} \)

b) Convert \(5.0 \times 10^{-8} \mathrm{~m}\) to nm

To convert meters to nanometers, use the following relationship: 1 m = \(1 \times 10^{9}\) nm The conversion can be done in one step: \(5.0 \times 10^{-8} \mathrm{~m}\) × \( \frac{1 \times 10^{9} \mathrm{~nm}}{1 \mathrm{~m}} \)

c) Convert \(6.88 \times 10^{5} \mathrm{~ns}\) to s

To convert nanoseconds to seconds, use the following relationship: 1 s = \(1 \times 10^{9}\) ns The conversion can be done in one step: \(6.88 \times 10^{5} \mathrm{~ns}\) × \( \frac{1 \mathrm{~s}}{1 \times 10^{9} \mathrm{~ns}} \)

d) Convert 0.50 lb to g

To convert pounds to grams, use the following relationship: 1 lb = 453.592 g The conversion can be done in one step: 0.50 lb × \( \frac{453.592 \mathrm{~g}}{1 \mathrm{~lb}} \)

e) Convert \(1.55 \mathrm{~kg/m^{3}}\) to \(\mathrm{g/L}\)

To convert kg/m³ to g/L, we need to use the following relationships: 1 kg = 1000 g 1 m³ = 1000 L The conversion can be done in two steps: 1. Convert kg to g: \(1.55 \mathrm{~kg/m^{3}}\) × \( \frac{1000 \mathrm{~g}}{1 \mathrm{~kg}} \) = 1550.00 \(\mathrm{g/m^{3}}\) 2. Convert m³ to L: \(1550.00 \mathrm{~g/m^{3}}\) × \( \frac{1 \mathrm{~m^{3}}}{1000 \mathrm{~L}}\)

f) Convert \(5.850 \mathrm{~gal/hr}\) to \(\mathrm{L/s}\)

To convert gal/hr to L/s, we need to use the following relationships: 1 gal = 3.78541 L 1 hr = 3600 s The conversion can be done in two steps: 1. Convert gal to L: \(5.850 \mathrm{~gal/hr}\) × \( \frac{3.78541 \mathrm{~L}}{1 \mathrm{~gal}} \) 2. Convert hr to s: \(5.850 \mathrm{~L/hr}\) (Result from Step 1) × \( \frac{1 \mathrm{~hr}}{3600 \mathrm{~s}}\)

Key concepts

Understanding the Metric System

The metric system is an international system of measurement that is widely used for scientific and everyday purposes. It's considered incredibly user-friendly because it's based on powers of ten. This means that conversions between different units within the metric system often involve simple multiplication or division by 10, 100, or 1000. For example, when converting liters (L) to milliliters (mL), you multiply by 1000, because 1 liter equals 1000 milliliters.
This simplicity is due to the use of common prefixes, such as "kilo-" for a thousand times a unit, "centi-" for one hundredth of a unit, and "milli-" for one thousandth of a unit. These prefixes make it easy to understand magnitude differences:

• Kilo-: 1 kilometer (km) = 1000 meters (m)
• Centi-: 1 meter (m) = 100 centimeters (cm)
• Milli-: 1 meter (m) = 1000 millimeters (mm)

Understanding these prefixes and the base ten nature of the metric system can help you quickly perform conversions and understand measurements in different contexts.

Demystifying Dimensional Analysis

Dimensional analysis is a powerful technique in physics and chemistry, used to convert quantities from one unit to another. It involves using equivalence relationships between units, known as conversion factors. These factors are essentially fractions that equal one. They are derived from defined or known relationships, such as 1 meter = 100 centimeters.
When performing dimensional analysis, you multiply the quantity you want to convert by the appropriate conversion factor(s). This process allows you to cancel units and ensure the conversion is correct. For example, converting 0.50 lbs to grams:

• The conversion factor between pounds and grams is 1 lb = 453.592 g.
• Multiply 0.50 lb by the conversion factor: \(0.50 \text{ lb} \times \frac{453.592 \text{ g}}{1 \text{ lb}} \)

The units of pounds cancel out, leaving the result in grams. This technique helps you handle more complex conversions, ensuring accuracy and clarity in your calculations.

Exploring Units of Measurement

Units of measurement are essential for describing quantities and comparing them across different contexts. They provide a standard way to convey amounts of physical quantities like length, mass, volume, and time. Understanding and using the correct units is crucial in scientific experimentation and day-to-day activities.
There are two main types of units: fundamental units and derived units. Fundamental units are the basic units of measurement for fundamental quantities, such as meters for length, kilograms for mass, and seconds for time. Derived units are created by combining fundamental units according to mathematical relationships. For example, speed is measured in meters per second (m/s), a derived unit from dividing length by time.
Remembering these two categories of units will help you better understand and perform conversions. Also, always keep conversion factors handy, as they are key to transitioning between related units and maintaining consistency across measurements.