Question

Complete the following metric conversions using the correct number of significant figures: (a) \(4.5 \mathrm{~cm}\) to \(\hat{\mathrm{A}}\) (b) \(12 \mathrm{~nm}\) to \(\mathrm{cm}\) (c) \(8.0 \mathrm{~km}\) to \(\mathrm{mm}\) (d) \(164 \mathrm{mg}\) to \(\mathrm{g}\) (e) \(0.65 \mathrm{~kg}\) to \(\mathrm{mg}\) (f) \(5.5 \mathrm{~kg}\) to g (g) \(0.468 \mathrm{~L}\) to \(\mathrm{mL}\) (h) \(9.0 \mu \mathrm{L}\) to \(\mathrm{mL}\)

Short answer

(a) 4.5×10⁸ Å, (b) 1.2×10⁻⁶ cm, (c) 8.0×10⁶ mm, (d) 0.164 g, (e) 6.5×10⁵ mg, (f) 5500 g, (g) 468 mL, (h) 0.0090 mL.

Step-by-step solution

Understanding Conversion Factors

Before converting, it's important to know the conversion factors between units. For example: 1 cm = 10 mm, 1 nm = 1×10⁻⁹ m, 1 km = 10⁶ mm, 1 mg = 0.001 g, 1 kg = 10⁶ mg, 1 kg = 1000 g, 1 L = 1000 mL, and 1 μL = 0.001 mL.

Converting 4.5 cm to Å

1 cm = 10⁸ Å. Therefore, 4.5 cm = 4.5 × 10⁸ Å = 4.5 × 10⁸ Å (with 2 significant figures).

Converting 12 nm to cm

1 nm = 1×10⁻⁷ cm. Therefore, 12 nm = 12 × 1×10⁻⁷ cm = 1.2 × 10⁻⁶ cm (with 2 significant figures).

Converting 8.0 km to mm

1 km = 10⁶ mm. Therefore, 8.0 km = 8.0 × 10⁶ mm = 8.0 × 10⁶ mm (with 2 significant figures).

Converting 164 mg to g

1 mg = 0.001 g. Therefore, 164 mg = 164 × 0.001 g = 0.164 g (with 3 significant figures).

Converting 0.65 kg to mg

1 kg = 10⁶ mg. Therefore, 0.65 kg = 0.65 × 10⁶ mg = 6.5 × 10⁵ mg (with 2 significant figures).

Converting 5.5 kg to g

1 kg = 1000 g. Therefore, 5.5 kg = 5.5 × 1000 g = 5500 g (with 2 significant figures).

Converting 0.468 L to mL

1 L = 1000 mL. Therefore, 0.468 L = 0.468 × 1000 mL = 468 mL (with 3 significant figures).

Converting 9.0 µL to mL

1 µL = 0.001 mL. Therefore, 9.0 µL = 9.0 × 0.001 mL = 0.0090 mL (with 2 significant figures).

Key concepts

significant figures

Significant figures (sig figs) are the digits that carry meaning in a number, contributing to its precision. When we perform metric conversions in chemistry, it's crucial to maintain the correct number of significant figures to ensure accuracy. Sig figs start from the first non-zero digit on the left and include all subsequent digits, including zeroes that come after a non-zero digit. For example, the number 0.0080 has two significant figures: '8' and '0'. Understanding and applying sig figs correctly in conversions and calculations helps maintain consistency and precision in experimental results and data interpretation.

conversion factors

Conversion factors are ratios used to express a quantity in different units. They are fundamental in chemistry calculations and metric conversions.
To use a conversion factor, you multiply the quantity you want to convert by the appropriate ratio. For example, converting 5 meters to centimeters involves using the factor 1 meter = 100 centimeters. Therefore, you multiply 5 meters by 100 to get 500 centimeters. Recognizing and applying the correct conversion factors is essential in transforming units accurately while keeping the magnitude of the quantity unchanged.

unit conversion

Unit conversion is the process of changing a measurement from one unit to another while maintaining the same value. It's a critical skill in chemistry as measurements often need to be expressed in different units to compare results or perform calculations.
The steps for unit conversion usually involve:

• Identifying the starting unit and the unit you want to convert to.
• Using conversion factors to bridge the two units.
• Multiplying the original measurement by the conversion factors.

For instance, to convert 1000 milligrams to grams, recognize that 1 gram = 1000 milligrams. Multiply 1000 milligrams by the factor (1 gram / 1000 milligrams) to get 1 gram, ensuring the units cancel out correctly and the quantity remains the same.

metric system

The metric system is an international decimalized system of measurement based on powers of ten. It's widely used in science, especially chemistry, because of its simplicity and ease of converting between units.
Key units in the metric system include:

• Meter (m) for length
• Gram (g) for mass
• Liter (L) for volume
• Second (s) for time

Prefixes such as kilo-, centi-, milli- are used to denote multiples or fractions of these base units. For example, 'kilo-' means 1000 times (1 kilometer = 1000 meters), while 'milli-' means one-thousandth (1 milligram = 0.001 grams). Understanding and using the metric system correctly is crucial for precise scientific measurements.

chemistry calculations

Chemistry calculations involve mathematical operations used to analyze chemical reactions and properties. They often require unit conversions and a deep understanding of significant figures to ensure precision.
Common chemistry calculations include:

• Converting between different units of measurement (e.g., liters to milliliters).
• Calculating molar masses and concentrations.
• Determining yield and stoichiometry in chemical reactions.

Mastering these calculations helps in accurately interpreting experimental data, ensuring that the results are reliable and reproducible. For instance, converting 0.5 kilometers to millimeters involves multiple conversion steps, applying both conversion factors and sig fig rules to get the precise result: 0.5 km = 500,000 mm.