Problem 73 - Chemistry Principles and Practice

Question

Perform the conversions needed to fill in the blanks. Use scientific notation where appropriate. Do the operations first without a calculator or spreadsheet, to check your understanding of SI prefixes. (a) \(6.39 \mathrm{~cm}=\mathrm{m}=\stackrel{\mathrm{mm}}=\underline{\mathrm{nm}}\) (b) \(55.0 \mathrm{~cm}^{3}=\mathrm{dm}^{3}=\overline{\mathrm{mL}}=\underline{\mathrm{L}}=\underline{\mathrm{m}^{3}}\) (c) \(23.1 \mathrm{~g}=\underline{\mathrm{mg}}=\) \(-\mathrm{kg}\) (d) \(98.6^{\circ} \mathrm{F}=\quad{ }^{\circ} \mathrm{C}=\) \(\mathrm{K}\)

Step-by-Step Solution

Answer

(a) 0.0639 m, 63.9 mm, 6.39×10⁷ nm; (b) 0.055 L, 55.0 mL, 5.5×10⁻⁵ m³; (c) 23100 mg, 0.0231 kg; (d) 37.0°C, 310.15 K.

Step 1: Convert cm to m

To convert from centimeters (cm) to meters (m), divide by 100 because 1 meter equals 100 centimeters. Thus, \( 6.39 \text{ cm} = \frac{6.39}{100} \text{ m} = 0.0639 \text{ m} \).

Step 2: Convert m to mm

To convert from meters (m) to millimeters (mm), multiply by 1000 because 1 meter equals 1000 millimeters. Thus, \( 0.0639 \text{ m} = 0.0639 \times 1000 \text{ mm} = 63.9 \text{ mm} \).

Step 3: Convert mm to nm

To convert from millimeters (mm) to nanometers (nm), multiply by \( 10^6 \) because 1 millimeter equals \( 10^6 \) nanometers. Thus, \( 63.9 \text{ mm} = 63.9 \times 10^6 \text{ nm} = 6.39 \times 10^7 \text{ nm} \).

Step 4: Convert cm³ to dm³

To convert from cubic centimeters (cm³) to cubic decimeters (dm³), divide by 1000 since 1 dm³ (or 1 L) equals 1000 cm³. Thus, \( 55.0 \text{ cm}^3 = \frac{55.0}{1000} \text{ dm}^3 = 0.055 \text{ dm}^3 \).

Step 5: Convert dm³ to mL

Since 1 dm³ equals 1 liter and 1 liter equals 1000 milliliters, \( 0.055 \text{ dm}^3 = 0.055 \times 1000 \text{ mL} = 55.0 \text{ mL} \).

Step 6: Convert mL to L

Given that 1000 mL is equivalent to 1 L, so divide 55.0 mL by 1000 to convert to liters: \( 55.0 \text{ mL} = \frac{55.0}{1000} \text{ L} = 0.055 \text{ L} \).

Step 7: Convert L to m³

To convert from liters (L) to cubic meters (m³), divide by 1000 because 1 m³ equals 1000 L. Thus, \( 0.055 \text{ L} = \frac{0.055}{1000} \text{ m}^3 = 5.5 \times 10^{-5} \text{ m}^3 \).

Step 8: Convert g to mg

To convert grams (g) to milligrams (mg), multiply by 1000 since 1 gram equals 1000 milligrams. Thus, \( 23.1 \text{ g} = 23.1 \times 1000 \text{ mg} = 23100 \text{ mg} \).

Step 9: Convert g to kg

To convert grams (g) to kilograms (kg), divide by 1000 because 1 kilogram equals 1000 grams. Thus, \( 23.1 \text{ g} = \frac{23.1}{1000} \text{ kg} = 0.0231 \text{ kg} \).

Step 10: Convert Fahrenheit to Celsius

To convert Fahrenheit (°F) to Celsius (°C), use the formula \( \°C = (\mathrm{°F} - 32) \times \frac{5}{9} \). Substituting, we have \( (98.6 - 32) \times \frac{5}{9} = 37.0^{\circ} \text{C} \).

Step 11: Convert Celsius to Kelvin

To convert Celsius (°C) to Kelvin (K), add 273.15 to the Celsius temperature. So, \( 37.0 + 273.15 = 310.15 \text{ K} \).

Key Concepts

Metric System

The metric system is a decimal-based system of measurement, universally used in the scientific community. It simplifies calculations by scaling units by powers of ten. Each metric prefix represents a power of ten and connects smaller units to larger ones seamlessly. For example:

• Kilo- (k) means 1,000 or \( 10^3 \)
• Centi- (c) means 0.01 or \( 10^{-2} \)
• Milli- (m) means 0.001 or \( 10^{-3} \)

By using these prefixes, conversions such as from centimeters to meters or meters to millimeters involve simple multiplications or divisions by powers of ten. This system makes it easier to conduct scientific calculations without complex arithmetic.

Scientific Notation

Scientific notation is a method of writing very large or very small numbers compactly. This involves using powers of ten. It is essential when dealing with measurements in science, such as conversions requiring large or small unit changes, like nanometers to meters.

• Numbers are expressed as \( a \times 10^n \), where 1 ≤ \( a \) < 10 and \( n \) is an integer.
• For example, 63,900,000 nm can be written as \( 6.39 \times 10^7 \) nm.

This notation not only makes numbers easier to read and write but also reduces error during calculations, facilitating clearer presentation of data and results.

Temperature Conversion

Temperature scales like Celsius and Fahrenheit measure thermal energy. Converting between these scales is important, especially in scientific experiments involving temperature control. To convert Fahrenheit to Celsius use the formula:

• \( \text{°C} = (\text{°F} - 32) \times \frac{5}{9} \)

For example, converting 98.6°F to Celsius:\[ 98.6 - 32 = 66.6 \]\[ 66.6 \times \frac{5}{9} = 37.0 \text{°C} \]Similarly, to convert Celsius to Kelvin, add 273.15:

• \( \text{K} = \text{°C} + 273.15 \)

Thus, 37°C becomes 310.15 K. These conversions are crucial for consistency in scientific discussions.

Mass Conversion

In science, converting between different units of mass—like grams to milligrams or kilograms—is common practice. Understanding these conversions relies on knowing that:

• 1 gram (g) equals 1,000 milligrams (mg)
• 1 kilogram (kg) equals 1,000 grams (g)

To convert grams to milligrams, multiply by 1,000. For instance, 23.1 g translates to \( 23.1 \times 1,000 = 23,100 \) mg. Conversely, converting grams to kilograms involves dividing by 1,000: \( 23.1 \) g equals \( \frac{23.1}{1,000} = 0.0231 \) kg. These conversions are straightforward due to the metric system's foundation in powers of ten, allowing for easy manipulation of units in scientific calculations.