Question

Carry out the following conversions: (a) \(22.6 \mathrm{~m}\) to decimeters, (b) \(25.4 \mathrm{mg}\) to kilograms, (c) \(556 \mathrm{~mL}\) to liters, (d) \(10.6 \mathrm{~kg} / \mathrm{m}^{3}\) to \(\mathrm{g} / \mathrm{cm}^{3}\).

Short answer

(a) 226 dm, (b) 0.0000254 kg, (c) 0.556 L, (d) 0.0106 g/cm³.

Step-by-step solution

Convert Meters to Decimeters

To convert from meters to decimeters, recall that 1 meter equals 10 decimeters. Multiply the length in meters by 10 to get the length in decimeters.\[ 22.6 \text{ m} \times 10 = 226 \text{ dm} \]

Convert Milligrams to Kilograms

To convert from milligrams to kilograms, remember that 1 kilogram equals 1,000,000 milligrams. Divide the mass in milligrams by 1,000,000 to get the mass in kilograms.\[ 25.4 \text{ mg} \div 1,000,000 = 0.0000254 \text{ kg} \]

Convert Milliliters to Liters

To convert from milliliters to liters, note that 1 liter is equivalent to 1,000 milliliters. Divide the volume in milliliters by 1,000 to get the volume in liters.\[ 556 \text{ mL} \div 1,000 = 0.556 \text{ L} \]

Convert Kilograms per Cubic Meter to Grams per Cubic Centimeter

To convert from kg/m³ to g/cm³, use the conversion factors: 1 kg = 1,000 g and 1 m³ = 1,000,000 cm³. Multiply the density by 1,000 and divide by 1,000,000.\[ 10.6 \text{ kg/m}^3 = 10.6 \times \frac{1,000}{1,000,000} \text{ g/cm}^3 = 0.0106 \text{ g/cm}^3 \]

Key concepts

Metric System Conversions

The metric system is an internationally adopted system of measurement based on multiples of ten. This simplifies calculations and conversions, making it a preferred system in science, especially in chemistry. Metric conversions can involve various physical quantities such as length, mass, volume, and density. These conversions are vital in chemistry, where precise measurements are necessary.
To perform metric conversions:

• Use conversion factors based on the multiples of ten.
• Set up conversion equations where units cancel out until the desired unit is achieved.

For example, when converting meters to decimeters or milligrams to kilograms, maintaining awareness of the power of ten can guide proper conversions and reduce errors. Each step in solving a conversion involves understanding both the units and the relationships between them.

Length Conversion

Length is a fundamental measurement in science. The metric system makes it straightforward to convert between different length units. For example, to convert meters to decimeters, you must know that 1 meter equals 10 decimeters. This relationship is straightforward because both units are linear measurements within the same system and require simple multiplication or division by powers of ten.
Here's how you can think of length conversion:

• Recognize the base unit, in this case, the meter.
• Use the conversion factor, such as 1 meter = 10 decimeters.
• Multiply or divide by the factor as appropriate for your conversion needs.

In practice: Convert 22.6 meters to decimeters by multiplying 22.6 by 10, resulting in 226 decimeters. This simple arithmetic makes length conversion within the metric system intuitive.

Mass Conversion

Mass conversion in the metric system often involves understanding the relation between grams, milligrams, and kilograms. As one of the broader use cases, converting milligrams to kilograms is necessary in scientific contexts to ensure consistency in reporting and analysis.
Here's an easy guide:

• Remember that 1 kilogram equals 1,000,000 milligrams.
• To convert from a smaller to a larger unit, divide the given mass by the appropriate conversion factor.

This means converting 25.4 milligrams into kilograms requires dividing by 1,000,000. Thus, 25.4 mg equals 0.0000254 kg. Understanding these conversions is critical since scientific measurements and calculations hinge on precise and consistent units.

Volume Conversion

Volume conversion is crucial in the study of liquids and gases. The metric system's simplicity is evident here, as it enables easy conversion between milliliters and liters.

• 1 liter equals 1,000 milliliters.
• To convert from milliliters (a smaller unit) to liters (a larger unit), divide by 1,000.

In practical terms, converting 556 milliliters to liters is a simple division task: 556 mL divided by 1,000 equals 0.556 liters. Volume conversions are essential for experiments and chemical reactions, where precise quantities of substances are required to achieve accurate results.

Density Conversion

Density involves both mass and volume, denoted in the metric system as kilograms per cubic meter or grams per cubic centimeter. Understanding how to convert these is vital in chemistry for determining substance properties.
Here's a step-by-step conversion approach:

• Recall that 1 kilogram = 1,000 grams and 1 cubic meter = 1,000,000 cubic centimeters.
• To convert kg/m³ to g/cm³, multiply by the conversion factors for mass and divide by the conversion for volume.

For instance, converting 10.6 kg/m³ into g/cm³ involves multiplying 10.6 by 1,000 and then dividing by 1,000,000, resulting in 0.0106 g/cm³. These conversions allow chemists to compare densities effectively and utilize data in various analyses and research.