Problem 100 - Chemistry Matter and Change

Question

Convert each quantity to the indicated units. a. 3.01 \(\mathrm{g} \rightarrow \mathrm{cg} \quad\) d. 0.2 \(\mathrm{L} \rightarrow \mathrm{dm}^{3}\) b. 6200 \(\mathrm{m} \rightarrow \mathrm{km} \quad\) e. 0.13 \(\mathrm{cal} / \mathrm{g} \rightarrow \mathrm{kcal} / \mathrm{g}\) c. 6.24 \(\times 10^{-7} \mathrm{g} \rightarrow \mu \mathrm{g} \quad\) f. 3.21 \(\mathrm{mL} \rightarrow \mathrm{L}\)

Step-by-Step Solution

Answer

a. 301 cg; b. 6.2 km; c. 0.624 µg; d. 0.2 dm³; e. 0.00013 kcal/g; f. 0.00321 L.

Step 1: Understand the Conversion Table

When converting units, you can use conversion factors. For metric conversions, remember that the prefixes denote powers of ten. For example, 1 gram (g) equals 100 centigrams (cg), 1 liter (L) equals 1 decimeter cubed (dm³), 1,000 meters (m) equals 1 kilometer (km), 1 calorie (cal) per gram equals 0.001 kilocalories (kcal) per gram, 1 gram equals 1,000,000 micrograms (µg), and 1 milliliter (mL) equals 0.001 liters (L).

Step 1: Convert Grams to Centigrams

To convert 3.01 grams to centigrams, multiply by the conversion factor of 100 centigrams per gram: \[ 3.01 ext{ g} imes 100 rac{ ext{cg}}{ ext{g}} = 301 ext{ cg} \]

Step 2: Convert Meters to Kilometers

To convert 6200 meters to kilometers, divide by 1,000 since 1 kilometer is 1,000 meters:\[ 6200 ext{ m} imes rac{1 ext{ km}}{1000 ext{ m}} = 6.2 ext{ km} \]

Step 3: Convert Grams to Micrograms

To convert 6.24 × 10⁻⁷ grams to micrograms, multiply by 1,000,000 since there are 1,000,000 micrograms in 1 gram:\[ 6.24 imes 10^{-7} ext{ g} imes 1,000,000 rac{ ext{µg}}{ ext{g}} = 0.624 ext{ µg} \]

Step 4: Convert Liters to Cubic Decimeters

0.2 liters is already equivalent to 0.2 dm³ because 1 liter equals 1 dm³:\[ 0.2 ext{ L} = 0.2 ext{ dm}^3 \]

Step 5: Convert Calories per Gram to Kilocalories per Gram

To convert 0.13 cal/g to kcal/g, multiply by the conversion factor of 0.001 kcal per cal:\[ 0.13 rac{ ext{cal}}{ ext{g}} imes 0.001 rac{ ext{kcal}}{ ext{cal}} = 0.00013 rac{ ext{kcal}}{ ext{g}} \]

Step 6: Convert Milliliters to Liters

To convert 3.21 mL to liters, divide by 1,000 since 1 liter is 1,000 mL:\[ 3.21 ext{ mL} imes 0.001 rac{ ext{L}}{ ext{mL}} = 0.00321 ext{ L} \]

Key Concepts

Metric System

The metric system is a standardized system of measurement used worldwide with a base 10 structure. This means that unit conversion is primarily about moving the decimal point.
This system makes metric conversions straightforward and efficient, as each 'step' up or down in the system corresponds to powers of ten.

• For example, 1 meter (m) is divided into 100 centimeters (cm), or further into 1,000 millimeters (mm).
• Conversely, 1 kilometer (km) is equal to 1,000 meters (m).

In chemistry and science in general, using the metric system simplifies equations and communication across international boundaries. It is essential for students and professionals alike to grasp metric conversions for accuracy and efficiency in problem solving.

Conversion Factors

Conversion factors are essential for converting between units in any system of measurement. These factors serve as multiplying coefficients that seamlessly change one unit into another.
In the metric system, conversion factors are often simple multiples of 10. Let's break these down:

• To convert from grams to centigrams, the conversion factor is 100 because 1 gram equals 100 centigrams.
• When converting meters to kilometers, the conversion factor is 1/1,000 since a kilometer is 1,000 meters.

Using conversion factors correctly involves multiplying or dividing the original measurement by the appropriate factor, ensuring that your units cancel properly and yield the desired units in the solution.

Grams to Centigrams

Converting grams to centigrams is a basic yet crucial conversion in chemistry and many scientific fields. For this:

• Recall that 1 gram (g) equals 100 centigrams (cg).
• Therefore, to convert grams to centigrams, you multiply by the conversion factor of 100.
• For example, to convert 3.01 grams to centigrams, you calculate: \[ 3.01 \, \text{g} \times 100 \frac{\text{cg}}{\text{g}} = 301 \, \text{cg} \]

This easy multiplication reflects how the metric system simplifies unit conversion by using consistent relationships between different units.

Liters to Cubic Decimeters

The conversion between liters and cubic decimeters is particularly straightforward due to their equal value.
This is because both measure volume. A liter (L) is, in fact, defined as a cubic decimeter \((\text{dm}^3)\). By definition:

• \[ 1 \, \text{L} = 1 \, \text{dm}^3 \]
• This makes calculations involving these two units extremely simple, often eliminating the need for lengthy computations.
• For practical purposes and initial understanding, remember that 0.2 liters is \[ 0.2 \, \text{dm}^3 \]

Understanding this equivalence helps alleviate confusion in measurements in chemistry and physics.

Meters to Kilometers

When converting meters to kilometers, you simplify distance measurement by understanding their relation. A meter is significantly smaller than a kilometer. Here's how you convert them:

• 1 kilometer (km) equals 1,000 meters (m).
• This means to convert meters to kilometers, divide the number of meters by 1,000.
• As an example, converting 6,200 meters into kilometers involves the calculation:\[ 6,200 \, \text{m} \times \frac{1 \, \text{km}}{1,000 \, \text{m}} = 6.2 \, \text{km} \]

Through mastering these simple conversions, you're equipped to handle vast distances or measurements presented in different units more naturally.