Problem 85 - Introductory Chemistry Atoms First

Question

Convert each length to meters. Report your answers in scientific notation and watch your significant figures. (a) \(2.31\) gigameters \((\mathrm{Gm})\) (b) \(5.00\) micrometers \((\mu m)\) (c) 1004 millimeters \((\mathrm{mm})\) (d) \(5.00\) picometers \((\mathrm{pm})\) (e) \(0.25\) kilometer \((\mathrm{km})\)

Step-by-Step Solution

Answer

(a) \(2.31 \times 10^9\) m (b) \(5.00 \times 10^{-6}\) m (c) \(1.004 \times 10^{0}\) m (d) \(5.00 \times 10^{-12}\) m (e) \(2.5 \times 10^{2}\) m

Step 1: 1. Converting gigameters (Gm) to meters (m)

To convert gigameters to meters, we will multiply the given length (in gigameters) by the conversion factor: 1 Gm = \(10^9\) m. So, for 2.31 Gm, we have: \(2.31\,\mathrm{Gm} \times 10^9\,\mathrm{m/Gm}\) Now, multiply the numbers: \(2.31 \times 10^9\) m which is already in scientific notation and has the correct number of significant figures.

Step 2: 2. Converting micrometers (µm) to meters (m)

To convert micrometers to meters, we will multiply the given length (in micrometers) by the conversion factor: 1 µm = \(10^{-6}\) m. So, for 5.00 µm, we have: \(5.00\,\mathrm{µm} \times 10^{-6}\,\mathrm{m/µm}\) Now, multiply the numbers: \(5.00 \times 10^{-6}\) m which is already in scientific notation and has the correct number of significant figures.

Step 3: 3. Converting millimeters (mm) to meters (m)

To convert millimeters to meters, we will multiply the given length (in millimeters) by the conversion factor: 1 mm = \(10^{-3}\) m. So, for 1004 mm, we have: \(1004\,\mathrm{mm} \times 10^{-3}\,\mathrm{m/mm}\) Now, multiply the numbers: \(1.004 \times 10^{3}\) m \(\times 10^{-3}\,\mathrm{m/mm}\) When simplified, we get: \(1.004 \times 10^{0}\) m, which is already in scientific notation and has the correct number of significant figures.

Step 4: 4. Converting picometers (pm) to meters (m)

To convert picometers to meters, we will multiply the given length (in picometers) by the conversion factor: 1 pm = \(10^{-12}\) m. So, for 5.00 pm, we have: \(5.00\,\mathrm{pm} \times 10^{-12}\,\mathrm{m/pm}\) Now, multiply the numbers: \(5.00 \times 10^{-12}\) m is already in scientific notation and has the correct number of significant figures.

Step 5: 5. Converting kilometers (km) to meters (m)

To convert kilometers to meters, we will multiply the given length (in kilometers) by the conversion factor: 1 km = \(10^{3}\) m. So, for 0.25 km, we have: \(0.25\,\mathrm{km} \times 10^{3}\,\mathrm{m/km}\) Now, multiply the numbers: \(2.5 \times 10^{-1}\) m \(\times 10^{3}\,\mathrm{m/km}\) When simplified, we get: \(2.5 \times 10^{2}\) m, which is in scientific notation and has the correct number of significant figures.

Key Concepts

Unit Conversion in Chemistry

Unit conversion is a vital skill in chemistry. It allows you to express measurements in different units. Chemistry often deals with very large or small numbers, which makes conversion crucial. For example, converting gigameters to meters is simply multiplying by the conversion factor since 1 gigameter is equal to \(10^9\) meters. So, if you have 2.31 gigameters, you perform the conversion like this:

• Multiply: \(2.31 \times 10^9\) meters, giving you the number in meters.

Each time you convert a unit, remember to use the appropriate conversion factor—be it from gigameters, micrometers, kilometers, etc. The conversion factors are powers of ten, making them simple but crucial for accurate chemistry measurements.

Significant Figures

Significant figures are important for expressing the precision of measurements. They indicate which digits in a number are meaningful. It's crucial to understand how many significant figures to use after a unit conversion. If a number like 2.31 has three significant digits, then the result of a conversion should also reflect the same level of precision. Let's try it with micrometers:

• A measurement like 5.00 micrometers (\(\mu m\)) should be converted to meters with its three significant figures intact: \(5.00 \times 10^{-6}\) meters.

In scientific notation, maintaining the correct number of significant figures ensures you are reporting the measurement accurately and appropriately, based on the original data's precision.

Metric Units Conversion

The metric system relies on a base unit and powers of ten for conversion, making it logical and straightforward. Converting between these units involves multiplying by a power of ten. For example, to convert millimeters to meters:

• 1004 millimeters is converted by multiplying 1004 by \(10^{-3}\), because 1 millimeter equals \(10^{-3}\) meters.
• Executing this gives us: \(1.004 \times 10^{0}\) meters, showcasing an understanding of powers of ten.

Similarly, consider converting kilometers to meters—it's just a matter of multiplying by 1000 or \(10^3\), because there are 1000 meters in a kilometer. The simplicity of moving between units by simple multiplication or division with powers of ten makes metric conversions both accessible and consistent across different quantities.