Problem 49 - Chemistry

Question

Which quantity in each of the following pairs is smaller? (a) \(154 \mathrm{pm}\) or \(7.7 \times 10^{-9} \mathrm{~cm}\) (b) \(1.86 \times 10^{11} \mu \mathrm{m}\) or \(2.02 \times 10^{2} \mathrm{~km}\) (c) \(2.9 \mathrm{GA}\) or \(3.1 \times 10^{15} \mu \mathrm{A}\)

Step-by-Step Solution

Answer

(a) 7.7 × 10⁻⁹ cm; (b) 1.86 × 10¹¹ μm; (c) 2.9 GA.

Step 1: Conversion of Units for (a)

First, we convert picometers to centimeters to compare the two quantities directly. \( 1 \text{ pm} = 10^{-12} \text{ m} = 10^{-10} \text{ cm} \), so \( 154 \text{ pm} = 154 \times 10^{-10} \text{ cm} = 1.54 \times 10^{-8} \text{ cm} \). Now compare \( 1.54 \times 10^{-8} \text{ cm} \) with \( 7.7 \times 10^{-9} \text{ cm} \).

Step 2: Comparison for (a)

Now that both quantities are in the same unit (cm), compare them: \( 1.54 \times 10^{-8} \text{ cm} \) is larger than \( 7.7 \times 10^{-9} \text{ cm} \). Thus, \( 7.7 \times 10^{-9} \text{ cm} \) is smaller.

Step 3: Conversion of Units for (b)

Convert micrometers to kilometers. \( 1 \mu\text{m} = 10^{-6} \text{ m} \), so \( 1.86 \times 10^{11} \mu\text{m} = 1.86 \times 10^{11} \times 10^{-6} \text{ m} = 1.86 \times 10^{5} \text{ m} = 186 \text{ km} \). Now compare \( 186 \text{ km} \) with \( 2.02 \times 10^{2} \text{ km} \), which is 202 km.

Step 4: Comparison for (b)

Now that both quantities are in the same unit (km), compare them: \( 186 \text{ km} \) is smaller than \( 202 \text{ km} \). Thus, \( 1.86 \times 10^{11} \mu\text{m} \) is smaller.

Step 5: Conversion of Units for (c)

Convert gigamperes to microamperes to compare the two quantities directly. \( 1 \text{ GA} = 10^9 \text{ A} = 10^{15} \mu\text{A} \), so \( 2.9 \text{ GA} = 2.9 \times 10^{15} \mu\text{A} \). Compare this with \( 3.1 \times 10^{15} \mu\text{A} \).

Step 6: Comparison for (c)

Now that both quantities are in the same unit (microamperes), compare them: \( 2.9 \times 10^{15} \mu\text{A} \) is smaller than \( 3.1 \times 10^{15} \mu\text{A} \). Thus, \( 2.9 \text{ GA} \) is smaller.

Key Concepts

Picometers to Centimeters

Understanding the conversion of picometers (pm) to centimeters (cm) is crucial for comparing measurements with vastly different scales.

• Picometer Basics: One picometer is a trillionth of a meter, making it an extremely small unit often used in atomic-level measurements.
• Conversion Formula: To convert from picometers to centimeters, recall that one picometer is equal to \(10^{-12}\) meters, and one meter is \(10^{2}\) centimeters. Thus, \(1 \, \text{pm} = 10^{-12} \times 10^{2} = 10^{-10} \, \text{cm}\).

For instance, to convert \(154 \, \text{pm}\) to centimeters, you multiply: \[154 \, \text{pm} = 154 \times 10^{-10} \, \text{cm} = 1.54 \times 10^{-8} \, \text{cm} \]This allows for direct comparison to quantities already expressed in centimeters, such as \(7.7 \times 10^{-9} \, \text{cm}\). By doing this, you can easily determine which is the smaller or larger value by comparing their powers of ten and coefficients.

Micrometers to Kilometers

Converting micrometers (µm) to kilometers (km) may seem daunting, but it becomes easy with a few simple conversion steps.

• Micrometer Overview: A micrometer is one-millionth of a meter, typically used for measuring small objects like cells and bacteria.
• Conversion Steps: To convert from micrometers to kilometers, remember that \(1 \, \mu\text{m} = 10^{-6} \, \text{m}\) and \(1 \, \text{m} = 10^{-3} \, \text{km}\). Combining these gives \(1 \, \mu\text{m} = 10^{-6} \times 10^{-3} = 10^{-9} \, \text{km}\).

For example, if we start with \(1.86 \times 10^{11} \, \mu\text{m}\), the conversion process involves: \[1.86 \times 10^{11} \, \mu\text{m} = 1.86 \times 10^{11} \times 10^{-9} \, \text{km} = 186 \, \text{km} \]This enables a straightforward comparison to other distances in kilometers. By comparing to a value such as \(202 \, \text{km}\), you can clearly see which measurement indicates the shorter length.

Gigamperes to Microamperes

Converting gigamperes (GA) to microamperes (µA) allows us to handle large electrical currents within the same scale as much smaller currents.

• Understanding Gigamperes: One gigampere signifies a billion amperes, often used in describing large-scale electrical flows.
• Conversion Insight: The conversion involves knowing that \(1 \, \text{GA} = 10^9 \, \text{A}\) and \(1 \, \text{A} = 10^6 \, \mu\text{A}\), which together give \(1 \, \text{GA} = 10^9 \times 10^6 = 10^{15} \, \mu\text{A}\).

For example, consider \(2.9 \, \text{GA}\). The conversion to microamperes is: \[2.9 \, \text{GA} = 2.9 \times 10^{15} \, \mu\text{A} \]This facilitates direct comparison with another current such as \(3.1 \times 10^{15} \, \mu\text{A}\). Such comparisons are key in determining which current is smaller or larger, providing clearer insights into electrical measurements.