Problem 55 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) A bumblebee flies with a ground speed of \(15.2 \mathrm{~m} / \mathrm{s}\). Calculate its speed in \(\mathrm{km} / \mathrm{hr}\). (b) The lung capacity of the blue whale is \(5.0 \times 10^{3} \mathrm{~L}\). Convert this volume into gallons. (c) The Statue of Liberty is \(151 \mathrm{ft}\) tall. Calculate its height in meters. (d) Bamboo can grow up to \(60.0 \mathrm{~cm} /\) day, Convert this growth rate into inches per hour.

Step-by-Step Solution

Answer

(a) \( 54.72 \frac{\mathrm{km}}{\mathrm{hr}} \) (b) \( 1320.86 \mathrm{~gal} \) (c) \( 46.02 \mathrm{~m} \) (d) \( 0.9843 \frac{\mathrm{in}}{\mathrm{hr}} \)

Step 1: Problem (a): Convert m/s to km/hr

To convert meters per second to kilometers per hour, we need to multiply by a conversion factor that cancels out the meters and seconds, and introduces kilometers and hours. We know that \( 1 \mathrm{~km} = 1000 \mathrm{~m} \) and \( 1 \mathrm{~hr} = 3600 \mathrm{~s} \). So, we can set up the conversion as follows: \[ 15.2 \frac{\mathrm{m}}{\mathrm{s}} \times \frac{1 \mathrm{~km}}{1000 \mathrm{~m}} \times \frac{3600 \mathrm{~s}}{1 \mathrm{~hr}} \] Now perform the calculations and cancel out the units step by step.

Step 2: Problem (b): Convert liters to gallons

To convert liters to gallons, we need to know the conversion factor that relates liters and gallons. We know that \( 1 \mathrm{~L} \approx 0.264172 \mathrm{~gal} \). So, we can set up the conversion as follows: \[ 5.0 \times 10^3 \mathrm{~L} \times \frac{0.264172 \mathrm{~gal}}{1 \mathrm{~L}} \] Now perform the calculations and cancel out the units step by step.

Step 3: Problem (c): Convert feet to meters

To convert feet to meters, we need to know the conversion factor that relates feet and meters. We know that \( 1 \mathrm{~ft} \approx 0.3048 \mathrm{~m} \). So, we can set up the conversion as follows: \[ 151 \mathrm{~ft} \times \frac{0.3048 \mathrm{~m}}{1 \mathrm{~ft}} \] Now perform the calculations and cancel out the units step by step.

Step 4: Problem (d): Convert cm/day to in/hr

To convert centimeters per day to inches per hour, we need to know the conversion factors that relate centimeters, inches, days, and hours. We know that \( 1 \mathrm{~in} \approx 2.54 \mathrm{~cm} \) and \( 1 \mathrm{~day} = 24 \mathrm{~hr} \). So, we can set up the conversion as follows: \[ 60.0 \frac{\mathrm{cm}}{\mathrm{day}} \times \frac{1 \mathrm{~in}}{2.54 \mathrm{~cm}} \times \frac{1 \mathrm{~day}}{24 \mathrm{~hr}} \] Now perform the calculations and cancel out the units step by step.

Key Concepts

Metric System

The metric system is a standard system of measurement used globally, known for its simplicity and ease of use. It is based on multiples of ten, making calculations straightforward.
For example, the unit of length in the metric system is the meter, with common conversions including millimeters (mm), centimeters (cm), and kilometers (km).
Understanding the metric system is essential for performing conversions, as it provides a universal language of measurement.

• Length: meter (m)
• Mass: kilogram (kg)
• Volume: liter (L)

These base units are easy to scale up or down using prefixes like kilo- (1000), centi- (0.01), and milli- (0.001).
This uniformity simplifies complex mathematical operations and communication across different regions.

Conversion Factors

Conversion factors allow us to change a measurement from one unit to another without altering the value. They are based on equivalencies between units.
For example, to convert from meters to kilometers, we use the conversion factor: \(1 \text{ km} = 1000 \text{ m}\).
These factors are essential in dimensional analysis, enabling the cancellation of units.

• Always set up conversion factors as fractions.
• Ensure that the unit you are converting from is canceled out.
• Multiply your original measurement by the appropriate conversion factors sequentially.

Using conversion factors correctly ensures accurate and efficient unit transformations.

Dimensional Analysis

Dimensional analysis is a method to convert units systematically, employing conversion factors at each step.
This technique helps track and cancel units, ensuring that the final result is in the desired unit. For example, converting meters per second to kilometers per hour involves:

• Multiplying by \(\frac{1\text{ km}}{1000\text{ m}}\) to change meters to kilometers.
• Multiplying by \(\frac{3600 \text{ s}}{1 \text{ hr}}\) to change seconds to hours.

Dimensional analysis is a powerful tool in physics and chemistry, where precise measurements and conversions are crucial.
It relies heavily on logical reasoning, using chains of conversion factors to systematically achieve the correct unit.

SI Units

SI units, or the International System of Units, form the foundation of modern scientific measurements.
Adopted worldwide, they provide consistency and reliability in data reporting and interpretation. Core SI units include:

• Length: meter (m)
• Time: second (s)
• Mass: kilogram (kg)
• Temperature: kelvin (K)

Using SI units simplifies international collaboration, as researchers and professionals speak the same language.
The universality of SI units minimizes errors in measurement and ensures that calculations are reproducible and understandable.