Problem 67
Question
Using scientific notation, convert: a \(6.15 \mathrm{ps}\) to \(\mathrm{s}\) b \(3.781 \mu \mathrm{m}\) to \(\mathrm{m}\) c \(1.546 \AA\) to \(\mathrm{m}\) d \(9.7 \mathrm{mg}\) to \(\mathrm{g}\)
Step-by-Step Solution
Verified Answer
a) \(6.15 \times 10^{-12} \text{ s}\), b) \(3.781 \times 10^{-6} \text{ m}\), c) \(1.546 \times 10^{-10} \text{ m}\), d) \(9.7 \times 10^{-3} \text{ g}\).
1Step 1 - Convert picoseconds to seconds
1 picosecond is equal to \(10^{-12}\) seconds. Therefore, to convert 6.15 picoseconds to seconds, you multiply by \(10^{-12}\): \[ 6.15 ext{ ps} = 6.15 imes 10^{-12} ext{ s} = 6.15 imes 10^{-12} ext{ s}\]
2Step 2 - Convert micrometers to meters
1 micrometer is equal to \(10^{-6}\) meters. To convert 3.781 micrometers to meters, multiply by \(10^{-6}\): \[ 3.781 ext{ μm} = 3.781 imes 10^{-6} ext{ m} = 3.781 imes 10^{-6} ext{ m}\]
3Step 3 - Convert ångströms to meters
1 ångström is equal to \(10^{-10}\) meters. To convert 1.546 ångströms to meters, multiply by \(10^{-10}\): \[ 1.546 ext{ ext{ ext{Å} }} = 1.546 imes 10^{-10} ext{ m} = 1.546 imes 10^{-10} ext{ m}\]
4Step 4 - Convert milligrams to grams
1 milligram is equal to \(10^{-3}\) grams. To convert 9.7 milligrams to grams, multiply by \(10^{-3}\): \[ 9.7 ext{ mg} = 9.7 imes 10^{-3} ext{ g} = 9.7 imes 10^{-3} ext{ g}\]
Key Concepts
Understanding Unit ConversionThe Metric SystemConverting Ångströms to MetersConverting Picoseconds to SecondsConverting Milligrams to Grams
Understanding Unit Conversion
Unit conversion is a fundamental concept in science and everyday life that involves changing a measurement from one unit to another. This process is crucial when working with different measurement systems or scales. Whether you're baking a cake and converting cups to milliliters, or solving a physics problem that requires converting kilometers to miles, understanding how to switch between units ensures accuracy and proper communication of quantities.
There are several strategies to tackle unit conversions, including using conversion factors or dimensional analysis. A conversion factor is a ratio that expresses how many of one unit is equivalent to another unit. Dimensional analysis involves using these factors to ensure that units cancel appropriately, leading to the desired outcome.
By becoming proficient in converting units, you build a solid foundation for handling real-world situations and scientific problems easily.
There are several strategies to tackle unit conversions, including using conversion factors or dimensional analysis. A conversion factor is a ratio that expresses how many of one unit is equivalent to another unit. Dimensional analysis involves using these factors to ensure that units cancel appropriately, leading to the desired outcome.
By becoming proficient in converting units, you build a solid foundation for handling real-world situations and scientific problems easily.
The Metric System
The metric system is an international decimalized system of measurement based on powers of ten. It's widely used around the globe due to its simplicity and ease of conversion. The basic units of the metric system include the meter for length, the kilogram for mass, and the second for time.
It simplifies unit conversion with its logical structure; each step in the scale is designed to be a power of ten. For example, when moving from millimeters (mm) to meters (m), you factor by 1,000, since there are 1,000 mm in a meter. Similarly, for moving from meters to kilometers (km), you factor by 1,000 again, as there are 1,000 meters in a kilometer.
It simplifies unit conversion with its logical structure; each step in the scale is designed to be a power of ten. For example, when moving from millimeters (mm) to meters (m), you factor by 1,000, since there are 1,000 mm in a meter. Similarly, for moving from meters to kilometers (km), you factor by 1,000 again, as there are 1,000 meters in a kilometer.
- Ease of use: All units are decimal-based.
- Universality: Predominantly used worldwide.
- Consistency: Logical and streamlined conversion process.
Converting Ångströms to Meters
An ångström (Å) is a unit of length used mainly to express sizes of atoms and molecules. It's especially convenient in fields like physics and chemistry. One ångström is equal to \(10^{-10}\) meters. Therefore, when converting ångströms to meters, you multiply the number of ångströms by \(10^{-10}\).
This conversion can help bridge the gap between microscopic scales and conventional metric measurements, making it easier to visualize atomic sizes or distances in a metric context. For instance, converting 1.546 Å to meters involves: \[1.546 \text{ Å} \times 10^{-10} = 1.546 \times 10^{-10} \text{ m}\]
Understanding this conversion is particularly useful in nanotechnology and materials science, where precise measurements at nanoscale and atomic levels are crucial.
This conversion can help bridge the gap between microscopic scales and conventional metric measurements, making it easier to visualize atomic sizes or distances in a metric context. For instance, converting 1.546 Å to meters involves: \[1.546 \text{ Å} \times 10^{-10} = 1.546 \times 10^{-10} \text{ m}\]
Understanding this conversion is particularly useful in nanotechnology and materials science, where precise measurements at nanoscale and atomic levels are crucial.
Converting Picoseconds to Seconds
A picosecond (ps) is a trillionth of a second, or \(10^{-12}\) seconds. This unit is used when discussing very brief time intervals, such as the time it takes for light to travel a small distance. Converting picoseconds to seconds is simple: multiply the number of picoseconds by \(10^{-12}\).
For example, converting 6.15 ps to seconds: \[6.15 \text{ ps} = 6.15 \times 10^{-12} \text{ s}\]
This can be essential in fields such as electronics, where signals and circuits operate within incredibly fast timeframes. The conversion helps scientists and engineers express these durations in more familiar time scales like seconds.
For example, converting 6.15 ps to seconds: \[6.15 \text{ ps} = 6.15 \times 10^{-12} \text{ s}\]
This can be essential in fields such as electronics, where signals and circuits operate within incredibly fast timeframes. The conversion helps scientists and engineers express these durations in more familiar time scales like seconds.
Converting Milligrams to Grams
In the metric system, the conversion between milligrams (mg) and grams (g) is straightforward thanks to the factor of 1,000. One milligram is equal to \(10^{-3}\) grams, meaning that to convert milligrams to grams, you multiply the milligram value by \(10^{-3}\).
Take, for example, the conversion of 9.7 mg to grams: \[9.7 \text{ mg} = 9.7 \times 10^{-3} \text{ g}\]
This conversion is often used in chemistry and medicine, where precise measurements of mass or dosage are crucial.
Take, for example, the conversion of 9.7 mg to grams: \[9.7 \text{ mg} = 9.7 \times 10^{-3} \text{ g}\]
This conversion is often used in chemistry and medicine, where precise measurements of mass or dosage are crucial.
- MG to G conversion factor: \(10^{-3}\).
- Application fields include pharmaceuticals and laboratory measurements.
- Enhances precision and ensures correct quantity measurement.
Other exercises in this chapter
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