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Chapter 1: Problem 5

Express each of the following quantities in micrograms: (a) \(10 \mathrm{mg}\), (b) \(10^{4} \mathrm{~g}\), (c) \(10 \mathrm{~kg}\), (d) \(100 \times 10^{3} \mathrm{~g}\), (e) \(1000 \mathrm{ng}\). (answer check available at lightandmatter.com)

Short Answer

Expert verified
(a) 10000 碌g, (b) 10鹿鈦� 碌g, (c) 10鹿鈦� 碌g, (d) 10鹿鹿 碌g, (e) 1 碌g.

Step by step solution

01

Understand the Units

We need to convert each given unit into micrograms. Here are the conversion factors to be aware of: 1 mg = 1000 micrograms, 1 g = 1000000 micrograms, 1 kg = 1000000000 micrograms, and 1 ng = 0.001 micrograms.
02

Convert Milligrams to Micrograms

For part (a), we are given 10 milligrams (mg). Using the conversion factor, we calculate \(10 \times 1000 = 10000\) micrograms.
03

Convert Grams to Micrograms

Part (b) requires converting 10000 grams (g) into micrograms. Using the conversion factor, we calculate \(10^4 \times 1000000 = 10^4 \times 10^6 = 10^{10}\) micrograms.
04

Convert Kilograms to Micrograms

In part (c), we need to convert 10 kilograms (kg) to micrograms. Since 1 kg = 1000000000 micrograms, for 10 kg: \(10 \times 1000000000 = 10^{10}\) micrograms.
05

Convert Large Grams to Micrograms

For part (d), 100 脳 10鲁 grams need to be converted to micrograms. Simplifying 100 脳 10鲁 gives 100000 grams, so \(100000 \times 1000000 = 10^5 \times 10^6 = 10^{11}\) micrograms.
06

Convert Nanograms to Micrograms

Lastly, for part (e), convert 1000 nanograms (ng) to micrograms using the conversion where 1 ng = 0.001 micrograms. Thus, \(1000 \times 0.001 = 1\) microgram.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Micrograms
Micrograms, abbreviated as \( \mu g \), is a unit of mass in the metric system used to measure very light weights. The prefix "micro" denotes a factor of one-millionth \((10^{-6})\), meaning that one microgram is one-millionth of a gram. While it's a very small unit, it鈥檚 essential for measuring mass when precision is critical.
  • One microgram \( (\mu g) \) is equal to \(0.000001\) grams \((g)\).
  • It is also equivalent to \(0.001\) milligrams \((mg)\).

Given its minuscule value, micrograms are often used in contexts like medicine or science, where even tiny differences in mass can be significant. These settings require accurate measurements to ensure safety, quality, and effectiveness.
Navigating Metric Conversions
Metric conversions play a vital role in translating measurements from one unit into another within the metric system. This system uses base units and prefixes to define quantities via powers of ten, making conversions very straightforward. The metric system is universal, making it widely used in scientific and daily contexts.
  • Common conversions include milligrams (mg) to micrograms \((1 \, mg = 1000 \, \mu g)\).
  • Converting grams (g) to micrograms involves multiplying by \(1,000,000\) because \(1 \, g = 1,000,000 \, \mu g\).
  • For kilograms (kg), the conversion is even larger: \(1 \, kg = 1,000,000,000 \, \mu g\).

It's essential to be comfortable with these conversions for effective problem-solving in physics, chemistry, and other sciences, as well as in everyday scenarios like cooking or baking, where precise ingredient measurements are necessary.
Precision in Mass Measurement
Mass measurement involves determining the quantity of matter in an object, typically using units like grams (g), kilograms (kg), and micrograms (\(\mu g\)). It's crucial for various applications, from industrial manufacturing to biological research. Accurate mass measurements ensure that products and experiments meet standards and expectations.
  • In everyday use, grams and kilograms are the standard units.
  • Micrograms are used in high-precision scenarios such as pharmacology and environmental science where trace amounts matter significantly.
  • The choice of unit often depends on the required precision and the nature of the task.

Using appropriate units and accurate measuring tools is integral to obtaining reliable results in mass measurement tasks. From weighing ingredients in a recipe to measuring compounds in a laboratory, understanding these fundamentals is key.

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