Question

Write two unit factors for each of the following metric relationships: (a) \(\mathrm{m}\) and \(\mathrm{Tm}\) (b) \(g\) and \(G g\) (c) \(\mathrm{L}\) and \(\mathrm{mL}\) (d) \(\mathrm{s}\) and \(\mu \mathrm{s}\)

Short answer

(a) \(\frac{1 \text{ Tm}}{10^{12} \text{ m}}, \frac{10^{12} \text{ m}}{1 \text{ Tm}}\); (b) \(\frac{1 \text{ Gg}}{10^{9} \text{ g}}, \frac{10^{9} \text{ g}}{1 \text{ Gg}}\); (c) \(\frac{1 \text{ L}}{10^{3} \text{ mL}}, \frac{10^{3} \text{ mL}}{1 \text{ L}}\); (d) \(\frac{1 \text{ s}}{10^{6} \mu \text{ s}}, \frac{10^{6} \mu \text{ s}}{1 \text{ s}}\).

Step-by-step solution

Understand Metric Prefixes

To write unit factors, we need to understand the metric prefixes involved. For example, Tera (T) means \(10^{12}\), Giga (G) means \(10^{9}\), milli (m) means \(10^{-3}\), and micro (\(\mu\)) means \(10^{-6}\).

Determine Unit Factors for Meters (m) and Terameters (Tm)

The relationship between meters and terameters is given by the fact that \(1 \text{ Tm} = 10^{12} \text{ m}\). Using this, we can write two unit factors: 1. \(\frac{1 \text{ Tm}}{10^{12} \text{ m}}\)2. \(\frac{10^{12} \text{ m}}{1 \text{ Tm}}\)

Determine Unit Factors for Grams (g) and Gigagrams (Gg)

The relationship between grams and gigagrams is based on the prefix Giga, which is \(10^{9}\). Therefore, \(1 \text{ Gg} = 10^{9} \text{ g}\). The unit factors are: 1. \(\frac{1 \text{ Gg}}{10^{9} \text{ g}}\)2. \(\frac{10^{9} \text{ g}}{1 \text{ Gg}}\)

Determine Unit Factors for Liters (L) and Milliliters (mL)

Milli (m) means \(10^{-3}\), so \(1 \text{ L} = 10^{3} \text{ mL}\). The unit factors are:1. \(\frac{1 \text{ L}}{10^{3} \text{ mL}}\)2. \(\frac{10^{3} \text{ mL}}{1 \text{ L}}\)

Determine Unit Factors for Seconds (s) and Microseconds (μs)

Micro (\(\mu\)) means \(10^{-6}\), so \(1 \text{ s} = 10^{6} \mu \text{ s}\). Thus, the unit factors are:1. \(\frac{1 \text{ s}}{10^{6} \mu \text{ s}}\) 2. \(\frac{10^{6} \mu \text{ s}}{1 \text{ s}}\)

Key concepts

Unit Conversion

Unit conversion is a vital skill when working with different measurement systems. In the metric system, it's especially straightforward, thanks to its base-10 structure. Conversion is needed when you move between units of different sizes, such as meters to kilometers or liters to milliliters. To convert between units, you use unit factors, which are ratios that express how many smaller units make up a larger unit.

For instance, when converting from meters to terameters, you apply the fact that one terameter (Tm) equals \(10^{12}\) meters (m). You would use one of two unit factors: either \(\frac{1 \text{ Tm}}{10^{12} \text{ m}}\) or \(\frac{10^{12} \text{ m}}{1 \text{ Tm}}\), depending on which direction you're converting. Each factor is simply a different representation of the same conversion relationship.

When converting, always multiply or divide by the appropriate unit factor. Remember, the goal is to cancel out the unwanted unit, leaving you with the desired unit. This method can be applied universally across metric conversions.

Metric Prefixes

Metric prefixes simplify the expression of quantities by providing shorthand for powers of ten. They are part of the International System of Units (SI), offering a consistent way to represent large and small numbers. Here is how you can understand some common prefixes:

• Tera (T): Represents \(10^{12}\) or one trillion. Example: 1 terameter (Tm) = \(10^{12}\) meters.
• Giga (G): Represents \(10^{9}\) or one billion. Example: 1 gigagram (Gg) = \(10^{9}\) grams.
• Milli (m): Represents \(10^{-3}\) or one thousandth. Example: 1 liter (L) = \(10^{3}\) milliliters (mL).
• Micro (\(\mu\)): Represents \(10^{-6}\) or one millionth. Example: 1 second (s) = \(10^{6}\) microseconds (\(\mu s\)).

These prefixes make it easier to communicate measurements without long chains of zeros. They are essential for scientific communication, ensuring precise and efficient data reporting. By mastering prefixes, you simplify unit conversion and enhance your understanding of the metric system.

SI Units

The International System of Units (SI) is the foundation for scientific measurement worldwide. This system is coherent and comprehensive, designed to work seamlessly with metric prefixes. Key characteristics of SI units include:

• Base Units: These are the fundamental units from which all other SI units are derived. Examples include the meter (m) for length, kilogram (kg) for mass, and second (s) for time.
• Consistency: SI units are interconnected, meaning that they easily convert into each other through defined relationships. For instance, the liter, which is for volume, directly corresponds to the cubic meter (\(m^3\)).
• Universality: SI units are used globally, facilitating international collaboration and understanding in science and engineering.

The SI system simplifies the sharing of scientific data across disciplines and borders. By using well-defined units and prefixes, it allows for precise communication and reduces the potential for error, making it the cornerstone of scientific measurement.