Question

Write two unit factors for each of the following metric relationships: (a) \(\mathrm{m}\) and \(\mathrm{Mm}\) (b) \(\mathrm{g}\) and \(\mathrm{kg}\) (c) \(L\) and \(n L\) (d) \(\mathrm{s}\) and \(\mathrm{ps}\)

Short answer

Two unit factors for each: (a) \( \frac{1 \, \mathrm{Mm}}{1,000,000 \, \mathrm{m}}, \frac{1,000,000 \, \mathrm{m}}{1 \, \mathrm{Mm}} \); (b) \( \frac{1 \, \mathrm{kg}}{1,000 \, \mathrm{g}}, \frac{1,000 \, \mathrm{g}}{1 \, \mathrm{kg}} \); (c) \( \frac{1 \, \mathrm{nL}}{10^{-9} \, \mathrm{L}}, \frac{10^{-9} \, \mathrm{L}}{1 \, \mathrm{nL}} \); (d) \( \frac{1 \, \mathrm{ps}}{10^{-12} \, \mathrm{s}}, \frac{10^{-12} \, \mathrm{s}}{1 \, \mathrm{ps}} \).

Step-by-step solution

Understanding Unit Factors

Unit factors are conversion factors used to convert one unit to another. They are equivalent to one, allowing us to multiply measurements by them without changing their value, just their units.

Analyzing the relationship between m and Mm

1 Mm (megameter) is equal to 1,000,000 m (meters). Thus, two unit factors can be written as: \( \frac{1 \, \mathrm{Mm}}{1,000,000 \, \mathrm{m}} \) and \( \frac{1,000,000 \, \mathrm{m}}{1 \, \mathrm{Mm}} \).

Analyzing the relationship between g and kg

1 kg (kilogram) is equal to 1,000 g (grams). Hence, the two unit factors are: \( \frac{1 \, \mathrm{kg}}{1,000 \, \mathrm{g}} \) and \( \frac{1,000 \, \mathrm{g}}{1 \, \mathrm{kg}} \).

Analyzing the relationship between L and nL

1 nL (nanoliter) is equal to \(10^{-9}\) L (liters). Therefore, the unit factors are: \( \frac{1 \, \mathrm{nL}}{10^{-9} \, \mathrm{L}} \) and \( \frac{10^{-9} \, \mathrm{L}}{1 \, \mathrm{nL}} \).

Analyzing the relationship between s and ps

1 ps (picosecond) is equal to \(10^{-12}\) s (seconds). Thus, the unit factors are: \( \frac{1 \, \mathrm{ps}}{10^{-12} \, \mathrm{s}} \) and \( \frac{10^{-12} \, \mathrm{s}}{1 \, \mathrm{ps}} \).

Key concepts

Metric Relationships

The metric system is an international system of measurement based on multiples of ten. This system allows for easy conversion between different units of measurement. All metric units are related by powers of ten, which makes it simple to move between a larger unit and a smaller unit. For instance, when dealing with length, the basic metric unit is the meter. A kilometer is 1000 meters, and likewise, a megameter is 1,000,000 meters.
Understanding these relationships is crucial when performing conversions, as knowing how units relate to one another helps us establish accurate conversion factors. Conversion factors allow you to switch from one unit of measurement to another without altering the actual size or amount of the substance being measured. This understanding of metric relationships is fundamental for correct scientific calculations and practical applications in everyday life.

Unit Conversion

Unit conversion is the process of converting a measurement from one unit to another. In the metric system, this is achieved easily through multiplying or dividing by powers of ten. This systematic approach ensures that the conversion factors always equate to one.
For instance, to convert meters to megameters, we divide the number of meters by 1,000,000 because 1 megameter equals 1,000,000 meters. The unit factor for this conversion is \( \frac{1 \, \text{Mm}}{1,000,000 \, \text{m}} \). Conversely, converting back from megameters to meters involves multiplying by 1,000,000, using the unit factor \( \frac{1,000,000 \, \text{m}}{1 \, \text{Mm}} \).
By consistently using unit factors, you ensure that your calculations are precise and consistent, no matter the conversion. This principle applies across all metric conversions, including those for mass (grams to kilograms) and volume (liters to nanoliters).

Measurement Units

Measurement units are the standardized quantities used to express physical quantities. They allow for a universal understanding of measurements across different disciplines and locations. In the metric system, units are well-organized and intuitive.
For mass, the base unit is the gram, and for weightier objects, we use kilograms. This relationship is standardized as 1 kilogram equals 1,000 grams. Similarly, in volume measurements, the basic unit is the liter. Smaller volumes are measured in milliliters or even nanoliters, where 1 nanoliter is \(10^{-9}\) liters. When expressing time, we might use seconds, but for much shorter intervals, picoseconds are used, where 1 picosecond equals \(10^{-12}\) seconds.
These units provide clarity and consistency, especially in scientific and technical fields. Knowing how to express measurements correctly is vital for effective communication of data and ensuring that measurements are both understood and accurate.