Question

Write a unit equation for each of the following metric equivalents: (a) \(\mathrm{m}\) and \(\mathrm{Mm}\) (b) \(g\) and \(k g\) (c) \(L\) and \(n L\) (d) \(\mathrm{s}\) and \(\mathrm{ps}\)

Short answer

(a) 1 Mm = 1,000,000 m, (b) 1 kg = 1,000 g, (c) 1 L = 1,000,000,000 nL, (d) 1 s = 1,000,000,000,000 ps.

Step-by-step solution

Understand unit conversion

We need to write equation conversions between metric units. This involves understanding how these units relate.

Convert meters to megameters

The conversion factor between meters (m) and megameters (Mm) is that 1 megameter equals 1,000,000 meters. Hence, the unit equation is: 1 Mm = 1,000,000 m.

Convert grams to kilograms

The conversion between grams (g) and kilograms (kg) is that 1 kilogram equals 1,000 grams. Thus, the unit equation is: 1 kg = 1,000 g.

Convert liters to nanoliters

For liters (L) and nanoliters (nL), 1 liter equals 1,000,000,000 nanoliters. Consequently, the unit equation is: 1 L = 1,000,000,000 nL.

Convert seconds to picoseconds

Seconds (s) can be converted to picoseconds (ps) with the conversion factor that 1 second equals 1,000,000,000,000 picoseconds. Therefore, the unit equation is: 1 s = 1,000,000,000,000 ps.

Key concepts

Understanding the Metric System

The metric system is a standardized system of measurement used around the world. It's based on multiples of ten, making calculations very straightforward. Most countries use the metric system because it's simpler compared to others, like the imperial system. Common measurements in the metric system include meters for length, grams for mass, and liters for volume. Each of these base units can be scaled up or down with prefixes that represent powers of ten, such as "kilo-" for one thousand times the base unit, or "milli-" for one-thousandth of the base unit. This system helps in maintaining consistency and simplicity in measurements.

Using Conversion Factors

A conversion factor is a number used to change one set of units to another, by multiplying or dividing. It's crucial in the metric system, as it allows for easy conversion between different metric units. For example, to convert from meters to megameters, we use the conversion factor where 1 megameter equals 1,000,000 meters. Thus, a quantity in meters can easily be converted to megameters by dividing by 1,000,000. This approach can be applied broadly across different measurements like grams to kilograms or liters to nanoliters. Conversion factors ensure that values are correctly scaled without altering the actual quantity.

Exploring Metric Units

Metric units are the building blocks of the metric system, each representing a specific type of measurement. Length is measured in meters, mass in grams, and volume in liters. These can be modified with prefixes, such as "mega-" for a million times the unit or "nano-" for a billionth of the unit, to represent larger or smaller quantities respectively. For example:

• 1 Megameter (Mm) is 1,000,000 meters (m).
• 1 Kilogram (kg) is 1,000 grams (g).
• 1 Nanoliter (nL) is 0.000000001 liters (L).

These units help standardize measurements, providing clear, consistent quantities that are easily understood and converted.

Converting Measurement Units

Measurement units in the metric system can easily be converted by applying the appropriate conversion factors. This involves multiplying or dividing by powers of ten.

• To convert meters to megameters, use the conversion: 1 Mm = 1,000,000 m. Divide the number of meters by 1,000,000 to find out how many megameters it equals.
• For grams to kilograms, 1 kg = 1,000 g. Divide the number of grams by 1,000 to obtain kilograms.
• When converting liters to nanoliters, 1 L = 1,000,000,000 nL. Multiply liters by 1,000,000,000 to get the value in nanoliters.
• Similarly, seconds can be converted to picoseconds, where 1 s = 1,000,000,000,000 ps. Multiply the number of seconds by 1,000,000,000,000 to find picoseconds.

This conversion process allows for precise measurement adjustments across different scales.