Question

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

Short answer

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

Step-by-step solution

Understand Metric Prefixes

Before writing unit equations, it's important to know common metric prefixes. Here are some relevant ones: - \(\text{Tera (T)} = 10^{12}\), meaning one terameter (Tm) is 10^{12} meters.- \(\text{Giga (G)} = 10^9\), meaning one gigagram (Gg) is 10^9 grams.- \(\text{milli (m)} = 10^{-3}\), meaning one milliliter (mL) is 10^{-3} liters.- \(\text{micro (}\mu\text{)} = 10^{-6}\), meaning one microsecond (1s) is 10^{-6} seconds.

Write Unit Equation for Meters and Terameters

For the metric equivalents of meters (m) and terameters (Tm): \[1\ \text{Tm} = 10^{12}\ \text{m}\] This equation shows the conversion factor between terameters and meters.

Write Unit Equation for Grams and Gigagrams

For the metric equivalents of grams (g) and gigagrams (Gg): \[1\ \text{Gg} = 10^9\ \text{g}\] This indicates how many grams are in one gigagram.

Write Unit Equation for Liters and Milliliters

To relate liters (L) and milliliters (mL), use the following equation: \[1\ \text{L} = 10^3\ \text{mL}\] This means there are 1000 milliliters in one liter.

Write Unit Equation for Seconds and Microseconds

For seconds (s) and microseconds (1s), the unit equation is: \[1\ \text{s} = 10^6\ \mu \text{s}\] This shows there are one million microseconds in one second.

Key concepts

Metric Prefixes

Metric prefixes are essential for understanding the metric system. They help us represent large or small numbers efficiently by attaching specific prefixes to the base units. Each prefix signifies a power of ten, allowing for easier calculations and conversions. Here are some common ones:

• Tera (T): Represents \(10^{12}\). So, one terameter (\(\text{Tm}\)) equals one trillion meters.
• Giga (G): Stands for \(10^9\). One gigagram (\(\text{Gg}\)) is equivalent to one billion grams.
• Milli (m): Denotes \(10^{-3}\). Therefore, one milliliter (\(\text{mL}\)) equals a thousandth of a liter.
• Micro (\(\mu\)): Means \(10^{-6}\). Thus, one microsecond (\(\mu\text{s}\)) equals one millionth of a second.

Understanding these prefixes is crucial for converting and comprehending metric measurements.

Unit Conversion

Unit conversion involves changing a measure expressed in one set of units to an equivalent measure in another set, without altering its actual value. This process is particularly important when working with metric prefixes.
To convert between units with different metric prefixes, use these simple steps:

• Identify the prefix associated with your starting unit and its power of ten.
• Determine the prefix of the unit you are converting to and its power of ten.
• Calculate the difference in the powers of ten between the two prefixes.
• Apply the conversion factor to change from one unit to another.

For example, when converting from terameters to meters, recognize that \(1\ \text{Tm} = 10^{12}\ \text{m}\). Hence, multiply the number of terameters by \(10^{12}\) to convert to meters. Understanding the unit conversion process simplifies changing measurements in the metric system.

Metric Equivalents

Metric equivalents are specific equations that define the relationship between units with different metric prefixes. These equations allow for accurate conversion between units. Let's see some examples of these metric equivalents:

• Meters and Terameters: The unit equation \(1\ \text{Tm} = 10^{12}\ \text{m}\) tells us that one terameter equals one trillion meters.
• Grams and Gigagrams: For this, \(1\ \text{Gg} = 10^9\ \text{g}\) indicates that a gigagram amounts to one billion grams.
• Liters and Milliliters: Given \(1\ \text{L} = 10^3\ \text{mL}\), this means there are 1000 milliliters in one liter.
• Seconds and Microseconds: The equation \(1\ \text{s} = 10^6\ \mu\text{s}\) shows that a second contains one million microseconds.

Having these equivalents at your fingertips makes precise calculations and conversions straightforward in scientific and everyday contexts.