Problem 32 - Chemistry The Central Science In Si Units

Question

Use appropriate metric prefixes to write the following measurements without use of exponents: (a) \(7.29 \times 10^{6} \mathrm{~g}\) (b) \(6.1 \times 10^{-10} \mathrm{~m}\) (c) \(1.828 \times 10^{-3} \mathrm{~s}\) (d) \(3.523 \times 10^{9} \mathrm{~m}^{3}\) (e) \(9.62 \times 10^{2} \mathrm{~m} / \mathrm{s}\) (f) \(8.923 \times 10^{-12} \mathrm{~kg}(\mathrm{~g}) 3.552 \times 10^{12} \mathrm{~L}\).

Step-by-Step Solution

Answer

(a) 7.29 Mg, (b) 6.1 nm, (c) 1.828 ms, (d) 3.523 GL, (e) 0.962 km/s, (f) 8.923 pg.

Step 1: Convert grams to kilograms for (a)

The given measurement is \(7.29 \times 10^{6} \text{ g}\). To convert grams to kilograms, we use the fact that \(1 \text{ kg} = 10^3 \text{ g}\). Therefore, \(7.29 \times 10^6\text{ g} = \frac{7.29 \times 10^6}{10^3} \text{ kg} = 7.29 \times 10^3 \text{ kg}\), which is equal to \(7.29 \text{ Mg (megagrams)}\).

Step 2: Convert meters to nanometers for (b)

Given \(6.1 \times 10^{-10} \text{ m}\), to convert to nanometers, note that \(1 \text{ m} = 10^9 \text{ nm}\). Thus, \(6.1 \times 10^{-10} \text{ m} = 6.1 \text{ nm}\).

Step 3: Convert seconds to milliseconds for (c)

The value is \(1.828 \times 10^{-3} \text{ s}\). Since \(1 \text{ ms} = 10^{-3} \text{ s}\), we have \(1.828 \times 10^{-3} \text{ s} = 1.828 \text{ ms}\).

Step 4: Convert cubic meters to gigaliters for (d)

The given measurement is \(3.523 \times 10^{9} \text{ m}^3\). Note that \(1 \text{ m}^3 = 1 \text{ kL}\) and \(1 \text{ GL} = 10^9 \text{ L}\). Therefore, \(3.523 \times 10^9 \text{ m}^3 = 3.523 \text{ GL}\).

Step 5: Convert meters per second to kilometers per second for (e)

For \(9.62 \times 10^{2} \text{ m/s}\), note \(1 \text{ km/s} = 10^3 \text{ m/s}\). Thus, \(9.62 \times 10^{2} \text{ m/s} = 0.962 \text{ km/s}\).

Step 6: Convert kilograms to picograms for (f)

For \(8.923 \times 10^{-12} \text{ kg (g)}\), converting to picograms involves using \(1 \text{ pg} = 10^{-12} \text{ kg}\). Thus, \(8.923 \times 10^{-12} \text{ kg} = 8.923 \text{ pg}\). Note: There seems to be a mistake in the expression given in (f); correcting this part only covers the kilogram portion.

Key Concepts

Measurement Conversion

Measurement conversion is a process used for converting a quantity from one unit to another. This is crucial because it allows us to consistently compare and express quantities, regardless of the unit used originally.
To perform a measurement conversion, you'll often rely on known conversion factors, which are ratios expressing how many of one unit equal another. For instance, in the metric system, conversions among mass units might include converting grams to kilograms or kilograms to megagrams by simply multiplying or dividing by pertinent powers of ten.

• Grams (g) to Kilograms (kg): 1 kg = 1000 g
• Meters (m) to Nanometers (nm): 1 m = 1,000,000,000 nm
• Seconds (s) to Milliseconds (ms): 1 ms = 0.001 s

For example, when converting 7.29 million grams into kilograms, you would divide by 1000 (or multiply by 10^-3), resulting in 7.29 Megagrams. Making conversions like these becomes second nature with practice, particularly when you're consistently dealing with SI units in science.

Scientific Notation

Scientific notation is a method of writing numbers that accommodates very large or very small values in a succinct way. It utilizes powers of ten, making calculations manageable and easy to follow.
For example, a number like 7,290,000 can be written as 7.29 × 10⁶. This format indicates that the decimal point of 7.29 is to be moved 6 places to the right to reach the number originally given. Similarly, a tiny number like 0.000061 can be expressed as 6.1 × 10^-5, where the decimal moves to the left.

• This expression helps avoid errors that occur when writing long sequences of digits.
• The exponent indicates how many places the decimal should move.

In scientific fields, it's crucial to use scientific notation to easily communicate and compute with figures like distance of astronomical bodies in meters or mass of atomic particles in kilograms.

Metric System

The metric system is an internationally agreed decimal system of measurement that is extensively used around the world. It relies on base units and prefixes which modify the base unit by a power of ten.
This system simplifies both written communication and mental math estimations, and substantially aids in the standardization of scientific data globally.

• Length: meter (m)
• Mass: gram (g)
• Volume: liter (L)

In the context of the problem in question, converting from meters to nanometers, grams to kilograms, or seconds to milliseconds, incorporates various metric prefixes like nano-, kilo-, and milli-, each representing distinct powers of ten. The metric system's consistent structure and factor-of-ten scaling shear through complex arithmetic, making quick conversions straightforward.

SI Units

SI units, which stand for the International System of Units, form the foundation of the metric system and are used globally for scientific measurements. They ensure uniformity and precision in expressing quantities across scientific disciplines.

• Mass is measured in kilograms (kg).
• Length is measured in meters (m).
• Time is measured in seconds (s).

Each of these base units can be further refined with metric prefixes to represent larger or smaller quantities, such as nanometers or megagrams, effectively simplifying the recording and interpretation of experimental data. In a classroom setting, learning interoperability of these units helps students grasp scientific concepts efficiently. An understanding of SI units bolsters a student's ability to conduct research, as the standardization of these units forms the worldwide language of science.