Problem 24 - Chemistry The Central Science

Question

Use appropriate metric prefixes to write the following measurements without use of exponents: (a) \(2.3 \times 10^{-10} \mathrm{~L}\) (b) \(4.7 \times 10^{-6} \mathrm{~g}\), (c) \(1.85 \times 10^{-12} \mathrm{~m}\) (d) \(16.7 \times 10^{6} \mathrm{~s}\); (e) \(15.7 \times 10^{3} \mathrm{~g}\) (f) \(1.34 \times 10^{-3} \mathrm{~m},(\mathrm{~g}) 1.84 \times 10^{2} \mathrm{~cm}\)

Step-by-Step Solution

Answer

(a) 230 pL (b) 4.7 µg (c) 1.85 pm (d) 16.7 Ms (e) 15.7 kg (f) 1.34 mm, 1.84 m

Step 1: Finding SI prefix for (a)

For (a), we have \(2.3 \times 10^{-10} L\). The exponent is -10, which corresponds to the SI prefix "pico" (p, \(10^{-12}\)). Therefore, we can rewrite this as 230 pL.

Step 2: Finding SI prefix for (b)

For (b), we have \(4.7 \times 10^{-6} g\). The exponent is -6, which corresponds to the SI prefix "micro" (µ, \(10^{-6}\)). Therefore, we can rewrite this as 4.7 µg.

Step 3: Finding SI prefix for (c)

For (c), we have \(1.85 \times 10^{-12} m\). The exponent is -12, which corresponds to the SI prefix "pico" (p, \(10^{-12}\)). Therefore, we can rewrite this as 1.85 pm.

Step 4: Finding SI prefix for (d)

For (d), we have \(16.7 \times 10^{6} s\). The exponent is 6, which corresponds to the SI prefix "mega" (M, \(10^6\)). Therefore, we can rewrite this as 16.7 Ms.

Step 5: Finding SI prefix for (e)

For (e), we have \(15.7 \times 10^{3} g\). The exponent is 3, which corresponds to the SI prefix "kilo" (k, \(10^3\)). Therefore, we can rewrite this as 15.7 kg.

Step 6: Finding SI prefix for (f)

For (f), we have two quantities: \(1.34 \times 10^{-3} m\) and \(1.84 \times 10^{2} cm\). For the first quantity, the exponent is -3, which corresponds to the SI prefix "milli" (m, \(10^{-3}\)). Therefore, we can rewrite this as 1.34 mm. For the second quantity, note the unit is cm, and the exponent is 2, which corresponds to the SI prefix "hecto" (h, \(10^2\)). To stay in the base unit of meters, we convert 1 cm in meters first, which is equal to \(1 \times 10^{-2} m\). Then, we can rewrite this as \( 1.84 \times 10^{2} \times 10^{-2} m = 1.84 m\). Final answers: (a) 230 pL (b) 4.7 µg (c) 1.85 pm (d) 16.7 Ms (e) 15.7 kg (f) 1.34 mm, 1.84 m

Key Concepts

SI Units

SI Units, or the International System of Units, provide a standardized way to measure quantities globally. This system ensures consistency whether we’re measuring time or length.

SI units are separate from traditional or local measurement systems by having a clear definition for each unit:

• Meter (m): the base unit for length.
• Kilogram (kg): the base unit for mass.
• Second (s): the base unit for time.
• Liter (L): often used for volume, though not officially an SI unit.

To work effectively with SI units, it is helpful to understand their compatibility with other units and how they integrate into the metric system. Recognizing SI units using appropriate abbreviations ensures proper communication and documentation in science and engineering fields.

Scientific Notation

Scientific notation is a method of expressing numbers that are too large or small for everyday writing. It condenses numbers into a simple format,

making them easier to read and use in calculations. We represent a number in scientific notation as:\[a \times 10^n\]where \( a \) is a number (usually between 1 and 10) and \( n \) is an integer representing the power of ten. For example,

• \(2.3 \times 10^{-10}\) is a small number expressed in scientific notation.
• \(16.7 \times 10^{6}\) is a large number expressed in scientific notation.

When converting from scientific notation to standard form, you simply move the decimal point left or right depending on the sign and magnitude of \( n \). This practice helps in converting large quantities into more manageable metric prefixes using SI Units.

Metric System

The metric system is a decimal-based system of measurement that is widely used around the world. It is especially prevalent in scientific and engineering disciplines for its simplicity and consistency.

The metric system includes a set of base units and prefixes that create multiples and submultiples of these units. Everything in the metric system is based on powers of ten:

• Base Unit: Each type of measurement has a base unit that measures one unit of quantity.
• Prefixes: Used to express larger or smaller quantities relative to the base unit. Examples include "kilo" for \(10^3\) and "milli" for \(10^{-3}\).

This method of using prefixes greatly simplifies unit conversions and allows for easier calculation across various fields of study. Understanding commonly used prefixes can help you better grasp the metric system essential for academia and industry.

Unit Conversion

Unit conversion is the process of converting a given value in one measurement unit to its equivalent in another unit. It is an essential skill in science and everyday life, allowing for proper interpretation and analysis of data.

To perform a unit conversion, you must be familiar with both the start and target units, as well as the mathematical relationship between them. Key steps include:

• Identify the initial unit and the target unit.
• Find or calculate the conversion factor, which is the ratio that converts one unit to another.
• Multiply the initial numeric value by the conversion factor.

In terms of metric conversion, it often involves shifting a decimal point based on the prefix relationship, for instance, converting grams to kilograms by recognizing that 1 kg equals 1000 g. Mastery of unit conversion streamlines calculations across all disciplines.