Problem 114 - General Chemistry

Question

Write each of the following in terms of the SI base unit (that is, express the prefix as the power of 10 ). a \(6.6 \mathrm{mK} \quad\) b \(275 \mathrm{pm}\) c \(22.1 \mathrm{~ms}\) d \(45 \mu \mathrm{m}\)

Step-by-Step Solution

Answer

a) \(6.6 \times 10^{-3}\) K, b) \(275 \times 10^{-12}\) m, c) \(22.1 \times 10^{-3}\) s, d) \(45 \times 10^{-6}\) m.

Step 1: Understanding the Units and Prefixes

In this exercise, we are tasked with converting measurements with certain prefixes into plain units using powers of ten. This is done by replacing the prefixes with the appropriate power of ten representation.

Step 2: Convert Millikelvin to Kelvin

For part a, the prefix 'milli' (m) in millikelvin (mK) means \(10^{-3}\). Therefore, \(6.6\) mK is converted to Kelvin as follows: \(6.6 \times 10^{-3}\) K.

Step 3: Convert Picometers to Meters

For part b, the prefix 'pico' (p) in picometers (pm) means \(10^{-12}\). Therefore, \(275\) picometers is expressed in meters as: \(275 \times 10^{-12}\) m.

Step 4: Convert Milliseconds to Seconds

For part c, the prefix 'milli' (m) in milliseconds (ms) means \(10^{-3}\). Therefore, \(22.1\) ms is converted to seconds as: \(22.1 \times 10^{-3}\) s.

Step 5: Convert Micrometers to Meters

For part d, the prefix 'micro' (\(\mu\)) in micrometers (\(\mu m\)) means \(10^{-6}\). Therefore, \(45\) micrometers is represented in meters as: \(45 \times 10^{-6}\) m.

Key Concepts

Metric Prefixes

Metric prefixes are incredibly useful when dealing with the International System of Units (SI) because they help denote very large or small quantities in a manageable form. Each prefix represents a specific power of ten. For instance, the prefix *"milli"* represents \(10^{-3}\), which is one-thousandth of a unit. Common metric prefixes include:

• Kilo-: \(10^3\) or 1,000
• Centi-: \(10^{-2}\) or 0.01
• Milli-: \(10^{-3}\) or 0.001
• Micro- (\(\mu\)): \(10^{-6}\) or 0.000001
• Nano-: \(10^{-9}\) or 0.000000001
• Pico-: \(10^{-12}\) or 0.000000000001

Each prefix helps make sense of how much larger or smaller a measurement is relative to the base unit. Once these prefixes are learned, converting between sizes becomes simple.

Scientific Notation

Scientific notation is a method of writing numbers that accommodates both very large and extremely small values in a concise form. This notation makes arithmetic operations like multiplication and division much simpler to manage. Typically, scientific notation expresses numbers as a product of a number (often between 1 and 10) and a power of ten. For example, the number 3,000 can be written as \(3 \times 10^3\).

In our exercise, when converting millikelvin to kelvin, the small temperature value was expressed as \(6.6 \times 10^{-3}\) K. This means that the measurement is a tiny fraction of 1 K. Understanding scientific notation is crucial for interpreting metric conversions, as it clearly illustrates the magnitude of the quantities involved.

Base Units

In the SI system, there are seven base units from which all other units are derived. The base units are fundamental to science and engineering, encapsulating measurements like length, mass, time, and temperature. These base units are:

• Meter (m) for length
• Kilogram (kg) for mass
• Second (s) for time
• Ampere (A) for electric current
• Kelvin (K) for temperature
• Mole (mol) for the amount of substance
• Candela (cd) for luminous intensity

These units allow scientists to communicate measurements worldwide without confusion. Any conversion or measurement begins from these fundamental units, ensuring consistency and accuracy. When prefixes are added to base units, they indicate specific multiples or fractions, modifying the scale while keeping the fundamental unit intact.

Measurement Conversion

Measurement conversion is the process of expressing a quantity in one unit in terms of another unit. This is essential when comparing different quantities, performing calculations, or standardizing data from various systems to the SI system. The key to successful measurement conversion lies in understanding the relationship between the units involved.

For example, in the original problem, converting 22.1 milliseconds to seconds involved acknowledging that 1 ms is \(10^{-3}\) s. Therefore, we expressed 22.1 ms as \(22.1 \times 10^{-3}\) s, simplifying analysis and comparison.

By consistently applying the power-of-ten representations of prefixes, you ensure precise and accurate conversions. This understanding allows for seamless navigation between varied measurements, aiding in the clarity and precision necessary for scientific exploration and practical applications.