Problem 65 - General Chemistry

Question

Write the following measurements, without scientific notation, using the appropriate SI prefix. a \(5.89 \times 10^{-12} \mathrm{~s} \quad\) b \(0.2010 \mathrm{~m}\) c \(2.560 \times 10^{-9} \mathrm{~g}\) d \(6.05 \times 10^{3} \mathrm{~m}\)

Step-by-Step Solution

Answer

a) 5.89 ps, b) 0.2010 m, c) 2.560 ng, d) 6.05 km.

Step 1: Convert and Write in Femto Prefix

The value given is \(5.89 \times 10^{-12} \text{ s}\). The SI prefix for \(10^{-12}\) is pico (p), but the prefix for \(10^{-15}\) is femto (f). Since femto is too small and doesn't fit well, we keep it in pico. The measurement is \(5.89 \text{ ps}\) (pico seconds).

Step 2: Standard Form without SI Prefix

The second measurement is \(0.2010 \text{ m}\). This is already given in meters without scientific notation. The SI prefix isn't necessary here since it's already simply 0.2010 meters.

Step 3: Convert and Write in Nano Prefix

For \(2.560 \times 10^{-9} \text{ g}\), the SI prefix for \(10^{-9}\) is nano (n). Rewrite the measurement by using the nano prefix: \(2.560 \text{ ng}\) (nanograms).

Step 4: Convert and Write in Kilo Prefix

Given \(6.05 \times 10^{3} \text{ m}\), the SI prefix for \(10^3\) is kilo (k). Hence, convert and write the measurement as \(6.05 \text{ km}\) (kilometers).

Key Concepts

Scientific Notation

Scientific notation is a clever way to write very large or very small numbers. It uses powers of ten to express numbers in a compact form. This method makes it easier to read, compare, and calculate with extremely large or tiny numbers.
Often, scientific notation looks like this: a number between 1 and 10 multiplied by ten raised to an exponent (like \(5.89 \times 10^{-12}\)). Here, the number 5.89 is the coefficient, while \(10^{-12}\) tells us how many places to move the decimal point.

• If the exponent is negative, it shows a very small number.
• If positive, it indicates a large number.

Learning how to convert scientific notation into regular numbers and vice versa is an essential skill, especially in subjects like physics and chemistry.

Measurement Units

Units of measurement are standard quantities used to specify and compare physical quantities like length, mass, or time. Consistent units allow everyone to understand exactly what is being measured.
For instance, when we speak about a 'meter', people worldwide recognize it as a standard unit of length. The International System of Units (SI) provides a well-organized set of measurement units to ensure clarity and uniformity.
If you have a measurement like 'seconds', which measures time, or 'grams', which measures mass, these are base units in SI. Converting between units, such as from grams to kilograms, assists in dealing with numbers more efficiently and meaningfully in everyday science.

Metric System

The metric system is a universal way of measuring things like distance, weight, and volume. It's very organized because it's based on factors of ten, making conversions straightforward yet logical.
This system includes units like meters for length, grams for mass, and liters for volume. They are easy to convert through prefixes like kilo, centi, and milli, for example:

• 1000 meters is a kilometer.
• 0.001 meters is a millimeter.

This simplicity is due to the metric system's use of the decimal system, which is familiar to anyone who uses numbers regularly. It's used in most of the world, which helps standardize measurements across different countries and fields.

SI Units

SI units, short for 'International System of Units', are the standard units used globally in science and industry. They are critical for ensuring accurate, repeatable measurements. The seven basic SI units include the meter (for length), the kilogram (for mass), and the second (for time).
Prefixes like kilo, milli, and micro help represent multiples or fractions of these base units without using scientific notation. For example, 1000 grams could be written as 1 kilogram, utilizing the kilo prefix.
Using SI units ensures that people from different parts of the world and various scientific fields can communicate measurements without misunderstandings. This universal applicability is crucial for collaborative research, manufacturing, and day-to-day activities.