Problem 24 - Chemistry: The Central Science

Question

Use appropriate metric prefixes to write the following measurements without use of exponents: (a) \(6.35 \times 10^{-2} \mathrm{~L}\), (b) \(6.5 \times 10^{-6} \mathrm{~s}\), (c) \(9.5 \times 10^{-4} \mathrm{~m}\), (d) \(4.23 \times 10^{-9} \mathrm{~m}^{3}\) (e) \(12.5 \times 10^{-8} \mathrm{~kg}\) (f) \(3.5 \times 10^{-10} \mathrm{~g}\) (g) \(6.54 \times 10^{9} \mathrm{fs}\)

Step-by-Step Solution

Answer

(a) 6.35 cL (b) 6.5 µs (c) 0.95 mm (d) 4.23 n\(m^{3}\) (e) 125 µg (f) 35 ng (g) 6.54 Gfs

Step 1: (a) Rewrite in terms of metric prefixes

6.35 × \(10^{-2}\) L can be rewritten by using the centi (c) prefix, which corresponds to \(10^{-2}\). Therefore, we get: 6.35 cL.

Step 2: (b) Rewrite in terms of metric prefixes

6.5 × \(10^{-6}\) s can be rewritten by using the micro (µ) prefix, which corresponds to \(10^{-6}\). Therefore, we get: 6.5 µs.

Step 3: (c) Rewrite in terms of metric prefixes

9.5 × \(10^{-4}\) m is in between milli and micro, so we can multiply by 10 to get the milli prefix. We now have: \(9.5 \times 10^{-1}\) mm, which can be rewritten using the deci (d) prefix to: 0.95 mm.

Step 4: (d) Rewrite in terms of metric prefixes

4.23 × \(10^{-9}\) m³ can be rewritten by using the nano (n) prefix, which corresponds to \(10^{-9}\). Therefore, we get: 4.23 n\(m^{3}\).

Step 5: (e) Rewrite in terms of metric prefixes

12.5 × \(10^{-8}\) kg is in between micro and nano, so we can multiply by 10 to get the micro prefix. We now have: \(12.5 \times 10^{-5}\) g, which can be rewritten using the micro (µ) prefix to: 125 µg.

Step 6: (f) Rewrite in terms of metric prefixes

3.5 × \(10^{-10}\) g is in between nano and pico, so we can multiply by 100 to get the nano prefix. We now have: \(3.5 \times 10^{-8}\) g, which can be rewritten using the nano (n) prefix to: 35 ng.

Step 7: (g) Rewrite in terms of metric prefixes

6.54 × \(10^{9}\) fs can be rewritten by using the giga (G) prefix, which corresponds to \(10^{9}\). Therefore, we get: 6.54 Gfs.

Key Concepts

Scientific Notation

Scientific notation is a way to express really big or really small numbers in a simple format. It uses powers of ten to do this. This method is incredibly helpful in making numbers more readable and manageable, especially in the world of science and engineering. For example, instead of writing 0.00000065 grams, you can write it as \(6.5 \times 10^{-7}\) grams. This format immediately shows you how many times you need to move the decimal point to get back to the original number.

• The sign in the exponent tells you whether the number is big or small. A negative sign means the number is less than one, while a positive sign means it's more than one.
• Scientific notation also helps in performing arithmetic operations like multiplication and division, making them much simpler.

Understanding scientific notation is important for diving deeper into measurements and conversions because it lays the foundation for using metric prefixes comfortably.

Unit Conversion

Unit conversion is the process of changing one unit of a measure to another. This is important for comparing and combining different types of measurements. With the metric system, conversions are straightforward because it's based on powers of ten. This means you just move the decimal point to convert between units.

Suppose you have a measurement in meters and you need it in centimeters. Since 1 meter equals 100 centimeters, you multiply the meter value by 100. Conversely, dividing by 100 will turn a centimeter value into meters.

• Always know the conversion factor. For example, 1 kilometer equals 1000 meters.
• Keep track of your units during conversion to avoid mistakes.

Using unit conversion correctly ensures accuracy in problem-solving, whether you're measuring liquids, time, mass, or length.

Metric System

The metric system is a universal way of measuring and is used all over the world. It is based on powers of ten, making it simple to understand and easy to convert. This system uses metric prefixes that correspond to different powers of ten. Familiar examples of these include kilo (\(10^3\)), centi (\(10^{-2}\)), and milli (\(10^{-3}\)). These prefixes help you know the scale of the measurement quickly.

• Kilo (k) means a thousand units of something, like a kilogram is a thousand grams.
• Centi (c) is one-hundredth, so a centimeter is one-hundredth of a meter.
• Prefixes like nano (\(10^{-9}\)) help describe very small amounts, such as in nanometers or nanograms.

The metric system not only standardizes measurements so they are the same around the world, but it also simplifies calculations and conversions. Learning this system makes it easier to understand scientific data and perform everyday tasks that involve measurements like cooking, traveling, and building.