Problem 134 - General Chemistry

Question

Convert: a \(127 \AA\) A to micrometers b \(21.0 \mathrm{~kg}\) to milligrams c \(1.09 \mathrm{~cm}\) to millimeters d \(4.6 \mathrm{~ns}\) to microseconds

Step-by-Step Solution

Answer

(a) 0.0127 µm, (b) 21,000,000 mg, (c) 10.9 mm, (d) 0.0046 µs.

Step 1: Understanding the problem

We need to convert different physical quantities into specified units. We'll perform these conversions by using the appropriate unit conversion factors.

Step 2: Convert Ångstroms to Micrometers for (a)

First, recognize that 1 Ångström (Å) is equal to \(10^{-10}\) meters. We want to convert this to micrometers. Since 1 micrometer (µm) is \(10^{-6}\) meters, we can set up the following conversion: \[127 \, Å = 127 \, \times \, 10^{-10} \, \text{m} = 127 \, \times \, 10^{-10} \, \times \, 10^{6} \, \mu m = 127 \, \times \, 10^{-4} \, \mu m = 0.0127 \, \mu m.\]

Step 3: Convert Kilograms to Milligrams for (b)

For the conversion from kilograms to milligrams, note that 1 kilogram = \(10^3\) grams and 1 gram = \(10^3\) milligrams, hence 1 kilogram = \(10^6\) milligrams. Therefore, \[21.0 \, ext{kg} = 21.0 \, \times \, 10^{6} \, ext{mg} = 21,000,000 \, ext{mg}.\]

Step 4: Convert Centimeters to Millimeters for (c)

Converting centimeters into millimeters is straightforward since 1 cm = 10 mm. Therefore, \[1.09 \, ext{cm} = 1.09 \, \times \, 10 \, ext{mm} = 10.9 \, ext{mm}.\]

Step 5: Convert Nanoseconds to Microseconds for (d)

To convert from nanoseconds to microseconds, we use the relationships that 1 nanosecond = \(10^{-9}\) seconds and 1 microsecond = \(10^{-6}\) seconds. Hence, \[4.6 \, ext{ns} = 4.6 \, \times \, 10^{-9} \, \text{s} = 4.6 \, \times \, 10^{-9} \, \times \, 10^{3} \, \mu s = 4.6 \, \times \, 10^{-3} \, \mu s = 0.0046 \, \mu s.\]

Key Concepts

Ångström

The Ångström, abbreviated as Å, is a unit of length used primarily in the fields of physics and chemistry, particularly when discussing wavelengths, atomic dimensions, and crystal structures. One Ångström is equal to \(10^{-10}\) meters. This makes it a suitable choice when measuring things on atomic scales, as atoms and molecules are often only a few Ångströms in diameter. To convert Ångströms to micrometers, it is essential to understand that 1 micrometer is equivalent to \(10^{-6}\) meters. So, when converting from Ångströms to micrometers, you multiply by \(10^{-4}\). For example: 127 Å becomes 0.0127 micrometers after conversion. This small number highlights the incredibly tiny scale that the Ångström measures.

Micrometers

Micrometers, indicated by the symbol µm, are one-millionth of a meter. These units are commonly used in fields like biology and materials science to express dimensions smaller than a millimeter but larger than a nanometer.
Micrometers are especially useful when describing cells, bacteria, and other small particles. Unlike Ångströms, which express the dimensions of single atoms, micrometers are more suitable for slightly larger objects, such as the length of a human hair (approximately 100 µm).
When you are converting from Ångströms to micrometers, remember that 1 micrometer is a thousand times larger than 1 Ångström.

Kilograms to Milligrams

Understanding the conversion from kilograms to milligrams is vital in scientific calculations when dealing with mass across different scales. A kilogram (kg) is a base unit of mass in the International System of Units (SI), equivalent to 1,000 grams. A milligram (mg) is equal to 1/1,000th of a gram, or \(10^{-3}\) grams.
So, to convert kilograms to milligrams, multiply the mass value in kilograms by \(10^6\).
This is because 1 kilogram equals \(10^6\) milligrams (or 1,000,000 mg).

• Example: Convert 21 kg to milligrams. \[21 \text{ kg} = 21 \times 10^6 \text{ mg} = 21,000,000 \text{ mg}.\]

This large number of milligrams underscores how milligram measurements can accurately represent very large or small quantities based on the context.

Centimeters to Millimeters

When converting between centimeters and millimeters, it’s important to recognize their relationship. A centimeter (cm) is a unit of length in the metric system equal to one hundredth of a meter. A millimeter (mm) is one tenth of a centimeter. Therefore, converting centimeters to millimeters involves a simple multiplication by 10.
This is because 1 cm equals 10 mm.

• For instance, converting 1.09 cm gives: \[1.09 \text{ cm} = 1.09 \times 10 \text{ mm} = 10.9 \text{ mm}.\]

This easy conversion is practical in everyday measurements like height, length, and width, extending its usability beyond the scientific community into daily life.

Nanoseconds to Microseconds

In time-sensitive applications, particularly in the fields of electronics and communications, understanding the conversion between nanoseconds and microseconds is crucial. One nanosecond (ns) is a billionth of a second \(10^{-9} \text{ s}\), while a microsecond (µs) is a millionth of a second \(10^{-6} \text{ s}\).
To convert nanoseconds to microseconds, you multiply by \(10^{-3}\).

• Consider the conversion of 4.6 ns: \[4.6 \text{ ns} = 4.6 \times 10^{-3} \text{ µs} = 0.0046 \text{ µs}.\]

This demonstrates how small these time intervals are and how they are utilized in high-speed technology contexts such as processors and communication devices.