Question

A dust particle measures \(14 \times 10^{-6} \mathrm{~m}\) in diameter. What is its diameter in nanometers?

Short answer

The diameter of the dust particle is 14,000 nanometers.

Step-by-step solution

Understand the Conversion

1 meter (m) is equivalent to 1,000,000,000 nanometers (nm). The task is to convert meters to nanometers.

Set Up the Conversion

Use the conversion factor between meters and nanometers to set up a multiplication that will cancel out the meters and leave the answer in nanometers.

Perform the Calculation

Multiply the diameter in meters by the conversion factor to find the diameter in nanometers. \(14 \times 10^{-6} \text{ m} \times \frac{1,000,000,000 \text{ nm}}{1 \text{ m}} = 14 \times 10^{3} \text{ nm}\)

Key concepts

Unit Conversion

Understanding unit conversion is crucial across various scientific disciplines, especially in chemistry where precise measurements dictate the outcomes of experiments. The process involves transforming a measurement in one unit to its equivalent in another unit.

When converting measurements, like the diameter of a dust particle from meters (m) to nanometers (nm), it's important to use the correct conversion factor. For meters to nanometers, the conversion factor is that 1 meter is equivalent to 1 billion nanometers. This factor is written as a ratio, \(\frac{1,000,000,000 \text{ nm}}{1 \text{ m}}\). By multiplying the initial measurement by this ratio, meters will cancel out, leaving the answer in nanometers.

To ensure accuracy, always double-check that you're using the correct conversion factor and that your units cancel out appropriately in your calculations. Unit conversion is not just a rote exercise but a fundamental skill that ensures clarity and uniformity in scientific communication.

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact form, which is useful in chemistry and other sciences where such numbers are common. It's written as a product of a number between 1 and 10 and a power of 10. For example, the number \(14 \times 10^{-6}\) signifies the coefficient 14 multiplied by \((10^{-6})\), indicating that the decimal point in 14 has moved 6 places to the left.

Steps to Convert Using Scientific Notation

To convert a number like \(14 \times 10^{-6} \text{ m}\) to nanometers using scientific notation:

• Find the conversion factor between meters and nanometers. In this case, 1 m = \(10^9\) nm.
• Write the measurement in meters using scientific notation: \(14 \times 10^{-6} \text{ m}\).
• Multiply the two numbers, taking care to apply the laws of exponents when multiplying powers of 10. The conversion then becomes: \((14 \times 10^{-6}) \times (10^9)\) nm.
• Finally, add the exponents on the powers of 10 to find the answer in scientific notation: \(14 \times 10^{3}\) nm.

Scientific notation simplifies calculations and allows scientists to easily compare orders of magnitude.

Measurement in Chemistry

Precise measurement is the backbone of chemistry, as it allows chemists to accurately understand and manipulate matter. When measuring any physical quantity like length, volume, mass, or concentration, chemists often deal with a wide range of magnitudes, from the subatomic scale (measured in picometers or femtometers) to macroscopic objects (measured in meters or kilometers).

For example, the diameter of a dust particle might seem insignificant, yet it's an essential measurement in fields like nanotechnology or environmental science. Expressing this diameter in nanometers, a suitable unit for such a small scale, makes it more comprehensible and relevant for analysis and practical application.

Chemistry also involves the use of significant figures and dimensional analysis, principles that ensure measurements are as accurate and meaningful as possible. A firm grasp of these concepts, along with unit conversion and scientific notation, ensures that measurements convey the correct amount of precision required for reliable results in experimentation and research.