Problem 79 - Chemistry The Molecular Science

Question

In addition to the metric units of \(\mathrm{nm}\) and \(\mathrm{pm}\), a commonly used unit is the angstrom, where \(1 \AA=1 \times 10^{-10} \mathrm{~m}\). If the distance between a \(\mathrm{Pt}\) atom and an \(\mathrm{N}\) atom in a compound is \(1.97 \AA\), what is the distance in \(\mathrm{nm}\) ? In pm?

Step-by-Step Solution

Answer

The distance is 0.197 nm and 197 pm.

Step 1: Understanding the Conversion from Angstroms to Nanometers

The relation between angstroms and meters is given as \(1 \AA = 1 \times 10^{-10} \text{ m}\). To convert from angstroms to nanometers, recall that \(1 \text{ nm} = 1 \times 10^{-9} \text{ m}\).

Step 2: Convert the Angstroms to Meters

First, convert the distance from angstroms to meters: \[1.97 \AA = 1.97 \times 10^{-10} \text{ m}\].

Step 3: Convert Meters to Nanometers

Now, convert meters to nanometers using the relation:\[1.97 \times 10^{-10} \text{ m} \times \frac{1 \text{ nm}}{1 \times 10^{-9} \text{ m}} = 0.197 \text{ nm}\].

Step 4: Understanding the Conversion from Angstroms to Picometers

To convert from angstroms to picometers, note that \(1 \text{ pm} = 1 \times 10^{-12} \text{ m}\).

Step 5: Convert the Angstroms to Picometers

Convert the distance from angstroms to meters (repeating from step 2 for clarity): \[1.97 \times 10^{-10} \text{ m}\]. Then, use the relation \(1 \text{ pm} = 1 \times 10^{-12} \text{ m}\) to find:\[1.97 \times 10^{-10} \text{ m} \times \frac{1 \text{ pm}}{1 \times 10^{-12} \text{ m}} = 197 \text{ pm}\].

Key Concepts

Angstrom to Nanometer Conversion

Understanding how to convert between different units like an angstrom (\( 1 \AA = 1 \times 10^{-10} \text{ m} \)) and a nanometer is essential in chemistry, especially when dealing with atomic and molecular distances.
One nanometer is defined as \( 1 \text{ nm} = 1 \times 10^{-9} \text{ m} \). This means that one nanometer is ten times larger than one angstrom.

• To convert from angstroms to nanometers, first convert the measure in angstroms to meters.
• Next, convert meters into nanometers using the relationship above.

For instance, if the distance is \( 1.97 \AA \), first rewrite this as \( 1.97 \times 10^{-10} \text{ m} \). Then, use the conversion factor\[1.97 \times 10^{-10} \text{ m} \times \frac{1 \text{ nm}}{1 \times 10^{-9} \text{ m}} = 0.197 \text{ nm}.\]Here, we used the fact that \( 10^{-10} \div 10^{-9} = 10^{-1} \), knowing that dividing exponents means subtracting them.

Angstrom to Picometer Conversion

Similarly, converting from angstroms (\( \AA \)) to picometers involves using another scale of measurement in the metric system.
Since \( 1 \text{ pm} = 1 \times 10^{-12} \text{ m} \), it implies that a single angstrom is equivalent to 100 picometers.

• First, convert the distance in angstroms to meters as before.
• Then, use the relation above to convert meters to picometers.

For our example of \( 1.97 \AA \), this translates initially into \( 1.97 \times 10^{-10} \text{ m} \). Now, multiply by the conversion factor:\[1.97 \times 10^{-10} \text{ m} \times \frac{1 \text{ pm}}{1 \times 10^{-12} \text{ m}} = 197 \text{ pm}.\]Since \( 10^{-10} \div 10^{-12} = 10^2 \), which equals 100, it shows that each angstrom equals 100 picometers.

Metric System in Chemistry

The metric system is a universal language in science that simplifies how we measure and express quantities like distance, volume, and mass.
In chemistry, it's especially crucial because molecules and atoms have dimensions on a very small scale.

• The base unit of length is the meter (m). Other units such as nanometers (nm), picometers (pm), and angstroms (\( \AA \)) are derived from this.
• Conversions within the metric system usually involve multiplying or dividing by powers of ten.
• This is useful because it makes calculations straightforward and less prone to errors.

For instance, converting distances measured in angstroms, a commonly used unit in chemistry for measuring atomic scales, involves understanding these relations:

• \( 1 \AA = 0.1 \text{ nm} \)
• \( 1 \AA = 100 \text{ pm} \)

Whether you're examining the structure of a compound or determining the distance between atoms, the metric system provides a clear and efficient framework for measurement.