Problem 1 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

The \(C-C\) bond length in a crystal of diamond is 0.154 nm. What is this distance in (a) metres, (b) picometres, (c) angstroms? (Section \(1.2)\)

Step-by-Step Solution

Answer

0.154 nm is 1.54 x 10^-10 m, 154 pm, and 1.54 Å.

Step 1: Understanding the Given Information

We are initially given that the carbon-carbon bond length in a diamond crystal is 0.154 nm. The goal is to convert this length into metres, picometres, and angstroms.

Step 2: Conversion to Metres

One nanometre (nm) is equal to \(10^{-9}\) metres. Hence, to convert 0.154 nm to metres, multiply by \(10^{-9}\):\[ 0.154 \text{ nm} = 0.154 \times 10^{-9} \text{ m} = 1.54 \times 10^{-10} \text{ m} \]

Step 3: Conversion to Picometres

One nanometre is also equivalent to 1000 picometres (pm). To convert 0.154 nm to picometres, multiply by 1000:\[ 0.154 \text{ nm} = 0.154 \times 1000 \text{ pm} = 154 \text{ pm} \]

Step 4: Conversion to Angstroms

One nanometre is equal to 10 angstroms (Å). Thus, to convert 0.154 nm to angstroms, multiply by 10:\[ 0.154 \text{ nm} = 0.154 \times 10 \text{ Å} = 1.54 \text{ Å} \]

Key Concepts

Nanometre to Metre Conversion

To convert nanometres to metres, it's important to recognize the scale difference between these two units. A metre is a much larger unit compared to a nanometre. In fact, one nanometre is equal to one-billionth of a metre. This can be expressed in scientific notation as:\[ 1 \text{ nm} = 10^{-9} \text{ m} \]The conversion is straightforward. You multiply the nanometre value by \(10^{-9}\). For instance, if you have 0.154 nm, and you wish to convert it to metres, the calculation would be:\[ 0.154 \times 10^{-9} = 1.54 \times 10^{-10} \text{ m} \]Understanding this conversion is crucial in chemistry, where dealing with atomic and molecular scales makes metres impractical for direct use. By converting nanometres to metres, we align measurements with the metric system, which is widely used in scientific contexts.

Nanometre to Picometre Conversion

When converting from nanometres to picometres, we're zooming into an even smaller scale. A picometre is one-thousandth of a nanometre:\[ 1 \text{ nm} = 1,000 \text{ pm} \]In the context of conversions, you can perform this by multiplying the nanometre value by 1,000. To illustrate, let's convert 0.154 nanometres to picometres:\[ 0.154 \times 1,000 = 154 \text{ pm} \]This conversion is useful in expressing dimensions of atomic bonds and measurements in fields like quantum mechanics, where precision at extremely small scales is required. By using picometres, scientists can communicate sizes and distances more efficiently at a level where nanometres may be too large a unit.

Nanometre to Angstrom Conversion

The angstrom is an older, yet still occasionally used unit of measure in scientific fields like chemistry and physics, particularly for atomic-scale dimensions. One angstrom is defined as 0.1 nanometres:\[ 1 \text{ nm} = 10 \text{ Å} \]Thus, converting nanometres to angstroms involves multiplying by 10. Consider the conversion of 0.154 nm to angstroms as follows:\[ 0.154 \times 10 = 1.54 \text{ Å} \]This conversion allows scientists to discuss microscopic distances in terms that might feel more intuitive due to the historical usage of angstroms in microscopy and crystallography. Though not an SI unit, the angstrom is still prevalent in literature, simplifying representation of atomic and molecular scales.