Problem 125 - Chemistry: The Central Science

Question

Look up the diameter of a silicon atom, in \(\hat{A}\). The latest semiconductor chips have fabricated lines as small as \(22 \mathrm{~nm}\). How many silicon atoms does this correspond to?

Step-by-Step Solution

Answer

In a 22 nm long fabricated line, there are approximately 104 silicon atoms. This is calculated by first converting the diameter of a silicon atom (2.10 \(\mathring{A}\)) to meters (\(2.1 \times 10^{-10}\) m) and then dividing the length of the fabricated line in meters (\(2.2 \times 10^{-8}\) m) by the diameter of a silicon atom in meters.

Step 1: Look up the diameter of a silicon atom

The diameter of a silicon atom is approximately 2.10 \(\mathring{A}\) (angstroms). Given that 1 \(\mathring{A}\) = \(1 \times 10^{-10}\) meters, we can convert the diameter of a silicon atom to meters.

Step 2: Convert the diameter of a silicon atom from angstroms to meters

To convert the diameter of a silicon atom from angstroms to meters, multiply it by the conversion factor: Diameter (m) = Diameter (\(\mathring{A}\)) x Conversion factor Diameter (m) = 2.10 \(\mathring{A}\) x \(1 \times 10^{-10}\) m/\(\mathring{A}\) Diameter (m) = \(2.1 \times 10^{-10}\) m So, the diameter of a silicon atom is \(2.1 \times 10^{-10}\) meters.

Step 3: Convert the fabricated line's length from nanometers to meters

The fabricated line is 22 nm long. To convert it to meters, multiply it by the conversion factor: Length (m) = Length (nm) x Conversion factor Length (m) = 22 nm x \(1 \times 10^{-9}\) m/nm Length (m) = \(2.2 \times 10^{-8}\) m So, the length of the fabricated line is \(2.2 \times 10^{-8}\) meters.

Step 4: Calculate the number of silicon atoms in the 22 nm line

Now that we have the diameter of a silicon atom and the length of the fabricated line in meters, we can calculate the number of silicon atoms that correspond to the 22 nm line using the formula: Number of atoms = Length (m) / Diameter (m) Number of atoms = \(\frac{2.2 \times 10^{-8}\ m}{2.1 \times 10^{-10}\ m}\) Number of atoms ≈ 104.76 Since we cannot have a fraction of an atom, we round down to the nearest whole atom: Number of atoms = 104

Step 5: State the final answer

In a 22 nm long fabricated line, there are approximately 104 silicon atoms.

Key Concepts

Angstrom to Meters Conversion

The angstrom (Å) is a unit of length commonly used in the field of chemistry and physics to express the sizes of atoms and molecules. However, the angstrom is not part of the SI (International System of Units) units of measurement, which predominantly use meters.
To convert from angstroms to meters, the conversion factor is essential: 1 angstrom is equal to \(1 \times 10^{-10}\) meters. This means that when you know the size of an object in angstroms, you can easily find its size in meters by multiplying by \(1 \times 10^{-10}\).

• For example, the diameter of a silicon atom is approximately 2.10 angstroms.
• To find its diameter in meters, you calculate \(2.10 \times 10^{-10} = 2.1 \times 10^{-10}\) meters.

This conversion is crucial in calculations where different units need to be compared, such as in semiconductor fabrication or when measuring atomic scales.

Nanometer to Meters Conversion

The nanometer (nm) is another vital unit of measurement, particularly prevalent in nanotechnology and semiconductor fabrication. A nanometer is one billionth of a meter, defined as: 1 nanometer = \(1 \times 10^{-9}\) meters.
In the context of semiconductor fabrication, we frequently need to convert measurements to meters to understand the physical dimensions more precisely.

• For instance, many modern semiconductor chips have features as small as 22 nm.
• To express this in meters, simply multiply by the conversion factor: \(22 \times 10^{-9} = 2.2 \times 10^{-8}\) meters.

Converting nanometers to meters not only helps in understanding sizes on a smaller scale but also enables seamless integration with lab measurements and industrial standards that typically use meters.

Semiconductor Fabrication

Semiconductor fabrication is the process used to create the integrated circuits that are essential to modern electronic devices. It involves numerous steps to form the silicon wafers into circuits. One of the most important aspects of this process is the scale of the circuit features, often measured in nanometers.
In this field, understanding size at the atomic and molecular scale is crucial because:

• Silicon, the base material for chips, has an atomic diameter of just a few angstroms.
• As technology advances, the size of transistors is decreasing, often reaching down to 22 nm or smaller.

Manufacturing with such precision involves meticulous control of conditions and cutting-edge technologies like photolithography. These intricacies highlight the importance of converting units like angstroms and nanometers into meters, ensuring accuracy in these high-tech processes.