Question

The diameter of a silicon atom is \(2.34 \times 10^{-8} \mathrm{~cm}\). If the silicon atoms were placed next to each other in a straight line across an 8.5-in. width of paper, how many silicon atoms would be there?

Short answer

An 8.5-inch width of paper is equivalent to 21.59 cm. Therefore, the number of silicon atoms that can fit across it is \frac{21.59 \text{ cm}}{2.34 \times 10^{-8} \text{ cm}} \text{ atoms}.

Step-by-step solution

Convert the width of the paper from inches to centimeters

Since the diameter of the silicon atom is given in centimeters, convert the width of the paper from inches to centimeters using the conversion rate that 1 inch is equal to 2.54 centimeters.

Calculate the width of the paper in centimeters

Multiply the width of the paper in inches by the conversion factor to find the width in centimeters: Width in cm = 8.5 inches * 2.54 cm/inch.

Calculate the number of silicon atoms

Divide the width of the paper in centimeters by the diameter of a single silicon atom to find the number of silicon atoms that would fit across the width of the paper: Number of atoms = Width in cm / Diameter of one silicon atom.

Key concepts

Unit Conversion

Unit conversion is a fundamental process in many scientific calculations, allowing us to convert measurements from one unit to another. In the context of calculating how many silicon atoms fit across a piece of paper, we must ensure that our units for measurement are consistent. For instance, if you're given the diameter of a silicon atom in centimeters but the paper's width in inches, you cannot directly compare these two measurements without converting them to the same unit. This is typically done through multiplication by a conversion factor, which is a ratio representing how many of one unit equal another unit.

For instance, since 1 inch is equivalent to 2.54 centimeters, we convert inches to centimeters by multiplying the number of inches by 2.54. Knowing how to perform unit conversions accurately is crucial, as it ensures the correctness of our calculations and thus the validity of our results.

Dimensional Analysis

Dimensional analysis, also known as the factor-label method or unit factor method, is a powerful tool used in problem-solving and unit conversion. It involves using the units of measurements as guides to solve complex problems. This is done by multiplying the quantity by one or more conversion factors so that unwanted units cancel out, leaving only the desired unit.

Using our silicon atoms example, once the paper's width is converted to centimeters, you can use dimensional analysis to calculate the number of atoms. To do this, you divide the total width by the diameter of one atom, which cancels out the 'centimeter' unit and leaves you with an unadorned number, representing the count of silicon atoms. An understanding of dimensional analysis is pivotal because it allows students to convert and manipulate units systematically, decreasing the chances of errors in scientific calculations.

Atomic Structure

The atomic structure is a key concept in both physics and chemistry, describing the composition and arrangement of atoms. Atoms are composed of a nucleus made up of protons and neutrons, surrounded by a cloud of electrons. Despite the complexity of the internal structure, when considering the size of an atom for exercises such as calculating how many would fit across a given space, we generally focus on the atom's diameter—essentially the space that an atom occupies.

The atomic diameter is particularly important in our problem because this measurement is what determines how many atoms can fit side by side along the length of paper. Much like beads on a string, you can only fit as many silicon atoms as the combined diameters will allow within the measured space. It's important for students to understand that while atoms are incredibly small (on the order of 10^-8 to 10^-10 meters in diameter for most elements), they still occupy physical space, and this concept is fundamental in fields ranging from material science to nanotechnology.