Physics 9702/12 — May/June 2018

Background Concept

For a very thin sheet made of atoms packed in layers, the number of atomic layers across the thickness is estimated by:

$$N \approx \frac{\text{sheet thickness}}{\text{thickness per atomic layer}}$$

If an atom is modelled as a sphere, one layer is approximately one atomic diameter thick (not the radius). So thickness per layer is about $2r$.

This is an order-of-magnitude estimate: real atoms are not hard spheres and packing can vary, but the calculation is designed to match one of the options.

Understanding the Question

You are given:

• Thickness of gold leaf: $0.125\,\mu\text{m}$
• Radius of a gold atom: $174\,\text{pm}$

You must estimate how many atoms fit through the thickness, i.e. the number of layers. The key is to convert both lengths into the same unit, then divide thickness by the atomic diameter.

Approach

1. Convert the sheet thickness from micrometres to metres.
2. Convert the atomic radius from picometres to metres.
3. Double the radius to get the diameter (one layer thickness).
4. Divide: $N = t / (2r)$.
5. Choose the closest option.

Step-by-Step Reasoning

1. Convert the thickness:

$$0.125\,\mu\text{m} = 0.125 \times 10^{-6}\,\text{m} = 1.25 \times 10^{-7}\,\text{m}$$

1. Find the diameter of a gold atom:

$$2r = 2(174\,\text{pm}) = 348\,\text{pm}$$

Convert to metres using $1\,\text{pm} = 10^{-12}\,\text{m}$:

$$348\,\text{pm} = 348 \times 10^{-12}\,\text{m} = 3.48 \times 10^{-10}\,\text{m}$$

1. Divide thickness by diameter:

$$N = \frac{1.25 \times 10^{-7}}{3.48 \times 10^{-10}} = \left(\frac{1.25}{3.48}\right) \times 10^{3} \approx 0.36 \times 10^{3} \approx 3.6 \times 10^{2}$$

So the number of layers is about $360$, which rounds to about $400$.

Therefore the correct option is C (400).

Key Takeaways

• Always convert to consistent units before dividing quantities.
• For an “atomic layers” estimate, use the diameter $2r$, not the radius.
• MCQ answers are typically based on approximate reasoning and rounding.

Common Mistakes

• Using $r$ instead of $2r$, giving roughly double the number of layers.
• Mixing prefixes (e.g. treating $\mu\text{m}$ as $10^{-9}\,\text{m}$ or $\text{pm}$ as $10^{-10}\,\text{m}$).
• Forgetting that $10^{-7}/10^{-10} = 10^{3}$, not $10^{-17}$ or $10^{-3}$.

Things to Be Careful About

• Prefixes: $\mu = 10^{-6}$ and $\text{p} = 10^{-12}$.
• Keep powers of ten separate from the decimal division to reduce arithmetic errors.
• Choose the nearest option; here $360$ is much closer to $400$ than to $700$.