Convert 3.00 µm² Graphene Area to m²

Answer

The area of the graphene layer is $3.00 \times 10^{-12} \text{ m}^2$. This result is obtained by squaring the linear conversion factor between micrometers and meters.

Explanation

Known:

• Initial Area ($A$): $3.00 \text{ µm}^2$
• Micro prefix ($\mu$): $10^{-6}$

Find:

• Area ($A$) in square meters ($\text{m}^2$)

---

1. Identify the linear conversion factor The prefix "micro" ($\mu$) represents one-millionth of a unit. Therefore, there are $10^6$ micrometers in 1 meter, or 1 micrometer is equal to $10^{-6}$ meters. $1 \text{ µm} = 10^{-6} \text{ m}$ A basic relationship showing that one micrometer is $10^{-6}$ meters.

2. Derive the area conversion factor ⚠️ This step is required on exams: When converting units of area, you must square the entire linear conversion factor. Failing to square the exponent is the most frequent source of error in dimensional analysis. $(1 \text{ µm})^2 = (10^{-6} \text{ m})^2$ Squaring both sides of the linear relationship to find the area ratio. $1 \text{ µm}^2 = 10^{-12} \text{ m}^2$ Developing the conversion factor where $10^{-6}$ becomes $10^{-12}$ through the power of a power rule.

3. Perform the substitution and calculation Multiply the given value by the derived conversion factor to cancel the square micrometers. $A = 3.00 \text{ µm}^2 \times \left( \frac{10^{-6} \text{ m}}{1 \text{ µm}} \right)^2$ Setting up the dimensional analysis calculation. $A = 3.00 \times 10^{-12} \text{ m}^2$ The numerical result after applying the squared conversion factor.

4. Unit and significant figure check The original measurement $3.00$ has three significant figures. Our conversion factor is exact, so the final answer must maintain three significant figures. The units $\text{µm}^2$ cancel out, leaving $\text{m}^2$. $\text{Units: } \text{µm}^2 \cdot \frac{\text{m}^2}{\text{µm}^2} = \text{m}^2$ Dimensional analysis proof showing that only square meters remain.

Final Answer

The area expressed in square meters to three significant figures is: $\boxed{3.00 \times 10^{-12} \text{ m}^2}$

Common Mistakes

• Forgetting to square the prefix: Students often multiply $3.00$ by $10^{-6}$ instead of $10^{-12}$. Remember: if the unit is squared, the conversion factor must be squared ($10^{-6} \times 10^{-6}$).
• Significant figure neglect: Ensure you keep the ".00" in $3.00 \times 10^{-12}$ to indicate that the measurement has a precision of three significant figures. Writing simply $3 \times 10^{-12}$ would be incorrect in a laboratory context.