Worksheet titled "Dimensional Analysis (Factor-Label Method) Practice Problems Level 3" with 12 questions covering various unit conversions and scientific calculations.

I will solve problem #12 from the worksheet: "A sample of air contains $2.33 \times 10^{-4}$ mg of lead per mL of gas. This air passes through an office, the volume of which is $3.25 \times 10^4$ L. Seven people normally work in this office. How many $\mu$g of lead will each person in the office receive from this sample of air?"

Step 1: Identify the given values and the goal.
* Concentration of lead = $2.33 \times 10^{-4}$ mg/mL
* Volume of office air = $3.25 \times 10^4$ L
* Number of people = 7
* Goal: Find the amount of lead per person in micrograms ($\mu$g).

Step 2: Convert units to be consistent.
The concentration is in milligrams (mg) and milliliters (mL), but the volume is in liters (L). I need to convert the volume to milliliters first so I can calculate the total mass of lead.
* Conversion factor: 1 Liter = 1000 milliliters ($10^3$ mL).
* Total Volume in mL = $(3.25 \times 10^4 \text{ L}) \times (1000 \text{ mL} / 1 \text{ L})$
* Total Volume = $3.25 \times 10^7$ mL

Step 3: Calculate the total mass of lead in the office.
Now multiply the total volume by the concentration.
* Total Mass (mg) = Concentration $\times$ Volume
* Total Mass (mg) = $(2.33 \times 10^{-4} \text{ mg/mL}) \times (3.25 \times 10^7 \text{ mL})$
* Calculation: $2.33 \times 3.25 = 7.5725$
* Exponents: $10^{-4} \times 10^7 = 10^3$
* Total Mass = $7.5725 \times 10^3$ mg
* Total Mass = 7572.5 mg

Step 4: Convert the total mass from milligrams (mg) to micrograms ($\mu$g).
The question asks for the answer in $\mu$g.
* Conversion factor: 1 mg = 1000 $\mu$g ($10^3$ $\mu$g).
* Total Mass ($\mu$g) = $7572.5 \text{ mg} \times 1000 \text{ }\mu\text{g/mg}$
* Total Mass = $7,572,500$ $\mu$g (or $7.5725 \times 10^6$ $\mu$g)

Step 5: Divide by the number of people to find the amount per person.
There are 7 people sharing this air.
* Lead per person = Total Mass / 7
* Lead per person = $7,572,500 \text{ }\mu\text{g} / 7$
* Calculation: $7,572,500 / 7 = 1,081,785.71...$

Step 6: Check significant figures.
The original values ($2.33$, $3.25$, $7$) generally have 3 significant figures. The count of people (7) might be considered exact, but usually in these problems, we stick to the precision of the measured data. Let's round to 3 significant figures.
* $1,081,785...$ rounded to 3 sig figs is $1,080,000$ or $1.08 \times 10^6$.

Let's re-verify the calculation chain using dimensional analysis all at once to ensure no errors.
$$ \frac{2.33 \times 10^{-4} \text{ mg}}{1 \text{ mL}} \times \frac{1000 \text{ mL}}{1 \text{ L}} \times 3.25 \times 10^4 \text{ L} \times \frac{1000 \text{ }\mu\text{g}}{1 \text{ mg}} \times \frac{1}{7 \text{ people}} $$

Group numbers:
$$ \frac{(2.33 \times 10^{-4}) \times (10^3) \times (3.25 \times 10^4) \times (10^3)}{7} $$
$$ \frac{2.33 \times 3.25 \times 10^{(-4 + 3 + 4 + 3)}}{7} $$
$$ \frac{7.5725 \times 10^6}{7} $$
$$ 1.08178... \times 10^6 $$
$$ \approx 1.08 \times 10^6 \text{ }\mu\text{g} $$

The math holds up.

Final Answer:
$1.08 \times 10^6$ $\mu$g