Problem 25 - Chemistry in Context

Question

Nail polish remover containing acetone was spilled in a room \(6 \mathrm{~m} \times 5 \mathrm{~m} \times 3 \mathrm{~m}\). Measurements indicated that \(3600 \mathrm{mg}\) of acetone evaporated. Calculate the acetone concentration in micrograms per cubic meter.

Step-by-Step Solution

Answer

The acetone concentration is 40,000 µg/m³.

Step 1: Understand the Problem

We need to find the concentration of acetone in the air of the room in micrograms per cubic meter. We are given the room dimensions and the amount of acetone that evaporated.

Step 2: Calculate the Volume of the Room

The room has dimensions of 6 meters by 5 meters by 3 meters. Calculate the volume by multiplying these dimensions: \(V = 6 \times 5 \times 3 = 90\) cubic meters.

Step 3: Convert Milligrams to Micrograms

The problem gives the mass of acetone as 3600 mg. Convert this to micrograms using the fact that 1 mg = 1000 µg. Thus, \(3600 \times 1000 = 3600000\) µg.

Step 4: Calculate Acetone Concentration

Now, calculate the concentration by dividing the total mass of acetone in micrograms by the volume of the room in cubic meters: \(\frac{3600000\, \text{µg}}{90\, \text{m}^3} = 40000\, \text{µg/m}^3\).

Key Concepts

Room Volume Calculation

To find how much space is inside a room, you calculate its volume. Think of the room as a box with three measurements: length, width, and height. Multiply these three measurements together.

• Length: how long the room is (6 meters in this case).
• Width: how wide the room is (5 meters).
• Height: how tall the room is (3 meters).

By multiplying these dimensions together, we find the volume: \[ V = ext{length} \times ext{width} \times ext{height} \] For this room: \[ V = 6 \times 5 \times 3 = 90 \, \text{cubic meters} \]
This means there are 90 cubic meters of space inside this room where the acetone vapor can spread out.

Unit Conversion

Unit conversion is like translating one quantity into another kind of measurement. In our case, we need to convert milligrams (mg) to micrograms (µg). Each milligram is 1,000 times bigger than a microgram, just as a meter is 1,000 times longer than a millimeter. When dealing with chemical spills like acetone, it is often practical to use smaller units like micrograms. Here's how you convert milligrams to micrograms:

• Take the number of milligrams you have (3600 mg in this case).
• Multiply it by 1,000 to get the number of micrograms.

So for 3600 mg: \[ 3600 \times 1000 = 3600000 \, \text{µg} \]Now we have the mass of acetone in the smaller and more precise unit of micrograms.

Micrograms per Cubic Meter Calculation

Calculating concentration involves figuring out how much acetone is in each cubic meter of the room. This tells us how spread out or concentrated the acetone is in the air.
To find the concentration of acetone in \,\text{µg/m}^3, take the total mass in micrograms and divide it by the room volume in cubic meters.

• Total mass of acetone: 3600000 µg.
• Room volume: 90 \, m^3.

The formula for concentration is: \[ ext{Concentration} = \frac{\text{Total mass in µg}}{\text{Volume in } \, m^3} \] Apply the known values: \[ \frac{3600000 \, \text{µg}}{90 \, \text{m}^3} = 40000 \, \text{µg/m}^3 \]This means there are 40,000 micrograms of acetone in every cubic meter of the room.