Problem 65 - Chemistry

Question

In the United States, the emissions limit for carbon monoxide in motorcycle engine exhaust is \(12.0 \mathrm{~g}\) of carbon monoxide per kilometer driven. What is this limit expressed in mg per mile with the correct number of significant figures?

Step-by-Step Solution

Answer

The limit is 7460 mg/mile.

Step 1: Convert grams to milligrams

To convert grams to milligrams, we need to multiply by 1000 since there are 1000 milligrams in a gram. The emissions limit is given as 12 grams per kilometer, so we convert by calculating: \(12 \, \text{g/km} \times 1000 \, \text{mg/g} = 12000 \, \text{mg/km}\).

Step 2: Convert kilometers to miles

Now we convert the distance from kilometers to miles. We use the conversion factor: 1 mile is approximately 1.60934 kilometers. Thus, we have: \(1 \, \text{km} = \frac{1}{1.60934} \, \text{miles}\).

Step 3: Convert emissions limit to mg/mile

Now, use the conversion factor to adjust the emissions limit from mg/km to mg/mile. Divide the mg/km value by the number of kilometers per mile: \(12000 \, \text{mg/km} \times \frac{1}{1.60934} \, \text{miles/km} \approx 7456 \, \text{mg/mile}\).

Step 4: Determine significant figures

The original value, \(12.0 \, \text{g/km}\), has 3 significant figures. Thus, the converted result \(7456 \, \text{mg/mile}\) should also be rounded to 3 significant figures, giving \(7460 \, \text{mg/mile}\).

Key Concepts

Understanding Significant Figures

Significant figures are crucial when handling measurements and calculations. They tell us about the precision of the number. In scientific and academic contexts, using the correct number of significant figures is important, as it reflects the accuracy of the measurement.
When you read a number like 12.0, each digit is a significant figure because it contributes to the precision of the number.

• The digit '1' is significant.
• The digit '2' is also significant.
• The '0' is significant because it's after the decimal point and included in the measurement.

In the emissions calculations, we started with 12.0 g/km. That means the number is given to three significant figures. When you convert or calculate values from this number, it's essential to maintain the same level of precision.
In our example, after converting to milligrams per mile, we get 7456 mg/mile. Since our initial value had three significant figures, the final answer should also be rounded to three significant figures, leading to 7460 mg/mile. This approach ensures consistency in precision across different units of measurement.

Using Conversion Factors Effectively

Conversion factors are vital tools in unit conversion, enabling us to transition between different measurement systems smoothly. A conversion factor is essentially a fraction that equals one and is used to convert a quantity from one unit to another.
In our example, we used the conversion factor for distance: 1 kilometer is approximately 0.621371 miles. This means:

• To convert kilometers to miles, you multiply by 0.621371.
• Alternatively, to convert miles to kilometers, you multiply by 1.60934.

We also needed to convert grams to milligrams, and since 1 gram equals 1000 milligrams, our conversion factor was 1000 mg/g.
When carrying out conversions, make sure to multiply or divide by the correct conversion factor to transition the units accurately. Consistency in conversion factors ensures you get the right values and maintains the integrity of the converted measurement.

Calculating Emissions Values

Emissions calculations in chemistry allow us to express the concentration of pollutants like carbon monoxide in different units easily. The emissions limit sets a cap on the amount of a specific pollutant that can be emitted by the source. In our problem, we began with a limit of 12.0 g of carbon monoxide per kilometer. To find the emission limit in mg/mile, several conversion steps were required:

• Firstly, convert grams to milligrams by multiplying by 1000.
• Then, convert kilometers to miles using the conversion factor 1 mile ≈ 1.60934 kilometers.
• Finally, compute the emissions limit using the conversions: Convert 12,000 mg/km to mg/mile by dividing by the km/mile conversion factor.

The resulting value required rounding due to significant figures, yielding a final value of 7460 mg/mile, accurately reflecting the initial data's precision after conversion. Understanding the steps and conversions in emissions calculations is essential in analyzing and applying emission data in real-world scenarios.