Problem 39 - Chemistry: The Central Science in SI Units, Global Edition

Question

If on a certain year, an estimated amount of 4 million metric tons ( 1 metric ton \(=1000 \mathrm{~kg}\) ) of nitrous oxide \(\left(\mathrm{N}_{2} \mathrm{O}\right)\) was emitted worldwide due to agricultural activities, express this mass of \(\mathrm{N}_{2} \mathrm{O}\) in grams without exponential notation, using an appropriate metric prefix.

Step-by-Step Solution

Answer

The mass of nitrous oxide emitted worldwide due to agricultural activities in the given year can be expressed as 4 Tg (tera grams).

Step 1: Convert million metric tons to metric tons

First, we need to convert the given amount from million metric tons to metric tons. As 1 million = \(10^6\), we can multiply the given amount (4 million) by \(10^6\). 4 million metric tons = 4 × \(10^6\) metric tons

Step 2: Convert metric tons to kilograms

Now, we need to convert metric tons to kilograms. Given that 1 metric ton = 1000 kg, we can multiply the result from Step 1 by 1000. 4 × \(10^6\) metric tons × 1000 kg/metric ton

Step 3: Convert kilograms to grams

Finally, we need to convert the result from Step 2 to grams. As 1 kg = 1000 grams, we can multiply the result by 1000. 4 × \(10^6\) metric tons × 1000 kg/metric ton × 1000 g/kg

Step 4: Calculate and express using an appropriate metric prefix

Now, we can calculate the result and express it using an appropriate metric prefix. 4 × \(10^6\) × 1000 × 1000 g = 4 × \(10^{12}\) g The metric prefix for \(10^{12}\) is "tera," represented by the letter "T." Therefore, we can express the given mass of nitrous oxide as: 4 Tg (tera grams)

Key Concepts

Metric Conversions

In science, especially in chemistry, converting between different units of measurement is crucial. When faced with a large number of units, like moving from metric tons to grams, understanding the method of conversion simplifies the process. In the example given, we used several conversions. First, converting millions to single units, and then systematically transitioning through metric ton to kilogram, and kilogram to gram conversions.

• The first step involved recognizing that 1 million is equivalent to \(10^6\). So, 4 million metric tons becomes 4 \(\times\) \(10^6\) metric tons.
• Next, noting that 1 metric ton equals 1000 kilograms helps in now multiplying the quantity by 1000 to switch units from tons to kilograms.
• Finally, with the knowledge that 1 kilogram is 1000 grams, we conclude with a final multiplication by 1000, achieving the desired unit of grams.

Each conversion multiplies by powers of 10, facilitating the expression in larger or smaller units as needed.

Mass Calculations

Mass calculations often require the systematic use of conversion factors to bridge different units. In this example, we start with 4 million metric tons of nitrous oxide, which we wish to express in grams.
By setting up our chain of multiplications using conversion factors, we effectively change each unit precisely:

• Given: 4 million metric tons.
• Convert metric tons to kilograms: Multiply by 1000 (since 1 metric ton = 1000 kg).
• Convert kilograms to grams: Multiply by another 1000 (because 1 kg = 1000 g).

Our final expression becomes 4 \(\times\) \(10^{12}\) grams. This calculated result signifies the enormity of the amount in grams, illustrating the importance of precise conversions in scientific calculations.

Metric Prefixes

Metric prefixes are shortcuts used to express very large or very small quantities succinctly. Each prefix correlates to a specific power of ten, simplifying tangible comprehension of vast numbers. In the context of our example, \(10^{12}\) grams is expressed as 4 teragrams.
The prefix 'tera' (T) stands for \(10^{12}\), offering a more digestible value without lengthy numerical expressions.

• 1 kilobyte (kB) equals \(10^3\) bytes.
• 1 megabyte (MB) equals \(10^6\) bytes.
• 1 gigabyte (GB) equals \(10^9\) bytes.
• 1 terabyte (TB) equals \(10^{12}\) bytes.

Understanding prefixes helps convert unwieldy numbers into manageable sizes quickly, aiding clearer communication in science and mathematics.