Question

As seen in Chapter 1, one teragram \((\mathrm{Tg})\) is equal to \(10^{12} \mathrm{~g} .\) In \(2000, \mathrm{CO}_{2}\) emissions from fuels used for transportation in the United States was \(1990 \mathrm{Tg}\). In 2020, it is estimated that \(\mathrm{CO}_{2}\) emissions from the fuels used for transportation in the United States will be \(2760 \mathrm{Tg}\). a. Calculate the number of kilograms of \(\mathrm{CO}_{2}\) emitted in the years 2000 and 2020 . b. Calculate the number of moles of \(\mathrm{CO}_{2}\) emitted in the years 2000 and \(2020 .\) c. What is the increase, in megagrams, for the \(\mathrm{CO}_{2}\) emissions between the years 2000 and \(2020 ?\)

Short answer

2000: \(1.99 \times 10^{12}\) kg and \(4.52 \times 10^{13}\) moles. 2020: \(2.76 \times 10^{12}\) kg and \(6.27 \times 10^{13}\) moles. Increase: \(7.7 \times 10^8\) Mg.

Step-by-step solution

- Convert Teragrams to Grams (2000)

First, convert the amount of CO\textsubscript{2} emissions in 2000 from teragrams (Tg) to grams (g). Since 1 Tg is equal to \(10^{12}\) grams, multiply 1990 Tg by \(10^{12}\).\[ 1990 \text{ Tg} \times 10^{12} \text{ g/Tg} = 1.99 \times 10^{15} \text{ g} \]

- Convert Grams to Kilograms (2000)

Next, convert grams to kilograms by dividing by \(10^3\) since 1 kg = \(10^3\) g.\[ 1.99 \times 10^{15} \text{ g} \times \frac{1 \text{ kg}}{10^3 \text{ g}} = 1.99 \times 10^{12} \text{ kg} \]

- Convert Teragrams to Grams (2020)

Now, repeat the process for the 2020 emissions. Convert 2760 Tg to grams.\[ 2760 \text{ Tg} \times 10^{12} \text{ g/Tg} = 2.76 \times 10^{15} \text{ g} \]

- Convert Grams to Kilograms (2020)

Convert the 2020 emissions from grams to kilograms.\[ 2.76 \times 10^{15} \text{ g} \times \frac{1 \text{ kg}}{10^3 \text{ g}} = 2.76 \times 10^{12} \text{ kg} \]

- Calculate Moles of CO\textsubscript{2} Emissions (2000)

The molar mass of CO\textsubscript{2} is 44 g/mol. To find the number of moles, divide the mass in grams by the molar mass.\[ \text{Moles of CO}_{2} = \frac{1.99 \times 10^{15} \text{ g}}{44 \text{ g/mol}} \approx 4.52 \times 10^{13} \text{ moles} \]

- Calculate Moles of CO\textsubscript{2} Emissions (2020)

Repeat the process for the 2020 emissions.\[ \text{Moles of CO}_{2} = \frac{2.76 \times 10^{15} \text{ g}}{44 \text{ g/mol}} \approx 6.27 \times 10^{13} \text{ moles} \]

- Calculate Increase in Megagrams

Convert the difference in CO\textsubscript{2} emissions between 2000 and 2020 from Tg to megagrams (Mg). To do this, subtract the 2000 emissions from the 2020 emissions, then note that 1 Tg = 1 million Mg.\[ 2760 \text{ Tg} - 1990 \text{ Tg} = 770 \text{ Tg} \]Since 1 Tg = \(10^6\) Mg, the increase is:\[ 770 \text{ Tg} \times 10^6 \text{ Mg/Tg} = 7.7 \times 10^8 \text{ Mg} \]

Key concepts

Unit Conversion

Unit conversion is a crucial concept. It involves converting a quantity from one unit to another. This is especially important in scientific calculations to ensure consistency and accuracy. When working with large numbers, it's often useful to convert between units such as grams (g), kilograms (kg), and teragrams (Tg).
For instance, knowing that 1 Tg equals to \(1 \times 10^{12}\) grams can simplify complex calculations. Always double-check conversion factors and understand the relationships between units.

Molar Mass

Molar mass is the mass of one mole of a substance. It is expressed in grams per mole (g/mol). Knowing the molar mass helps in converting between mass and moles. For example, the molar mass of CO\textsubscript{2} is 44 g/mol. This means one mole of CO\textsubscript{2} weighs 44 grams. When calculating the number of moles of a substance, you use the formula:
\[ \text{Number of moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \] By understanding this, you can easily move between mass and number of moles in your chemical calculations.

Teragram

A teragram (\text{Tg}) is a unit of mass in the metric system. It represents one trillion grams or \(1 \times 10^{12}\) grams. Teragrams are used for measuring extremely large quantities. For example, CO\textsubscript{2} emissions are often reported in teragrams because the amount of emissions is huge. To convert teragrams to grams, multiply by \(10^{12}\). For instance, 1990 Tg would be calculated as: \[ 1990 \text{ Tg} \times 10^{12} \text{ g/Tg} = 1.99 \times 10^{15} \text{ g} \]Understanding teragrams helps manage and interpret large-scale scientific data.

Kilogram

A kilogram (kg) is a unit of mass in the metric system. One kilogram equals to one thousand grams (1000 g or \(10^{3}\) g). Kilograms are commonly used in everyday life for measuring weight. In scientific contexts, it's important to be comfortable converting between grams and kilograms. For example, if you need to convert 1.99 × \(10^{15}\) grams to kilograms, you can use:
\[ 1.99 \times 10^{15} \text{ g} \times \frac{1 \text{ kg}}{10^3 \text{ g}} = 1.99 \times 10^{12} \text{ kg} \] This conversion is critical when dealing with large masses in scientific calculations.

Megagram

A megagram (Mg) is also known as a metric ton and equals one million grams (\(10^{6}\) g). It is used for measuring very large masses. For example, in reporting emissions or industrial output. Understanding megagrams is essential in large-scale applications.
In the given problem, we calculated the increase in CO\textsubscript{2} emissions between 2000 and 2020. This increase was initially in teragrams but was converted to megagrams because 1 Tg is equivalent to 1 million Mg: \[ \text{Difference in Tg} = 2760 \text{ Tg} - 1990 \text{ Tg} = 770 \text{ Tg} \] Then, convert to megagrams: \[ 770 \text{ Tg} \times 10^{6} \text{ Mg/Tg} = 7.7 \times 10^{8} \text{ Mg} \]