Problem 31 - Chemistry: An Atoms-Focused Approach

Question

Land Management It has been estimated that better management of cropland, grazing land, and forests could reduce the amount of carbon dioxide in the atmosphere by \(5.4 \times 10^{9}\) kilograms of carbon per year. a. How many moles of carbon are present in \(5.4 \times 10^{9}\) kilograms of carbon? b. How many kilograms of carbon dioxide does this quantity of carbon represent?

Step-by-Step Solution

Answer

Answer: 1.98 x 10^10 kg of carbon dioxide

Step 1: Convert the mass of carbon into moles

First, we need to convert the mass of carbon into moles. To do this, we will use the molar mass of carbon which is 12 g/mol. The given mass of carbon is \(5.4 \times 10^{9}\) kg, which is equal to \(5.4 \times 10^{12}\) grams. The formula to convert the mass of carbon into moles is: moles = mass / molar mass moles = \(\frac{5.4 \times 10^{12} grams}{12 grams/mol}\)

Step 2: Calculate the moles of carbon

Now, we just need to perform the calculation: moles = \(\frac{5.4 \times 10^{12} grams}{12 grams/mol}\) = \(4.5 \times 10^{11}\) moles So, there are \(4.5 \times 10^{11}\) moles of carbon present in \(5.4 \times 10^{9}\) kilograms of carbon.

Step 3: Convert the moles of carbon to moles of carbon dioxide

Since one mole of carbon combines with one mole of oxygen to form one mole of carbon dioxide, the moles of carbon dioxide will be equal to the moles of carbon. moles of carbon dioxide = moles of carbon = \(4.5 \times 10^{11}\) moles

Step 4: Calculate the mass of carbon dioxide

To find the mass of carbon dioxide, we will use the molar mass of carbon dioxide, which is 44 g/mol. We will multiply the moles of carbon dioxide by its molar mass: mass of carbon dioxide = moles of carbon dioxide × molar mass of carbon dioxide mass of carbon dioxide = \(4.5 \times 10^{11}\) moles × 44 g/mol

Step 5: Convert the mass of carbon dioxide into kilograms

Finally, let's perform the calculation and convert the mass into kilograms: mass of carbon dioxide = \(4.5 \times 10^{11}\) moles × 44 g/mol = \(1.98 \times 10^{13}\) grams mass of carbon dioxide (in kilograms) = \(\frac{1.98 \times 10^{13} grams}{1000}\) = \(1.98 \times 10^{10}\) kg So, the given quantity of carbon (\(5.4 \times 10^{9}\) kg) represents \(1.98 \times 10^{10}\) kg of carbon dioxide.

Key Concepts

Mole Conversion

Mole conversion is a crucial concept in stoichiometry, helping us to connect mass with the number of particles. When we talk about moles, we're referring to the number of atoms or molecules in a substance. One mole equals Avogadro’s number, which is approximately \(6.022 \times 10^{23}\) particles. This relationship allows chemists to calculate the amount of substance present based on its mass.

To convert grams into moles, we use the formula:

• moles = mass (in grams) / molar mass (g/mol)

In the original exercise, we needed to determine the moles of carbon from its given mass. Using the molar mass of carbon, which is 12 g/mol, helped perform this conversion accurately. It's essential to always have the correct molar mass when performing mole conversions to ensure precise calculations.

Carbon Dioxide

Carbon dioxide (CO₂) is a compound composed of one carbon atom and two oxygen atoms. It is a major greenhouse gas contributing to global warming through the greenhouse effect. Understanding how carbon turns into carbon dioxide is vital for environmental and chemistry studies.

In the exercise, one mole of carbon combined with one mole of O₂ forms one mole of CO₂. This conversion reflects the stoichiometry of their reaction, which is represented as:

• \(\text{C} + \text{O}_2 \rightarrow \text{CO}_2\)

The balance in the moles ensures that the number of carbon atoms is conserved during the conversion. This is why the moles of carbon directly translate to the moles of carbon dioxide in the solutions.

Molar Mass

Molar mass is the mass of one mole of a given substance, generally expressed in grams per mole (g/mol). It is a bridge between how much a substance weighs and the amount of its particles. Molar mass allows us to translate between the macroscopic world we can measure and the microscopic world of atoms and molecules.

For the problem provided, knowing the molar mass of carbon (12 g/mol) and carbon dioxide (44 g/mol) is essential:

• Molar mass of carbon: 12 g/mol
• Molar mass of carbon dioxide: 44 g/mol (12 from carbon + 16×2 from oxygen)

Understanding these values enabled the steps to convert mass into moles and further calculate the final mass of carbon dioxide produced. This connection is fundamental in chemical reactions and stoichiometric calculations.

Carbon Emissions

Carbon emissions refer to the release of carbon, particularly as carbon dioxide, into the atmosphere from various activities. They are significant contributors to global warming and climate change. Reducing carbon emissions is a critical environmental goal that can be achieved through better land management, as hinted in the exercise.

By converting the mass of carbon to carbon dioxide, we can visualize the impact of carbon emissions. In this exercise, understanding how much CO₂ a certain amount of carbon represents is key to assessing and controlling emissions. Quantifying these emissions in kilograms emphasizes the scale of the issue. An accurate measure helps in developing strategies to mitigate atmospheric CO₂ levels, reducing the adverse effects on the climate.