Question

A large tree might have a trunk \(0.5 \mathrm{~m}\) in diameter and be \(40 \mathrm{~m}\) tall. Even though it branches out many times, pretend all the wood fits into a cylinder maintaining this \(0.5 \mathrm{~m}\) diameter for the full height of the tree. Wood floats, \({ }^{33}\) so let's say it has a density around \(800 \mathrm{~kg} / \mathrm{m}^{3}\). How many kilograms of \(\mathrm{CO}_{2}\) did this tree pull out of the atmosphere to get its carbon, if we treat the tree's mass as \(50 \%\) carbon?

Short answer

Answer: To find the mass of CO₂ removed from the atmosphere, follow the steps in the solution: 1. Calculate the volume of the tree using the formula V = πr²h. 2. Calculate the mass of the tree using the formula m = ρ · V. 3. Calculate the mass of carbon in the tree by multiplying the mass of the tree by 50%. 4. Calculate the mass of CO₂ removed from the atmosphere by converting the mass of carbon to CO₂ using the molar ratio and molar masses. Once these calculations are completed, you will find the mass of CO₂ removed from the atmosphere by the pine tree.

Step-by-step solution

Calculate the volume of the tree

To find the volume of the tree, we will use the formula for the volume of a cylinder: \(V = \pi r^2h\), where \(V\) is the volume, \(r\) is the radius, and \(h\) is the height. We are given a diameter of \(0.5\,\mathrm{m}\), so the radius is \(0.25\,\mathrm{m}\). The height of the tree is given as \(40\,\mathrm{m}\). Thus, the volume is \(V = \pi (0.25\,\mathrm{m})^2 (40\,\mathrm{m})\).

Calculate the mass of the tree

To find the mass of the tree, we will use the formula: \(m = \rho \cdot V\), where \(m\) is the mass, \(\rho\) is the density, and \(V\) is the volume. We are given a density of \(800\,\mathrm{kg/m^3}\). Use the volume calculated in Step 1: \(m = (800\,\mathrm{kg/m^3}) \cdot V\).

Calculate the mass of carbon in the tree

We are given that the tree's mass is \(50\%\) carbon. To find the mass of carbon in the tree, we will multiply the mass of the tree by the percentage: \(\text{mass of carbon} = m \cdot 50\%\).

Calculate the mass of \(\mathrm{CO}_{2}\) removed from the atmosphere

To find the mass of \(\mathrm{CO}_{2}\) that was removed from the atmosphere, we need to convert the mass of carbon into the mass of \(\mathrm{CO}_{2}\). We can use the molar ratio for this conversion: \(1\) mol of carbon forms \(1\) mol of \(\mathrm{CO}_{2}\). The molar mass of carbon is approximately \(12\,\mathrm{g/mol}\), and the molar mass of \(\mathrm{CO}_{2}\) is approximately \(44\,\mathrm{g/mol}\). So, the mass of \(\mathrm{CO}_{2}\) removed from the atmosphere is: \(\frac{44\,\mathrm{g/mol}}{12\,\mathrm{g/mol}} \cdot \text{mass of carbon} = \frac{44}{12} \cdot \text{mass of carbon}\). Convert the result to kilograms.

Key concepts

Tree Biomass Calculation

Calculating the biomass of a tree is a fascinating part of understanding ecosystems. In this exercise, the tree's trunk is assumed to be a perfect cylinder, simplifying calculations. First, we need to understand the volume of the tree trunk. The cylinder volume formula, which we'll explore in the next section, helps us here. Once we calculate the volume, we multiply it by the density of wood to find the tree's mass. In this case, the density is given as 800 kg/m³. This mass gives us the tree's total weight in kilograms. But we're particularly interested in the carbon content, so we note that the tree is 50% carbon. By multiplying the total mass by 50%, we discover the amount of carbon stored in the tree's biomass.

Cylinder Volume Formula

The formula for the volume of a cylinder is crucial in this exercise. It is given by the equation \[ V = \pi r^2 h \]where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height of the cylinder. Here, the radius is half of the diameter, which is given as 0.5 m. Hence, our radius is 0.25 m. The height is specified as 40 m. Inserting these values into the formula allows us to compute the total volume of the tree trunk:

• The area of the base (cross-section) is \( \pi (0.25 \text{ m})^2 \)
• Multiplying this base area by the height (40 m) gives the total volume.

This calculation yields the space the tree trunk occupies, and is the basis for finding the biomass.

Molar Mass Conversion

Converting the mass of carbon to carbon dioxide is an essential step. This conversion uses the molar mass of carbon and carbon dioxide. The molar mass of carbon is 12 g/mol, and for CO₂, it's 44 g/mol. We need to use a molar ratio to determine how much CO₂ corresponds to the carbon in the tree. Since one mole of carbon transforms into one mole of CO₂ during respiration and photosynthesis, we can directly use the ratio:

• \( \frac{44 \text{ g/mol}}{12 \text{ g/mol}} \text{ mass of carbon} \)
• This simplifies to multiplying the mass of carbon by \( \frac{44}{12} \)

This calculation reveals how much CO₂ the tree has effectively removed from the atmosphere.

Atmospheric Carbon Dioxide Removal

Trees play a vital role in removing carbon dioxide from the atmosphere. By converting carbon into biomass, they store carbon naturally. When we calculate the carbon in a tree, we're essentially finding out how much CO₂ has been absorbed. This process benefits ecosystems by reducing atmospheric carbon dioxide levels.
Carbon sequestration, the process of storing carbon in natural "reservoirs" like trees, is a powerful tool against climate change. Each tree contributes to a healthier, cleaner environment by locking away carbon that would otherwise contribute to the greenhouse effect.
Trees like the one in our exercise act as carbon sinks. Over its lifetime, the tree has sequestered carbon equivalent to the mass of CO₂ we calculate. This emphasizes the importance of conserving and planting more trees to sustain our planet's health.