Example: Can trees save the planet?

Understanding the situation

Excess CO₂ in the atmosphere is a serious problem as it traps heat and warms the planet. Since we started measuring atmospheric CO₂ in the mid twentieth century, the concentration of CO₂ has been inexorably rising to levels associated historically with a much warmer planet. Such warming can cause serious problems for human civilization. 

One approach to solving this problem is explicitly trying to extract CO₂ from the atmosphere and storing it. One place it can be stored is in trees. The physicist Freeman Dyson once suggested we might solve our excess CO₂ problem by planting a lot of trees. Is this plausible?

Let's get some data and do some estimations.

Presenting a sample problem

We'll see what it takes to save the planet with trees in two steps. First, we'll estimate how much CO₂ a typical tree can bind, then we'll see how many trees we would need.

A. Trees extract carbon dioxide (CO₂) from the air and bind the carbon into their structure. Growing trees is therefore one way of reducing the amount of CO₂ in the atmosphere (and burning them is a way of increasing the amount of CO₂ in the atmosphere). About 45% of a tree’s dry mass is carbon and about 25% of the mass of a CO₂ molecule is carbon. Estimate the mass of CO₂ that is removed from the atmosphere by a single full-grown tree.

B. One report states that the human population of the earth was responsible for emitting 2.4 x 10¹³ kg of CO₂ into the atmosphere in the year 2000 as a result of burning fossil fuels. If we decided to capture all the CO₂ emitted from burning fossil fuels by planting new forests, estimate how many square kilometers we would have to convert to forest each year to achieve this goal.

Solving this problem

A. To do the first part, we have to estimate the mass of a full grown tree, calculate the amount of carbon in it, and then calculate the amount of CO₂ required to provide this much carbon.

Mass of a full-grown tree

I have full-grown trees in my backyard. A typical oak is about 20 m tall and has a diameter of about 50 cm. This gives the volume of the trunk as being about

(20 m) x (1/2 m) x (1/2 m) = 5 m³.

This assumes a square trunk. I could also use the radius and do πr², but this is an estimation and the area of a circle and a square are not that different.

I know there are also roots, branches, and leaves. Estimating by thinking what it looks like and assuming that the roots are a bit smaller than the leaves, I know it’s more than just the trunk but I think the mass of roots and leaves are less. Let’s take the whole volume as 8 m³.

I now have to get the mass. I know that a dry log floats in water so its density is less than water. If it’s very dry I would estimate that it floats half-in half-out of the water, so the density might be about half that of water - 500 kg/m³ (water is 1000 kg/m³ = 1 g/cm³). This gives the tree a mass of (8 m³) x (500 kg/m³) = 4000 kg.

Amount of carbon in tree

Since 45% of the tree’s mass is carbon, this is 0.45 x 4000 kg = 1800 kg.

Amount of CO₂ to put this much C in the tree

Only ¼ of the mass of CO₂ is carbon, so to get 1800 kg of carbon I need to use 4 x 1800 kg = 7200 kg. Assuming that all of the CO₂ in the tree comes from CO₂, this is the amount that is removed from the atmosphere by a tree.

This estimates the mass of a full-grown tree to be about 7000 kg (one sig fig).

B. To solve the second part, we need to figure out how many trees we would need, about how much land area a single tree would occupy, and infer how much total area we would need.

How many trees do we need?

To capture all that CO₂, we would need to create

(2.4 x 10¹³ kg) / (7.2 x 10³ kg/tree) = 3.5 x 10⁹ trees

or more than 3 billion full grown trees. Of course they wouldn’t grow to their full growth in a year, so we are only calculating the number we need to plant each year. We still have to cultivate them to their full growth. But when, say, the trees we planted in 2009 have reached their full growth (perhaps in 2039) they will have removed all the CO₂ we put out in 2009.

How much space do we need for one tree?

In my backyard I have a small bit of forest. Trees don’t grow right next to each other since their canopies (top leaves) need to not overlap. Looking out my window, I see that the distance between full-grown trees is about 10 meters. Let’s assume each tree needs 100 m².

How much space do we need for all our trees?

Then for 3.4 billion trees we would need

A = (100 m²/tree)(3.5 x 10⁹ trees) = 3.5 x 10¹¹ m².

Since a square km is (10³ m)² = 10⁶ m², the area A is 3.5 x 10⁵ km², or a square almost 600 km on a side every year.

Joe Redish 3/13/09