Balance of Rates: A Thought Experiment

Printer-friendly versionPrinter-friendly version Share this

We often look at the number of tonnes of carbon dioxide we emit into the atmosphere for a given activity. But what if it's not a question of how much- but rather, how fast?

Big Ideas: 
  • In a given system, if something- whether it be water in a bathtub or greenhouse gas in the atmosphere- is being added and taken away at the same rate, its amount in the system does not change.
  • If something's input rate is greater than the output rate, even just by a little bit, its amount in the system will increase over a period of time.
  • Positive feedback occurs when the increase of a particular thing causes that same thing to increase further.


The purpose of this experiment is to explore the effect of unbalanced rates and apply this idea to carbon dioxide concentrations in the Earth's atmosphere. Students will explore how even a small addition to an already balanced system can cause an overload with significant repercussions.


The concept of rates is a relatively simple one: if you add something faster than you remove it, that thing will accumulate.

Let us start with the analogy of a bathtub. A bathtub can have water flowing in, and water flowing out. Imagine you open the faucet and start adding water to your bathtub without plugging the drain: all the water drains away and almost no water is left in the tub, even if you run it for a very long time. This is because the water is leaving the tub faster than it is being put in. When you plug the drain, however, the amount of water being drained decreases greatly- or even stops. Because the water is entering the tub faster than it is being drained out, the water level will keep on rising.

Now imagine the bathtub as our atmosphere, and the water as the carbon dioxide in it. Like the faucet of the bathtub, the Earth's atmosphere has sources of emissions pumping carbon dioxide in. And, like the drain, there are also things that absorb the carbon dioxide, taking them out.

In nature, plant growth and decay and ocean temperature fluctuations are some of the things that cause carbon dioxide absorption and emission; these happen in the scale of hundreds of billions of tonnes per year! Emissions due to human activities, on the other hand, are in the scale of tens of billions of tonnes per year, an order of magnitude smaller than natural carbon dioxide exchanges.1. Human contribution to the carbon dioxide pool in the atmosphere (mostly from burning fossil fuels and deforestation), then, doesn't seem like that much at all.

However, for over a half a million years before the Industrial Revolution (when fossil fuels started to become a popular energy source), the concentration of carbon dioxide in the atmosphere had steadied at roughly 280 parts per million, with dips in concentration during the ice ages.2 This meant that nature's carbon dioxide emissions and absorptions were generally balanced: the rate at which carbon dioxide was emitted into the atmosphere was roughly equal to the rate at which it was being absorbed out. When humans add to the rate of carbon dioxide emissions, even by a little bit, it can put the whole system off balance.

Scientists have been able to monitor and track the changing level of carbon dioxide concentrations in the atmosphere at the Mauna Loa Observatory in Hawaii3 for the past 50 years or so:

The red line plots the monthly mean carbon dioxide concentration on Mauna Loa, showing that the carbon dioxide concentration in the atmosphere has been increasing since the measurements started. If we return to the bathtub analogy, it is as if humans have opened up the faucet only slightly more than the rate of draining water; but even that, after a while, will cause the water level to rise.

The problem with carbon dioxide is its property as a greenhouse gas, which means it absorbs and re-radiates long-wave thermal radiation inside the atmosphere. Too much of this in the atmosphere at one time contributes to the overall rising temperature of the Earth. (Read more on the greenhouse effect here.)

Plot twist: Positive Feedback
We have been comparing the sources of carbon dioxide emissions to the faucet of a bathtub, but the rate of the this so-called "faucet" does not stay steady- it gets more complicated than that. Greenhouse gases like carbon dioxide being in the atmosphere (water in the bathtub) actually causes more greenhouse gas to be released (causes the tap to flow faster). The increase in greenhouse gas in the atmosphere causes the Earth's average temperature to rise, which in turn causes an increased amount of water to be turned into water vapour, which itself is a greenhouse gas. An increase in ocean temperature also decreases its ability to absorb carbon dioxide from the atmosphere. This increases temperature again, causing more greenhouse gas release. This self-enforcing loop is called a "positive feedback loop". This is a process where the increase of a particular thing causes, through a series of events, itself to increase further. Can you spot this phenomenon in the graph of carbon dioxide concentrations above?

Rising atmospheric temperatures can also be considered as a positive feedback loop: the increase of temperature, through the melting of reflective ice caps, as well as the greenhouse gas increase- itself a positive feedback loop-, causes temperature to increase even further.

To watch a video on how humans affect the carbon dioxide cycle, take a look at this presentation by the Pacific Institute for Climate Solutions:


Follow these questions as a thought experiment, designed to go through the concept of balance of rates, positive feedback, and finally applying that to carbon dioxide concentrations in the atmosphere. Some questions may be omitted depending on the grade level:

  1. Imagine you have an empty bathtub, with an open drain. What happens when you open the tap?
    1. Can you fill the bathtub with water like this?
    2. Why not?
  2. How would you make the water level rise?
    1. Can you still make the water level rise when the drain is open? Partly open?
    2. How would you do this?
    3. Why does the water level rise now?
  3. Imagine your bathtub is half full, and the water level does not seem to be rising or falling (the amount of water in the tub stays the same)… but you notice that the tap is open, i.e. there is water coming out of the faucet and into the tub.
    1. What could be happening?
    2. What will happen if you turn the tap down (reduce the rate of water coming out)?
    3. If you turn the tap off?
  4. Now imagine the bathtub as our atmosphere, and the water as the carbon dioxide in it.
    1. What does the faucet represent?
    2. What does the drain represent?
  5. Usually, more water there is in your bathtub, the faster it will drain out (due to water pressure). What if it were the other way around?
    1. What if more water causes it to drain out slower? How is this situation like the greenhouse gases in the atmosphere?
    2. What if more water causes it to flow in faster? How is this situation like the greenhouse gases in the atmosphere?
    3. Hint: think about the relationship between greenhouse gases, the greenhouse effect, the temperature in the atmosphere, and what kinds of things that might cause.
  6. What are the limitations to the bathtub analogy?
  7. In December 2015, 195 countries agreed to reduce enough carbon emissions to keep global warming “to well below 2 degrees C” as part of the Paris Climate Conference.
    1. If all countries were to stop all their carbon dioxide and other greenhouse gas emissions (hypothetically), what would happen to the carbon dioxide concentrations in the atmosphere?
    2. Using what you know about the greenhouse effect, what would happen to the Earth’s global mean temperature? Would it keep on increasing?
    3. What if countries cut their carbon dioxide emissions in half?
  8. Challenge: Can you calculate the change in the amount of carbon dioxide in the atmosphere given a number of years and the total rates of absorption and emission for that period of time? (Assume these rates are constant.)
    1. Can you think of a way that will work if the rates are not constant?
    2. Can you express this change of carbon dioxide (gain or loss) in terms of a mathematical equation? (What does it mean if you get a negative number?)
    3. Hint: make up, or look up, some example values for the rates and number of years, and try it out!


Post new comment

Please note that these comments are moderated and reviewed before publishing.

The content of this field is kept private and will not be shown publicly.
By submitting this form, you accept the Mollom privacy policy.

a place of mind, The University of British Columbia

C21: Physics Teaching for the 21st Century
UBC Department of Physics & Astronomy
6224 Agricultural Road
Vancouver, BC V6T 1Z1
Tel 604.822.3675
Fax 604.822.5324

Emergency Procedures | Accessibility | Contact UBC | © Copyright The University of British Columbia