<p>As far as I can understand your question, There is not a direct mathematical relationship between CO<sub>2</sub> and warming, although there is an approximate mathematical relationship between atmospheric CO<sub>2</sub> concentration and <em><a href="https://en.wikipedia.org/wiki/Radiative_forcing" rel="nofollow noreferrer">radiative forcing</a></em> which was established in 1998 by <a href="https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98GL01908" rel="nofollow noreferrer">this study</a>:</p>
<p><span class="math-container">$${\displaystyle \Delta F=5.35\times \ln {C \over C_{0}}~~(\mathrm {W} ~\mathrm {m} ^{-2})\,}$$</span></p>
<p>Where C is the CO<sub>2</sub> concentration in question and C<sub>0</sub> is the baseline CO<sub>2</sub> concentration (typically 278 ppm is used as this was the CO2 concentration in the atmosphere in 1750 before significant anthropogenic CO<sub>2</sub> emissions).</p>
<p>Unfortunately, this is just an estimate, Earth's climate is extremely complex and the issue is that we still don't fully understand every variable and aspect of the system. The biggest of the uncertainties relates to <a href="https://en.wikipedia.org/wiki/Climate_sensitivity" rel="nofollow noreferrer">climate sensitivity</a> - this refers to the various earth processes that create feedback loops, which can both amplify and reduce changes in temperature. Without complete knowledge of all variables that influence the Earth's climate (and there are many) and how they all interact with each other, we cannot determine the precise mathematical relationship between CO<sub>2</sub> and warming, we can only make estimations.</p>

CC BY-SA 4.0
2022-03-01T03:33:51.123Z
If my calculus hasn't deserted me after all these years, I think that (taking the derivative), along with the "commonly accepted value of climate sensitivity parameter λ, 0.8 K /(W m−2)" (your Wikipedia link), leads to a resulting approximation that a 1 ppm increase = a $4.28/C$ Celsius change in temperature, so at around the current 415 ppm, that's 4.28/415 ≈ roughly a 0.01 °C per ppm increase... which seems to make sense a very very rough value to me?  So hopefully that finishes the idea to fullness for the question's focus about the unit increase.
JeopardyTempest
CC BY-SA 4.0
2022-03-01T03:46:17.143Z
Well I guess you'd wants tons as to match your question.  https://www.theworldcounts.com/challenges/climate-change/global-warming/global-co2-emissions/story suggests we release 43.1 billion tons a year, and https://www.climate.gov/news-features/understanding-climate/climate-change-atmospheric-carbon-dioxide looks like roughly a 2 ppm increase per year, so that yet more crudely offers suggestion that each ton equates to around a 5 x10^-13 degree Celsius change (aka half a trillionth of a degree Celsius)? Thankfully it takes a lot of CO2 to cause increases. Unfortunately there's a lot produced.
JeopardyTempest
CC BY-SA 4.0
2022-03-04T21:45:46.750Z
@JeopardyTempest  "Unfortunately there's a lot produced." Yes, more than all other waste combined, several times over. The only thing we make more of now is "aggregate" ie crushed rock. Leave out crushed rock and it gets close to as much as everything else we produce put together.
Ken Fabian
CC BY-SA 4.0
2022-03-01T00:56:05.987Z
2022-03-01T02:12:39.297Z
2022-03-01T02:12:39.297Z
Tim51092
<p>I'm trying to find the impact of X tonnes of CO<sub>2</sub> on global temperature so I can infer the unit impact of CO<sub>2</sub> on temperature. While we know there is a positive correlation, I can't seem to find a study that provides a coefficient.</p>
<p>Any guidance would be helpful.</p>

<climate-change><climate><temperature>
What is the unit impact of $\small\sf{CO_2}$ on global temperature?
Ali
# What is the unit impact of $\small\sf{CO_2}$ on global temperature?

(Asked by: username_0 on 2022-02-28)

I'm trying to find the impact of X tonnes of CO2 on global temperature so I can infer the unit impact of CO2 on temperature. While we know there is a positive correlation, I can't seem to find a study that provides a coefficient.

Any guidance would be helpful.

**Tags:** climate-change, climate, temperature


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## 1 answers

### Answer by username_1 on 2022-03-01. Score: 5 | Accepted answer

As far as I can understand your question, There is not a direct mathematical relationship between CO2 and warming, although there is an approximate mathematical relationship between atmospheric CO2 concentration and *[radiative forcing](https://en.wikipedia.org/wiki/Radiative_forcing)* which was established in 1998 by [this study](https://agupubs.onlinelibrary.wiley.com/doi/10.1029/98GL01908):

$${\displaystyle \Delta F=5.35\times \ln {C \over C\_{0}}~~(\mathrm {W} ~\mathrm {m} ^{-2})\,}$$

Where C is the CO2 concentration in question and C0 is the baseline CO2 concentration (typically 278 ppm is used as this was the CO2 concentration in the atmosphere in 1750 before significant anthropogenic CO2 emissions).

Unfortunately, this is just an estimate, Earth's climate is extremely complex and the issue is that we still don't fully understand every variable and aspect of the system. The biggest of the uncertainties relates to [climate sensitivity](https://en.wikipedia.org/wiki/Climate_sensitivity) - this refers to the various earth processes that create feedback loops, which can both amplify and reduce changes in temperature. Without complete knowledge of all variables that influence the Earth's climate (and there are many) and how they all interact with each other, we cannot determine the precise mathematical relationship between CO2 and warming, we can only make estimations.

> **Comments:**
> **username_2**: If my calculus hasn't deserted me after all these years, I think that (taking the derivative), along with the "commonly accepted value of climate sensitivity parameter λ, 0.8 K /(W m−2)" (your Wikipedia link), leads to a resulting approximation that a 1 ppm increase = a $4.28/C$ Celsius change in temperature, so at around the current 415 ppm, that's 4.28/415 ≈ roughly a 0.01 °C per ppm increase... which seems to make sense a very very rough value to me?  So hopefully that finishes the idea to fullness for the question's focus about the unit increase.  
> **username_2**: Well I guess you'd wants tons as to match your question.  https://www.theworldcounts.com/challenges/climate-change/global-warming/global-co2-emissions/story suggests we release 43.1 billion tons a year, and https://www.climate.gov/news-features/understanding-climate/climate-change-atmospheric-carbon-dioxide looks like roughly a 2 ppm increase per year, so that yet more crudely offers suggestion that each ton equates to around a 5 x10^-13 degree Celsius change (aka half a trillionth of a degree Celsius)? Thankfully it takes a lot of CO2 to cause increases. Unfortunately there's a lot produced.  
> **username_3**: @username_2  "Unfortunately there's a lot produced." Yes, more than all other waste combined, several times over. The only thing we make more of now is "aggregate" ie crushed rock. Leave out crushed rock and it gets close to as much as everything else we produce put together.  

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