<p>No. Not in a significant way.</p>

<p>The full width at half maximum (FWHM) of a spectral line widened by thermal <a href="https://en.wikipedia.org/wiki/Doppler_broadening" rel="nofollow noreferrer">Doppler broadening</a> is</p>

<p><span class="math-container">$\Delta f_{\text{FWHM}}={\sqrt{\frac{8kT\ln(2)}{mc^{2}}}}f_{0}$</span></p>

<p>where <span class="math-container">$m$</span> is the mass of the emitting particle, <span class="math-container">$T$</span> is the temperature, <span class="math-container">$f_{0}$</span> is the frequency of the spectral line, and <span class="math-container">$k$</span> is the <a href="https://en.wikipedia.org/wiki/Boltzmann_constant" rel="nofollow noreferrer">Boltzmann constant</a>.</p>

<p>Let's look at the CO<span class="math-container">$_2$</span> absorption lines:</p>

<p><a href="https://i.sstatic.net/REexQ.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/REexQ.png" alt="enter image description here"></a></p>

<p>(Image from Wikipedia commons, <a href="https://commons.wikimedia.org/wiki/File:Atmospheric_Transmission.png" rel="nofollow noreferrer">source</a>)</p>

<p>Let's take for example the band roughly between 12 and 20 <span class="math-container">$\mu m$</span>. That corresponds to frequencies from 15 to 25 THz, with a mean of 20 THz.</p>

<p>If we solve the above equation for the mass of a CO<span class="math-container">$_2$</span> molecule (<span class="math-container">$44.01/6.02\times 10^{23}$</span> g), a frequency of 20 THz and an Earth's mean temperature of let's say 15°C (288.15 K) we get 1158 kHz broadening, and for an scenario 2°C warmer it would be 1162 kHz.</p>

<p>Therefore, the width of the absorption band would grow 8 kHz (4 kHz on either side). Which considering the original width of 10 THz <strong>corresponds to a 0.00008% increase</strong>. This could be roughly equated to an analogous increase in absorbed energy, <strong>which is absolutely negligible for all practical effects</strong>.</p>

CC BY-SA 4.0
2019-03-03T03:18:37.837Z
2019-03-03T03:18:37.837Z
Camilo Rada
<p>I've read not long ago that the absorption line of the <span class="math-container">$\text{CO}_2$</span> becomes larger with temperature thanks to Doppler effect (if it's false it would be great to tell me). My question is the following: Does this imply, like I think, that more energy is "stored" on Earth (thanks to greenhouse effect) and so play a role in climate change? </p>

<p>If the answer is yes, how big is this role? </p>

<geophysics><climate-change><temperature><co2><greenhouse-gases>
Does the extension of the absorption line of $\text{CO}_2$ with temperature play a role in the climate change?
thephysics17
# Does the extension of the absorption line of $\text{CO}_2$ with temperature play a role in the climate change?

(Asked by: username_0 on 2018-06-02)

I've read not long ago that the absorption line of the $\text{CO}\_2$ becomes larger with temperature thanks to Doppler effect (if it's false it would be great to tell me). My question is the following: Does this imply, like I think, that more energy is "stored" on Earth (thanks to greenhouse effect) and so play a role in climate change?

If the answer is yes, how big is this role?

**Tags:** geophysics, climate-change, temperature, co2, greenhouse-gases


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## 1 answers

### Answer by username_1 on 2019-03-03. Score: 2

No. Not in a significant way.

The full width at half maximum (FWHM) of a spectral line widened by thermal [Doppler broadening](https://en.wikipedia.org/wiki/Doppler_broadening) is

$\Delta f\_{\text{FWHM}}={\sqrt{\frac{8kT\ln(2)}{mc^{2}}}}f\_{0}$

where $m$ is the mass of the emitting particle, $T$ is the temperature, $f\_{0}$ is the frequency of the spectral line, and $k$ is the [Boltzmann constant](https://en.wikipedia.org/wiki/Boltzmann_constant).

Let's look at the CO$\_2$ absorption lines:

[![enter image description here](https://i.sstatic.net/REexQ.png)](https://i.sstatic.net/REexQ.png)

(Image from Wikipedia commons, [source](https://commons.wikimedia.org/wiki/File:Atmospheric_Transmission.png))

Let's take for example the band roughly between 12 and 20 $\mu m$. That corresponds to frequencies from 15 to 25 THz, with a mean of 20 THz.

If we solve the above equation for the mass of a CO$\_2$ molecule ($44.01/6.02\times 10^{23}$ g), a frequency of 20 THz and an Earth's mean temperature of let's say 15°C (288.15 K) we get 1158 kHz broadening, and for an scenario 2°C warmer it would be 1162 kHz.

Therefore, the width of the absorption band would grow 8 kHz (4 kHz on either side). Which considering the original width of 10 THz **corresponds to a 0.00008% increase**. This could be roughly equated to an analogous increase in absorbed energy, **which is absolutely negligible for all practical effects**.


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