<p>The equation for relative humidity is <span class="math-container">$$RH=100\times \frac{e}{e_s(T)}=100\times \frac{e_s(T_d)}{e_s(T)} \tag{1}$$</span>
Where <span class="math-container">$T_d$</span> is the dewpoint temperature and <span class="math-container">$T$</span> is the temperature, <span class="math-container">$e$</span> is the water vapor pressure, and <span class="math-container">$e_s$</span> is the saturation vapor pressure, which is also known as the <a href="https://glossary.ametsoc.org/wiki/Clausius-clapeyron_equation" rel="noreferrer">Clausius Clapeyeron equation.</a> While the previous link has a decent definition and equation, my preferred equation (particularly because it is derivable) is the <a href="https://en.wikipedia.org/wiki/Clausius%E2%80%93Clapeyron_relation#Ideal_gas_approximation_at_low_temperatures" rel="noreferrer">low temperature approximation</a>:</p>
<p><span class="math-container">$$e_s(T)= e_s(273 \mathrm{K})\exp\left[\frac{L_v}{R_v}\left(273.15^{-1}-T^{-2}\right)\right] \tag{2}$$</span>
where <span class="math-container">$T$</span> is the temperature or dewpoint temperature <strong>in Kelvin</strong>, <span class="math-container">$L_v$</span> is the <a href="https://glossary.ametsoc.org/wiki/Latent_heat" rel="noreferrer">latent heat of vaporization</a>,<span class="math-container">$e_s(273 \mathrm{K})=6.11 hPa$</span>, and <span class="math-container">$R_v$</span> is the <a href="https://glossary.ametsoc.org/wiki/Gas_constant" rel="noreferrer">specific gas constant for water vapor</a>. Note that <span class="math-container">$(2)$</span> is for liquid water. You may be able to replace <span class="math-container">$L_v$</span> for <span class="math-container">$L_s$</span> for sublimation/deposition. Also note, that this is the &quot;pure&quot; form of the relative humidity equation. <a href="https://en.wikipedia.org/wiki/Raoult%27s_law" rel="noreferrer">The presence of solutes</a> (<a href="https://en.wikipedia.org/wiki/Cloud_condensation_nuclei" rel="noreferrer">Cloud condensation nuclei</a>) may lower the saturation vapor pressure, increasing the actual relative humidity. I'll let combining <span class="math-container">$(1)$</span> and <span class="math-container">$(2)$</span> be an exercise left for the reader.</p>

CC BY-SA 4.0
2020-12-15T12:19:22.987Z
Thanks, the first link did the trick. But just out of curiosity, what's with Lv from (2)? (I don't know meteorology/physics well enough, so I can only guess what's happening). Do I have to compute it for any particular temperature if it were to use the formula derived from (1) and (2)?
Darius V.
CC BY-SA 4.0
2020-12-15T16:39:48.893Z
$L_v$ is the latent heat of vaporization (the amount of energy it takes to turn water vapor into liquid water). Technically, [t does vary with temperature] [https://en.wikipedia.org/wiki/Latent_heat#Specific_latent_heat_for_condensation_of_water_in_clouds], but it is assumed to be a small amount. I usually use 2.5E6 J/kg, Obviously, if the temperature is below freezing, then it is more likely ice would form rather than liquid water (in which case, follow the hyperlinks for deposition).
BarocliniCplusplus
CC BY-SA 4.0
2020-12-14T16:54:38.980Z
2020-12-14T16:54:38.980Z
BarocliniCplusplus
<p>I would like to find a formula to approximate RH based on known <em>air temperature</em> and <em>dew point</em>. I need the approximation to be valid at least between -20...+40 degrees Celsius. I read the accepted answer on <a href="https://earthscience.stackexchange.com/questions/16570/how-to-calculate-relative-humidity-from-temperature-dew-point-and-pressure">this question</a>, but unfortunately</p>
<ul>
<li>the link for the document where the tabulated values were retrieved from is no longer working</li>
<li>using those values yield weird results. I tried it (the RH formula with the constants in the first row of the table) with T = 16.15 and TD (dew point) = -4.45, and it gives 215% RH (could it be <span class="math-container">$100^{m*...}$</span> instead of <span class="math-container">$100 *10^{m*...}$</span>?...)</li>
</ul>
<p>Any help would be appreciated. Thanks.</p>
<p><strong>UPDATE</strong></p>
<p>Thanks @BarocliniCplusplus for his answer. If anyone reading this needs an implementation in Python to approximate the RH, here is one:</p>
<pre><code>RH = 100*(math.exp((17.625*TD)/(243.04+TD))/math.exp((17.625*T)/(243.04+T)))
</code></pre>
<p>where T is the temperature in Celsius, and TD the dew point in Celsius. This approximation is derived from the &quot;Conclusions&quot; section of <a href="https://journals.ametsoc.org/view/journals/apme/35/4/1520-0450_1996_035_0601_imfaos_2_0_co_2.xml" rel="nofollow noreferrer">this article</a> and is suggested to have relative error of 0.384% or less when used between -40°C and 50°C.</p>

CC BY-SA 4.0
2021-10-04T19:20:23.747Z
The update takes a few leaps and would be best as an answer so people can vote upon it's validity.  That said, it appears to be right, as you usually can use the same vapor pressure equations with T and Td respectively to get e and es, and then you canceled the multiplier.  That said, explanation of how you got there helps others.  But I did get it to match other RH calculators, so seems it may be valid.
JeopardyTempest
CC BY-SA 4.0
2021-10-16T09:58:14.733Z
@JeopardyTempest I added back the *100 part, as RH is usually expressed as percentage rather than as fraction. The formula is the same as the one found [here](https://bmcnoldy.rsmas.miami.edu/Humidity.html). Validity range/error might be different than what you added, according to [this other resource](https://bmcnoldy.rsmas.miami.edu/humidity_conversions.pdf) found on the same website.
Darius V.
CC BY-SA 4.0
2021-10-16T10:14:42.737Z
As for the derivation of the formula, I suppose it's directly equation (21) in the [paper](https://journals.ametsoc.org/view/journals/apme/35/4/1520-0450_1996_035_0601_imfaos_2_0_co_2.xml) applied to the RH formula (see the accepted answer).
Darius V.
<meteorology><atmosphere><temperature><humidity>
Relative Humidity approximation from Dew Point and Temperature
Darius V.
# Relative Humidity approximation from Dew Point and Temperature

(Asked by: username_0 on 2020-12-14)

I would like to find a formula to approximate RH based on known *air temperature* and *dew point*. I need the approximation to be valid at least between -20...+40 degrees Celsius. I read the accepted answer on [this question](https://earthscience.stackexchange.com/questions/16570/how-to-calculate-relative-humidity-from-temperature-dew-point-and-pressure), but unfortunately

* the link for the document where the tabulated values were retrieved from is no longer working
* using those values yield weird results. I tried it (the RH formula with the constants in the first row of the table) with T = 16.15 and TD (dew point) = -4.45, and it gives 215% RH (could it be $100^{m\*...}$ instead of $100 \*10^{m\*...}$?...)

Any help would be appreciated. Thanks.

**UPDATE**

Thanks @username_2 for his answer. If anyone reading this needs an implementation in Python to approximate the RH, here is one:

```
RH = 100*(math.exp((17.625*TD)/(243.04+TD))/math.exp((17.625*T)/(243.04+T)))

```

where T is the temperature in Celsius, and TD the dew point in Celsius. This approximation is derived from the "Conclusions" section of [this article](https://journals.ametsoc.org/view/journals/apme/35/4/1520-0450_1996_035_0601_imfaos_2_0_co_2.xml) and is suggested to have relative error of 0.384% or less when used between -40°C and 50°C.

**Tags:** meteorology, atmosphere, temperature, humidity

> **Comments:**
> **username_1**: The update takes a few leaps and would be best as an answer so people can vote upon it's validity.  That said, it appears to be right, as you usually can use the same vapor pressure equations with T and Td respectively to get e and es, and then you canceled the multiplier.  That said, explanation of how you got there helps others.  But I did get it to match other RH calculators, so seems it may be valid.  
> **username_0**: @username_1 I added back the *100 part, as RH is usually expressed as percentage rather than as fraction. The formula is the same as the one found [here](https://bmcnoldy.rsmas.miami.edu/Humidity.html). Validity range/error might be different than what you added, according to [this other resource](https://bmcnoldy.rsmas.miami.edu/humidity_conversions.pdf) found on the same website.  
> **username_0**: As for the derivation of the formula, I suppose it's directly equation (21) in the [paper](https://journals.ametsoc.org/view/journals/apme/35/4/1520-0450_1996_035_0601_imfaos_2_0_co_2.xml) applied to the RH formula (see the accepted answer).  

---

## 1 answers

### Answer by username_2 on 2020-12-14. Score: 8 | Accepted answer

The equation for relative humidity is $$RH=100\times \frac{e}{e\_s(T)}=100\times \frac{e\_s(T\_d)}{e\_s(T)} \tag{1}$$
Where $T\_d$ is the dewpoint temperature and $T$ is the temperature, $e$ is the water vapor pressure, and $e\_s$ is the saturation vapor pressure, which is also known as the [Clausius Clapeyeron equation.](https://glossary.ametsoc.org/wiki/Clausius-clapeyron_equation) While the previous link has a decent definition and equation, my preferred equation (particularly because it is derivable) is the [low temperature approximation](https://en.wikipedia.org/wiki/Clausius%E2%80%93Clapeyron_relation#Ideal_gas_approximation_at_low_temperatures):

$$e\_s(T)= e\_s(273 \mathrm{K})\exp\left[\frac{L\_v}{R\_v}\left(273.15^{-1}-T^{-2}\right)\right] \tag{2}$$
where $T$ is the temperature or dewpoint temperature **in Kelvin**, $L\_v$ is the [latent heat of vaporization](https://glossary.ametsoc.org/wiki/Latent_heat),$e\_s(273 \mathrm{K})=6.11 hPa$, and $R\_v$ is the [specific gas constant for water vapor](https://glossary.ametsoc.org/wiki/Gas_constant). Note that $(2)$ is for liquid water. You may be able to replace $L\_v$ for $L\_s$ for sublimation/deposition. Also note, that this is the "pure" form of the relative humidity equation. [The presence of solutes](https://en.wikipedia.org/wiki/Raoult%27s_law) ([Cloud condensation nuclei](https://en.wikipedia.org/wiki/Cloud_condensation_nuclei)) may lower the saturation vapor pressure, increasing the actual relative humidity. I'll let combining $(1)$ and $(2)$ be an exercise left for the reader.

> **Comments:**
> **username_0**: Thanks, the first link did the trick. But just out of curiosity, what's with Lv from (2)? (I don't know meteorology/physics well enough, so I can only guess what's happening). Do I have to compute it for any particular temperature if it were to use the formula derived from (1) and (2)?  
> **username_2**: $L_v$ is the latent heat of vaporization (the amount of energy it takes to turn water vapor into liquid water). Technically, [t does vary with temperature] [https://en.wikipedia.org/wiki/Latent_heat#Specific_latent_heat_for_condensation_of_water_in_clouds], but it is assumed to be a small amount. I usually use 2.5E6 J/kg, Obviously, if the temperature is below freezing, then it is more likely ice would form rather than liquid water (in which case, follow the hyperlinks for deposition).  

---

