Why does relative humidity drop if I open the window in winter? What about the temperature drop?

Question

I understand that 40% rel. hum. at 0 °C is less absolute humidity than 40% at 20 °C. However, assume that we have 2$~m^3$ of air. Both are 40% humid at 20 °C. Now we exchange 1$~m^3$ with air that is 40% humid at 0°C. Then according to this picture we should get 7$~g\, m^{-3}$ and 2$~g\, m^{-3}$ of absolute humidity for each part. So that's 4.5$~g\, m^{-3}$ on average. Now the temperature should be 10 °C (right?). Then, 4.5$~g\, m^{-3}$ means a relative humidity of around 50%! So why does relative humidity drop in reality? Is it because the temperature would actually be higher somehow?

Answer 1

Let's do this with accurate equations. We know how to calculate the saturated water vapour pressure for temperature $T$:

$$e_s(T)=e_{s0}e^{\frac{h_i}{R_w}(\frac{1}{T_{s0}}-\frac{1}{T})}=610\text{ Pa} \times e^{5423(\frac{1}{273\text{ K}}-\frac{1}{T})}$$

$e_s$ is saturated water vapour pressure as function of $T$ in Pascals
$T$ is the temperature in Kelvins

So, the saturated water vapour pressure at $T=0°C=273K$ is equal to $e_s(273K)=610 Pa$. Similarly, $e_s(293K)=2360Pa$ and $e_s(283K)=1230Pa$.

So we have two parcels of air: 1. 20°C, 40% and 2. 0°C, 40%. What are the masses of the water vapour in both parcels? $$e=f\times e_s(T)$$ $f$ is relative humidity, $e$ is the water vapour pressure

$$m(T)=\frac{eV}{R_w T}=\frac{f\times e_s(T)\times V}{R_w\times T}=\frac{fe_s(T)V}{461\frac{J}{kg}T}$$

$$m(293K)=0.0070kg$$ $$m(273K)=0.0019kg$$

The next step is to combine these two masses:

$$M=m(293K)+m(273K)=0.0089kg$$

So the density is $\rho=\frac{M}{V}=0.0045\frac{kg}{m^3}$

The water vapour pressure is given by: $$e=\rho RT=587Pa$$ The relative humidity is then: $$f=\frac{e}{e_s(283K)}=0.477$$

So this is around 48% of the relative humidity.

Why did I put here so many equations? Simple! These equations hold on high degree of accuracy, so they can't be wrong. Why would the relative humidity drop in reality?

---

Also, for Skew-T Log-P lovers: This solution is even more concise and less error prone.