4.2: Saturation
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Imagine a closed jar filled halfway with water. At the initial time, more water molecules evaporate from the water surface than the number that return. However, after some time, the number of molecules evaporating from the surface will be equal to the number of molecules condensing back into the water surface. When condensation and evaporation are equal, this is called saturation.
Saturation occurs when air contains the maximum amount of water vapor possible for its given temperature. That is why condensation equals evaporation. If evaporation occurs, the air cannot contain more water vapor, so some must condense. Now let’s get quantitative.
Vapor pressure at saturation
Every gas in the atmosphere exerts pressure, for example, vapor pressure makes up a fraction of the total atmospheric pressure. In the following equation, all of the gases in Earth’s atmosphere contribute to the total atmospheric pressure \(P_{\text {atmosphere }}\).
\[P_{\text {atmosphere }}=P_{N_2}+P_{O_2}+P_{A r}+\ldots . . P_{H_2 O} \nonumber\]
Specifically for water vapor, the more water vapor that is added to the atmosphere, the higher the vapor pressure \(P_{H_2 O}\). The units for vapor pressure are the same as pressure and can be in Pascals, hectoPascals, or kiloPascals. Because we are staying consistent with Roland Stull’s Practical Meteorology textbook, we will use kiloPascals (kPa) throughout this chapter.
The amount of water vapor that the atmosphere can contain depends on temperature. Lower temperature air cannot contain as much water vapor as higher temperature air. If we think of this quantitatively in terms of pressure, saturation vapor pressure refers to the pressure exerted by the movement of water vapor molecules exerted over a surface of liquid water. When the partial pressure exerted by water vapor is equal to the saturation vapor pressure, the air is saturated.
The Clausius-Clapeyron equation gives the approximate relationship between saturation vapor pressure (\(e_s\)) and temperature in the atmosphere
\[e_s \approx e_0 \cdot \exp \left[\frac{L_v}{R_v} \cdot\left(\frac{1}{T_0}-\frac{1}{T}\right)\right] \nonumber \]
where the water-vapor gas constant \(\Re_v\) is 461 J·K–1·kg–1, \(T_0\) is 273.15 K, \(e_0\) is 0.6113 kPa, and \(L_v\) is the latent heat of vaporization, 2.5×106 J·kg–1. This results in \(L_v / \Re_v\) being equal to 5423 K. In this equation, units for temperature must be in Kelvin. Note that in the equation above, \(\exp[x] \) implies the exponential function \(e^x\), but it is written on one line for visual purposes.
The image shows the relationship between temperature and saturation vapor pressure based on the Clausius-Clapeyron equation. Lower temperatures are saturated with respect to water vapor at lower vapor pressures, while higher temperatures need higher vapor pressures to be saturated. Temperature is the primary factor determining water vapor saturation.
In the graph of saturation vapor pressure vs. temperature notice the saturation vapor pressure value at the boiling temperature, 100 °C. The saturation vapor pressure value \(e_s\)(100 °C)=101.325 kPa, is the same value as the atmospheric surface pressure. Water boils at the Earth’s surface when the saturation vapor pressure is equal to the atmospheric pressure, which is why water boils at 100 °C. Will water boil at the same temperature at the top of Mount Everest?