1.3: Equation of State — Ideal Gas Law
- Page ID
- 46136
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Air pressure is caused by the collisions of rapidly, randomly moving air molecules, so you might expect pressure to increase when there are more molecules in one place (higher density, \(\rho\)) and when they are moving faster (higher temperature, \(T\)). The relationship between pressure, density, and temperature is called the Equation of State.
The gases in the atmosphere have a simple equation of state called the Ideal Gas Law.
For dry air (no water vapor present), the ideal gas law is
\[P=\rho \times R_d \times T \nonumber \]
The ideal gas law assumes that atmospheric gases act in an ideal manner, meaning that there are few intermolecular forces, and that the size of the molecules is small compared to the space between them. In almost all cases, gases in the atmosphere can be assumed to be ideal gases. Again, this is another example of a simplifying assumption we use to be able to approximate relationships in the atmosphere.
The above figure illustrates the effect that air temperature has on its density and pressure. The two air columns above Cities 1 and 2 in (a) have the same temperature, contain the same amount of mass, and have the same volume, so their density is the same. The pressure exerted on the surface is the same in both cities because the pressure at the surface is related to the number of air molecules above.
In (b) the air temperature above City 1 is lower than the air temperature above City 2. Because of this the air molecules in the column above City 1 are moving more slowly, and take up a smaller volume. Likewise, in City 2 the air molecules are moving more quickly and the volume in the column is larger. It only takes a smaller cold air volume to apply the same pressure on the surface as a larger warm air volume. The surface pressures are the same but the temperatures are not.
The surface pressure may be the same in (b) but the pressure aloft is not, which results in air movement that can be seen in (c). At the same altitude above Cities 1 and 2 in (c), there is more air above the same level in City 2 than in 1. Locally this creates a higher pressure aloft over the warmer City 2. Because of the difference in pressure aloft, airflow is created that moves from the higher pressure toward the lower pressure. This airflow caused by a localized difference in air pressure is what we call wind.
The force that arises due to a difference in air pressure is called the pressure gradient force (PGF). The word “gradient” here refers to a change over a distance. A pressure gradient force is the force that drives the wind. This concept will be further explored in Chapter 10.
The movement of air from City 2 to City 1 creates a falling surface pressure in City 2 and a rising surface pressure in City 1.