B1 - variation of pressure with height

background

considering an ideal gas (well-mixed, non-interacting particles):

p V = n R T

in the atmosphere, volume is not well-defined
rewrite in terms of density:

p = \frac{ρ R T}{m_{r}} = ρ R^{'} T

where, $m_{r}$ is the relative (average) molecular mass, and $ρ = n m_{r} / V$

for the atmosphere, $m_{r} \approx 0.8 m_{r} (N_{2}) + 0.2 m_{r} (O_{2}) = 29 {g mol}^{- 1}$
so, $R^{'} = R / m_{r} \approx 287 {J kg}^{- 1} K^{- 1}$ is the gas constant for dry air
partial pressure is the pressure exerted by a particular gas in a mixture, $p_{i}$

p = \sum_{i} p_{i}

molar mixing ratio:

χ_{i, m} = \frac{n_{i}}{n_{t o t a l}} = \frac{p_{i}}{p_{t o t a l}}

mass mixing ratio:

χ_{i} = \frac{m_{i}}{m_{t o t a l}}

atmospheric pressure at sea level: $1 bar = 1000 mbar = 10^{5} Pa$
standard atmospheric pressure: $1 atm = 1.01325 bar = 101325 Pa = 760 torr = 760 mmg Hg$

variation with height

assuming the atmosphere to be isothermal
considering a slice of atmosphere at height $z$ , thickness $Δ z$ , and area $A$
the pressure at the bottom and the top of the slice are $p$ and $p + Δ p$ respectively
remember that the pressure is the force per unit area exerted by the weight of the atmosphere above a given height

p = \frac{F}{A} = \frac{m g}{A}

m g + A Δ p = 0 ⟹ Δ p = \frac{ρ V g}{A} = - ρ g Δ z

this results in the equation of hydrostatic equilibrium

\frac{\partial p}{\partial z} = - ρ g

recall that $p = ρ R^{'} T :$

\begin{matrix} \frac{\partial p}{\partial z} = - \frac{p g}{R^{'} T} \\ \int \frac{d p}{p} = - \int \frac{g}{R^{'} T} d z \\ \ln (p) = - \frac{g z}{R^{'} T} + k \\ p = p_{o} \exp (- \frac{g z}{R^{'} T}) = p_{o} \exp (- \frac{z}{H}) \end{matrix}

where, $H = R^{'} T / g$ is the scale height

the scale height is the height for the pressure to drop by a factor of $e$
it is approximately the measure of the thickness of the troposphere
the decadal scale height $(H_{10})$ is the height for pressure to drop by a factor of $10$

image: A Wright, lecture notes

note that isothermal atmosphere was assumed, but clearly, there is a temperature dependence
it is however very small, so can be ignored

image: A Wright, lecture notes