Question

Assume that the temperature remains essentially constant in the upper part of the atmosphere. Obtain an expression for the variation in pressure with height in the upper atmosphere. The mean molecular weight of air is M. Assume a variable $P_O$ for the pressure at a height h = 0, where h = 0 depends on the choice of origin (could very well be at the beginning of the stratosphere, for example).

• A. $P=P_0{e}^{-\frac{Mgh}{RT}}$
• B. $P=P_0{e}^{-\frac{RT}{Mgh}}$
• C. $P=P_0{e}^{\frac{Mgh}{RT}}$
• D. $P=P_0{e}^{\frac{RT}{Mgh}}$.

Answer 1

The correct option is A

$P=P_0{e}^{-\frac{Mgh}{RT}}$

Suppose the pressure at height h is P and that at (h+dh) is (P+dP). Then -
$dP=-\rho gdh$

Now considering any small volume $\Delta V$ of air of mass $\Delta m,$

$P\Delta V=nR\Delta T=\frac{\Delta m}{M}R\Delta T$

$\Rightarrow P=[\frac{\Delta m}{\Delta V}\times \frac{RT}{M}]=\frac{\rho RT}{M}$

$\Rightarrow \begin{array}{c}\rho =\frac{M}{RT}P\end{array}$.

Substituting -

$dP=-\frac{M}{RT}Pgdh$

$\Rightarrow \underset{P_0}{\overset{P}{\int }}\left(\frac{dP}{P}\right)=\underset{0}{\overset{h}{\int }}(-\frac{M}{RT}g)dh$

$\Rightarrow ln\left(\frac{P}{P_0}\right)=-\left(\frac{Mgh}{RT}\right)$,

where $P_0$ is the pressure at h = 0

Thus, $\begin{array}{c}P=P_0{e}^{-\left(\frac{Mgh}{RT}\right)}\end{array}$,