Question

Assume that the temperature remains essentially constant in the upper part of the atmosphere. Obtain an expression for the variation of pressure in the upper atmosphere with height. (The mean molecular weight of air is $M$.)

• A. $p=p_0{e}^{-\left(\frac{Mg}{RT}\right)h}$
  
  
• B. $p=p_0{e}^{\left(\frac{Mg}{RT}\right)\frac{h}{2}}$
  
  
• C. $p=p_0{e}^{-\left(\frac{Mg}{RT}\right)\frac{h}{2}}$
  
  
• D. $p=p_0{e}^{-\left(\frac{Mg}{RT}\right)h}$
  
  

Answer 1

The correct option is A $p=p_0{e}^{-\left(\frac{Mg}{RT}\right)h}$





$P\left(h\right),\rho \left(h\right)$
$\begin{array}{cc}dP& =-\rho gdh...\left(1\right)\\ P& =\frac{\rho RT}{M}\Rightarrow \rho =\frac{PM}{RT}\end{array}$

$dP=-\frac{PM}{RT}gdh$

$\begin{array}{cc}{\int }_{P_0}^P\frac{dP}{P}& =-\frac{Mg}{RT}{\int }_0^{h}dh\\ \ln P{|}_{P_0}^P& {=-\left(\frac{Mg}{RT}\right)h|}_0^h\\ \ln P-\ln P_0& =(-\frac{Mg}{RT})h\\ \ln \left(\frac{P}{P_0}\right)& =-\left(\frac{Mg}{RT}\right)h\end{array}$

$\frac{p}{p_0}=e^{-\left(\frac{Mq}{RT}\right)h}$
$p=p_0{e}^{-\left(\frac{Mg}{RT}\right)h}$