Question

A stationary vertical cylindrical container of very large height filled with a gas of molar mass $M$ at constant Temperature. The pressure at the bottom is $P_{1}$ and at the top is $P_{2}$. If the acceleration due to gravity is assumed to be constant for the whole cylinder, which is equal to $g$. Then the height of the cylinder is:

• A. $\frac{R T}{M g} \ln \frac{P_{1}}{P_{2}}$
• B. $\frac{R T}{2 M g} \ln \frac{P_{1}}{P_{2}}$
• C. $\frac{2 R T}{M g} \ln \frac{P_{1}}{P_{2}}$
• D. $\frac{R T}{3 M g} \ln \frac{P_{1}}{P_{2}}$

Answer 1

Answer: (A)

$d P=g \rho d y$
$\Rightarrow \int\limits_{P_{1}}^{P_{2}} \frac{d P}{P}=\frac{M g}{R T}$
$\Rightarrow H=\int\limits_{0}^{H} d y$
$\Rightarrow H=\frac{R T}{M g} \ln \frac{P_{1}}{P_{2}}$