Respuesta :
Explanation:
Expression for vertical fluid pressure is as follows.
[tex]\frac{dP}{dh} = \rho \times g[/tex] ........... (1)
where, p = pressure
[tex]\rho[/tex] = density
g = acceleration due to gravity
h = height
Since, g is negative then it means that an increase in height will lead to decrease in pressure.
Also, expression for density according to ideal gas law is as follows.
[tex]\rho = \frac{mP}{kT}[/tex] .......... (2)
where, m = average mass or air molecule
P = pressure at a given point
k = Boltzmann constant
T = temperature in kelvin
Substituting values from equation (2) into equation (1) as follows.
[tex]\frac{dP}{dh} = \rho \times g[/tex]
[tex]\frac{dP}{dh} = \frac{mP}{kT} \times g[/tex]
[tex]\frac{dP}{P} = \frac{mg}{kT}dh[/tex]
Now, on applying integration on both the sides we get the following.
[tex]ln\frac{P_{h}}{P^{o}} = \frac{-mgh}{kT}[/tex]
or, [tex]P_{h} = P_{o}e^{\frac{-mgh}{kT}}[/tex]
where, [tex]P_{h}[/tex] = pressure at height h
[tex]P_{o}[/tex] = pressure at reference point
Thus, we can conclude that the correlation between pressure and elevation for an ideal gas is [tex]P_{h} = P_{o}e^{\frac{-mgh}{kT}}[/tex].
