The density of the atmosphere can be estimated using the Boltzmann distribution
ID: 924095 • Letter: T
Question
The density of the atmosphere can be estimated using the Boltzmann distribution and the gravitational potential energy of the gas molecules. Assume that the air temperature is (20 degree C) and use the molecular weight of diatomic nitrogen (N_2). Calculate the following ratios: The probability that a molecule is at sea level to the probability that it is at 1600 meters (Deliver). The probability that a molecule is at sea level to the probability that it is at 320(1 meters (tall peak in Rockies). The probability that a molecule is at sea level to the probability that it is at 9000 meters (cruising altitude of commercial airplane).Explanation / Answer
Given:
T of the air is assumed to be 20 0C = 273.15 + 20.0 = 293.15 K
We are asked to use molar mass of N2.
a).
The probability that a molecule is at sea level to the probability that is at 1600 meter.
Lets p1 is the probability at see level.
P2 is at height h
We assumed the system has constant T so the difference in these two states are due to the energy. And in this case the potential energy.
Formula for ratio is as follow
(P2/P1) = Exp ( - mgh / kBT )
In this equation we have m is mass in kg ( we convert mass of N2)
h is height in meter, g is gravitational constant. KB is boltzman constant and its value is = 1.38 E-23 J/K T is temperature in K .
Mass in kg = 28 amu x 1.66 x 10-27 kg / 1 amu
= 4.65E-26 kg
Gravitational constant = 9.8 m/ s2
Lets plug all these value in order to get probability ratio.
First at height h = 1600 m
(P2/P1) = Exp [ -(4.65 E-26 x 9.8 x 1600 ) / ( 1.38 E-23 x 293.15 )]
= 0.84
So at height 1600 m the probability ratio (height to sea level ) is 0.84
b). At h = 3200
(P2/P1) = Exp [ -(4.65 E-26 x 9.8 x 3200 ) / ( 1.38 E-23 x 293.15 )]
= 0.70
So at height 3200 m the probability ratio (height to sea level ) is 0.70
c). At height h = 9000
(P2/P1) = Exp [ -(4.65 E-26 x 9.8 x 9000 ) / ( 1.38 E-23 x 293.15 )]
= 0.36
So at height 9000 m the probability ratio (height to sea level ) is 0.70