Earth’s atmosphere becomes more rarefied at higher elevations. One way to quanti

Question

Earth’s atmosphere becomes more rarefied at higher elevations. One way to quantify this is to use the number density of air particles nV , to wit, the number of particles per unit volume. Take the number density at sea level to be n0. (a) Calculate how this number density varies with increasing altitude y, measured from sea level. For simplicity, consider the atmosphere to be at a single uniform temperature T. Hint: Analyze a parcel of air at an arbitrary height. What’s it doing? What condition therefore applies? (b) What fraction of the value at sea level is the number density near the top of the troposphere, at about an altitude of 11.0 km? (c) Determine the average height of an “air particle.” Hint: The result from (a) can be thought of as a distribution function. How can these functions yield (weighted) averages? Refer to Appendix B as necessary. (d) What is the value for this average height for an atmospheric temperature of 10.0C. Recall that the effective molar mass for air is 28.9 u.

Explanation / Answer

a)Air density can be defined as the number of air molecules per unit volume (number density). Near sea level there are about 2.7x1019 molecules per cm3(cubic centimeter) or 4.4x1020 molecules per inch3(cubic inch). Air molecules are held near the earth by gravity. In other words, air has weight. Weigh an empty bag, then fill it with air, it now weighs more. In addition gases, like air, are easily compressed, i.e., squeeze a gas together and its number density increases. In other words, we say gases are compressible because they can easily be squeezed into a smaller volume. Solids and liquids on the other hand are not easily compressed.

The weight of all of the air above a given point in the atmosphere squeezes air molecules closer together, which causes their numbers in a given volume to increase (increase in number density). The more air above a level (and hence the more weight of air above a level), the greater the squeezing effect (or compression).

Since air density is the number of air molecules in a given space (volume), air density is typically greatest at the surface or sea level (where it is squeezed by the weight of the entire atmosphere above) and decreases as we move up in the atmosphere because the weight of air above becomes less and hence there is less of a squeezing effect

b)located the surfaceAtmospheric pressure is sometimes expressed in terms of the number of inches in a column of Mercury that are supported by the downward push of the atmosphere. Meteorologists and weather people often do it this way. If the pressure is 1 bar, then the atmosphere will support a column of Mercury which is 29.5 inches tall. Air pressure is also sometimes measured in units of psi where 1 psi = 1 pound per square inch. The inflation limit for automobile and bicycle tires is usually expressed in psi. There certainly are many ways to express air pressure!!! of the Earth to approximately 11 kilometers in altitude. This average as the depth of the troposphere varies with time and location, range 6 - 16 km. Troposphere literally means the "turning or changing sphere" in reference to the fact that this lowest layer of the atmosphere is characterized by strong vertical mixing of air. The upper boundary of the vertical mixing is at the top of the troposphere at the troposphere-tropopause boundary.
Essentially all weather and clouds occur in the troposphere or this overturing layer.
The troposphere contains about 75% of the total mass of the atmosphere.
Typically, the maximum air temperature occurs near the Earth's surface and drops with increasing height at an average rate of 6.5°C per 1000 meters (or 3.6°F per 1000 ft). The reason the temperature usually decreases with increasing height is because the atmosphere is mainly heated from below by the underlying ground surface. The gas molecules in the atmosphere are generally transparent for visible radiation coming from the Sun, i.e., if it is not cloudy, most of the Sun's radiation passes right through the atmosphere, and is absorbed by the ground surface, heating the ground up. Air in contact with the warm ground is heated from below, therefore, the air temperature typically lowers as you move up away from the warm ground.
However, it is not all that uncommon for there to be a shallow layer of air where the air temperature actually increases with increasing altitude. These layers are called temperature inversions and can be very important in weather forecasting

c)Atmospheric pressure is sometimes expressed in terms of the number of inches in a column of Mercury that are supported by the downward push of the atmosphere. Meteorologists and weather people often do it this way. If the pressure is 1 bar, then the atmosphere will support a column of Mercury which is 29.5 inches tall. Air pressure is also sometimes measured in units of psi where 1 psi = 1 pound per square inch. The inflation limit for automobile and bicycle tires is usually expressed in psi. There certainly are many ways to express air pressure!!

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