7) Numerous pions are created by cosmic rays in the upper atmosphere, but very f

Question

7) Numerous pions are created by cosmic rays in the upper atmosphere, but very few reach sea level. One reason is that most of the pions decay before they reach sea level, and another is that they are quickly absorbed by the nuclei in the atmosphere. To estimate this second effect, answer the following: a) About how many nuclei (of nitrogen and oxygen) are contained in a column of atmosphere of area 1 m2, extending from sea level to the top of the atmosphere? Use the fact that atmospheric pressure is 105 N/m2 b) About how many of these nuclei are needed to completely cover 1 m in a layer one nucleus thick? c) Use these two results to estimate the number of nuclei through which a pion would have to pass traveling from the upper atmosphere down to sea level.

Explanation / Answer

(a)

The force an air column of cross-sectional area A exerts on a surface of earth = Pressure x A

F = 105  x 1 = 105 Newtons

But this force is the weight of the air column. Hence

mg = 105

m = 104 kg taking g = 10 m/s2

Now,

fraction of nitrogen in air by mass is 0.78

fraction of nitrogen in air by mass is 0.21

Thus,

the mass of nitrogen in the column of air = 7.8 x 103 kg

  the mass of oxygen in the column of air = 2.1 x 103 kg

Now

the molecular molar mass of N(i.e., mass of N2) = 2.8 x 10-2 kg

the molecular molar mass of O(i.e., mass of O2) = 3.2 x 10-2 kg

Thus

the No. of moles of N2 = (7.8 x 103)/ (2.8 x 10-2)

= 2.78 x 105

thus the No. of moles of N nuclei = 5.56 x 105

Also,

the No. of moles of O2 = (2.1 x 103)/ (3.2 x 10-2)

= 6.6 x 104

thus the No. of moles of O nuclei = 1.32 x 105

So, the total No. of moles of nuclei = No. of moles of N nuclei + No. of moles of O nuclei

= 6.88 x 105

(b)

Radius of O nucleus = 2.6991 x 10-15 m

Radius of N nucleus = 2.5582 10-15 m

Cross-section area of O nucleus = x (2.6991 x 10-15)2

Cross-section area of N nucleus = x (2.5582 x 10-15)2  

The area of 1 m2 will contain = 0.78 x 2 =

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