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MATHEMATICS - CASE STUDY

    Introduction


Electron "Cloud" in an Atom
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A hydrogen atom is composed of one proton at the nucleus and one electron that moves around the nucleus. The electron does not move in a well-defined orbit but occupies a sphere shape "cloud" that surrounds the nucleus. At the lowest energy sate, the "cloud" can be assumed as a sphere centered at the nucleus. The way to describe the motion of the electron is to find the probability that the electron will be found within a certain area.

What is known:

The probability density function for the electron radial location is:

      p(r) = 4/a3r2e-2x/a

where a is a constant. Its integral gives the probability that the electron will be found within a certain area:

      

where r is the radius of the sphere centered at the nucleus where the electron will be found and a is a constant.

     
    Questions

 

  Determine the probability that the electron can be within the spherical volume of radius of 3a.
     
    Approach


 

Use integration by parts. Let:
      u = x2
      dv = e-2x/a dx