hey guys take a look at this very much interesting problem in a hypothetical experiment it established that the potential energy between an electron and a proton at a separation R is given by U equal K * the logarithm of the distance R where K is a real constant and all units are in the International System it can be state that b radius for an orbit n and the energy at level n are respectively how much so it was given the expression for the potential energy between the electron and a proton and this given expression is
absolutely different from the electric potential energy so we can realize this is not an electric force in order to find the expression of this Force we must calculate the derivative of this potential energy related to the distance R let's do it now at this point you must remember that the derivative of the natural log of x in respect to S it's 1 /x we're going to use this information so let's move on the derivative will be minus K Over the distance r at this point you must be worried about this Minos Sino but let me
explain this minus Sino only tell us this is a attrative system but this is not important to us at this moment now let's remember the second Newton's law this force will play the role of the centripetal force and from here we can find two important results so here is the expression of the orbital velocity of this electron from here we can also obtain the expression of the the kinetic energy of this electron let us write two Below in the both sides and here it is I know it's a bit strained but the kinetic energy of
this electron it independ of the distance R it's not my fault let's move on now as we are considering B's model we have to considerate the quantization of the angular momentum of this electron MVR equals NH over 2 pi so at this point we can find B's radius remember we have already found the expression of the orbital velocity of this electron so we must replace this expression here let's do it now so here it is the expression of Bar's radius let's move on now to calculate the total energy of the system we must add up
the kinetic and the potential energy of the system the potential energy was given K natural log of R plus the kinetic energy we have already found over here let's replace and when you take a look at the options of this problem we realize we got a I have some trick here to find one of these Expressions so at this point I will write a two below and a two above times the natural log of r + k / 2 so what is the trick at this point you must remember this property of the logarithms so
I'm going to move up this number two and it will become the exponent of this R so now I'm going to factorize this expression we have the common factor k/ 2 and then between the brackets we have but remember we have already found this r so we must replace it here squared let's do it now so we finally found the expression of the total energy of the system and if you take a look carefully in the options you're going to realize that the correct answer is letter b and if you enjoy this explanation click like
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