Concept explainers
(a)
The radius of the first Bohr orbit of the hydrogen atom considering that the electron is bound to the proton by gravitational force (rather than electrostatic force)
Answer to Problem 94QAP
The radius of the first Bohr orbit of the hydrogen atom is given by the following expression
Explanation of Solution
Given:
It is assumed that the electron is bound to the proton by gravitational force (rather than electrostatic force) in hydrogen atom.
Formula used:
In this case the
where me and mp are electron and proton mass respectively, G is the gravitational constant.
According to Bohr quantization principle
So mathematically we can write it as
where, me is the mass of the electron, v and r are the velocity of the electron and radius of the electron orbit respectively, n is a positive integer.
Calculation:
From Eq. (1.1) we can write
Putting n =1 for the 1st Bohr orbit in Eq. (1.2) we get
Using the expression for v in Eq. (1.3) we get
Conclusion:
Therefore, the radius of the first Bohr orbit of the hydrogen atom is given by the following expression
(b)
The energy of the electron in the first Bohr orbit
Answer to Problem 94QAP
The energy expression for the electron in the 1st Bohr orbit is given by
Explanation of Solution
Given:
It is assumed that the electron is bound to the proton by gravitational force (rather than electrostatic force) in hydrogen atom.
Calculation:
Now substituting the expression for the radius of the 1st Bohr orbit r given by Eq. 1.5 in Eq. 1.4 we get the expression for the velocity of the electron as
So the kinetic energy of the electron is
The gravitational potential energy of the electron is given by
Now substituting the expression for r from Eq. 1.5 in Eq. 1.8 we get
Therefore the total energy of the electron in the 1st Bohr orbit is given by
Now using the expressions for Ek (Eq. 1.7) and Ep (Eq. 1.9) we get the total energy as
Conclusion:
Therefore, the energy expression for the electron in the 1st Bohr orbit is given by
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Chapter 26 Solutions
COLLEGE PHYSICS
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