Concept explainers
(a)
The
Answer to Problem 77QAP
Mathematically the angular momentum of an electron can be written as
Explanation of Solution
Given:
Hydrogen atom model according to Bohr quantization principle
Calculation:
According to Bohr quantization principle angular momentum of an electron in an atom is an integral multiple of
So mathematically we can write it as,
(1)
where, m is the mass of the electron, vnis the velocity on the nthorbit, n is a positive integer.
Conclusion:
So mathematically the angular momentum of an electron can be written as
(b)
The radius of the electron orbit in hydrogen atom
Answer to Problem 77QAP
The radius of the electron orbit in hydrogen atom is given by
Explanation of Solution
Given:
Hydrogen atom model according to Bohr quantization principle
Calculation:
Electrostatic force between the orbiting electron and proton supplies the required
(2)
where
(3)
Now using equation (1) in equation (3) we can write,
(4)
Conclusion:
So, the radius of the nthelectron orbit in hydrogen atom is given by
(c)
The kinetic energy of an electron in hydrogen atom
Answer to Problem 77QAP
The kinetic energy of an electron in hydrogen atom is given by
Explanation of Solution
Given:
Hydrogen atom model according to Bohr quantization principle
Calculation:
Kinetic energy of the electron is given by
(5)
Now substituting the expression for rnin equation (1) we obtain
(6)
Now substituting the value of vnin equation (5) we obtain
(7)
Conclusion:
So, the kinetic energy of an electron in hydrogen atom is given by
(d)
The total energy of an electron in hydrogen atom
Answer to Problem 77QAP
The total energy of an electron in hydrogen atom is given by
Explanation of Solution
Given:
Hydrogen atom model according to Bohr quantization principle
Calculation:
Total energy of the electron is given by the sum of kinetic and potential energy
(8)
Knexpression we have already derived in part (c). Now we want to derive an expression for the potential energy Epwhich is the electro static energy between proton and the orbiting electron. So
(9)
Now substituting the value of rnusing equation (4) in equation (9) we get
(10)
Now putting equation (7) and (10) in equation (8) we get
Conclusion:
So, the total energy of an electron in hydrogen atom is given by
(e)
The speed of an electron in hydrogen atom
Answer to Problem 77QAP
The speed of an electron in hydrogen atom is given by
Explanation of Solution
Given:
Hydrogen atom model according to Bohr quantization principle
Calculation:
We have already derived the expression for speed of an electron in part (c) equation (6). The expression for speed in the nth orbit of the electron is given by
Conclusion:
So, the speed of an electron in hydrogen atom is given by
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