Show that the circumference of the Bohr orbit for the hydrogen atom is an integral multiple of the de Broglie wavelength associated with the electron revolving around the orbit.
Solution and Explanation
Since a hydrogen atom has only one electron, according to Bohr's postulate, the angular momentum of that electron is given by: mvr =\(\frac{nh}{2\pi}\) ....(1)
Where, n = 1, 2, 3, …
According to de Broglie's equation: \(\lambda=\frac{h}{mv}\)
or mv = \(\frac{h}{\lambda}\) ....(2)
Substituting the value of 'mv' from expression (2) in expression (1):
\(\frac{hr}{\lambda}\) = \(\frac{nh}{2\pi}\)
or 2πr = nλ ......(3)
Since '2πr' represents the circumference of the Bohr orbit (r), it is proved by equation (3) that the circumference of the Bohr orbit of the hydrogen atom is an integral multiple of de Broglie's wavelength associated with the electron revolving around the orbit.
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Concepts Used:
Structure of Atom
Atomic Structure:
The atomic structure of an element refers to the constitution of its nucleus and the arrangement of the electrons around it. Primarily, the atomic structure of matter is made up of protons, electrons and neutrons.
Dalton’s Atomic Theory
Dalton proposed that every matter is composed of atoms that are indivisible and indestructible.
The following are the postulates of his theory:
- Every matter is made up of atoms.
- Atoms are indivisible.
- Specific elements have only one type of atoms in them.
- Each atom has its own constant mass that varies from element to element.
- Atoms undergo rearrangement during a chemical reaction.
- Atoms can neither be created nor be destroyed but can be transformed from one form to another.
Cons of Dalton’s Atomic Theory
- The theory was unable to explain the existence of isotopes.
- Nothing about the structure of atom was appropriately explained.
- Later, the scientists discovered particles inside the atom that proved, the atoms are divisible.
Subatomic Particles
- Protons - are positively charged subatomic particles.
- Electron - are negatively charged subatomic particles.
- Neutrons - are electrically neutral particles and carry no charge
Atomic Structure of Isotopes
Several atomic structures of an element can exist, which differ in the total number of nucleons.These variants of elements having a different nucleon number (also known as the mass number) are called isotopes of the element. Therefore, the isotopes of an element have the same number of protons but differ in the number of neutrons. For example, there exist three known naturally occurring isotopes of hydrogen, namely, protium, deuterium, and tritium.