Kinetic energy of electron, proton and a particle is given as K, 2K and 4K respectively, then which of the following gives the correct order of De-Broglie wavelengths of electron, proton and a particle
λp > λα > λe
λα > λp > λe
λe > λp > λα
λe > λα > λp
The Correct Option is C
Approach Solution - 1
\(\lambda _{e}=\frac{h}{\sqrt{\frac{2\times m}{2000}}\times4K}=\frac{10\sqrt{20h}}{2\sqrt{2mK}}\)
\(\lambda _{p}=\frac{h}{\sqrt{2mK}}=\frac{h}{\sqrt{2mK}}\)
\(\lambda _{\alpha }=\frac{h}{\sqrt{2\times 4m\times 2K}}=\frac{h}{4\sqrt{mK}}\)
\(\lambda _{e}>\lambda _{p}>\lambda _{\alpha }\)
So, the correct option is (C): λe > λp > λα
Approach Solution -2
The De Broglie wavelength of a particle is given by the equation:
λ = \(\frac{h}{p}\)
where λ is the wavelength, h is the Planck constant, and p is the momentum of the particle.
Since the kinetic energy of the particles is given, we can use the relation:
K = \(\frac{p^2}{2m}\)
where K is the kinetic energy, p is the momentum, and m is the mass of the particle.
Solving for momentum, we get:
p = \(\sqrt{(2mK)}\)
Using this expression for each particle and substituting in the equation for De Broglie wavelength, we get:
\(\lambda_e=\frac{h}{\sqrt{(2mK)}}\)
\(\lambda_p=\frac{h}{\sqrt{(4mK)}}\) = \(\frac{h}{\sqrt{(2m)}\sqrt{(2K)}}\)
\(\lambda_\alpha=\frac{h}{\sqrt{(4\alpha mK)}}\)
where α is the ratio of the mass of the alpha particle to the mass of the hydrogen atom.
Since the mass of the proton is approximately the same as the mass of the hydrogen atom, we can write:
α = \(\frac{m\alpha}{mH}=\frac{4}{1}\) = 4
Therefore, we have:
\(\lambda_e=\frac{h}{\sqrt{(2mK)}}\)
\(\lambda_p=\frac{h}{\sqrt{(2m)}\sqrt{(2k)}}\)
\(\lambda_\alpha=\frac{h}{\sqrt{(16mK)}}\)
Simplifying these expressions, we get:
\(\lambda_e=\frac{h}{\sqrt{(2mK)}}\)
\(\lambda_p=\frac{h}{\sqrt{(4mK)}}\)
\(\lambda_\alpha=\frac{h}{4\sqrt{mK}}\)
Since \(\sqrt{(2mK)}>\sqrt{(4mK)}>\frac{1}{(4\sqrt{mK})}\),
We can conclude that: λe > λp > λα
Therefore, the correct option is (C).
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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.