Question

Which one of the following about an electron occupying the 1s orbital in a hydrogen atom is incorrect?
[The Bohr radius is represented by ${{a}_{0}}$]
(A) The electron can be found at a distance $2{{a}_{0}}$ from the nucleus
(B) The probability density of finding the electron is maximum at the nucleus
(C) The magnitude of potential energy is double that of its kinetic energy on an average
(D) The total energy of the electron is maximum when it is at a distance ${{a}_{0}}$ from the nucleus

Answer 1

Hint: Orbitals are those spaces in which there is maximum probability of finding an electron. Thus, the probability of finding the electron at any point in space will never be zero. By the quantum mechanical approach of the orbitals, we would be able to identify the incorrect statement about an electron occupying the 1s orbital in a hydrogen.

Complete step by step solution:
- Let’s start with the concepts of orbitals. As we know, an orbital can be defined as the three dimensional description of the most likely location of an electron around an atom. Or in other words orbital is the space in which there is maximum probability of finding an electron.
- In terms of quantum mechanics, an atomic orbital can be defined as the mathematical function describing the wave-like behavior and location of an electron in an atom and this function is used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus or in orbitals.
-As we know, the 1s orbital is spherically symmetrical and so the probability of finding an electron at any given point in 1s orbital depends only on its distance from the nucleus. At very large values of${{a}_{0}}$, the electron probability density might be small but not zero. Hence the first statement is true.
- The function ${{\Psi }^{2}}$ provides the probability of finding an electron in a given volume of space and ${{\Psi }^{2}}\propto r$ where r is the distance from the nucleus. So the probability of finding an electron is maximum at the nucleus. Hence the second statement is also true.
- The relation between the magnitude of potential energy and kinetic energy on an average in an orbital is given by the relation$\left| P.E \right|=2\left| K.E \right|$. That is on an average the potential energy is double that of its kinetic energy. Hence the third statement is also true.
- The total energy of an electron can be given as \[E=-13.6\dfrac{{{z}^{2}}}{{{n}^{2}}}ev\]
At the distance ${{a}_{0}}$, energy is ${{E}_{1}}=\dfrac{-13.6}{{{a}_{o}}}$
At the distance $2{{a}_{0}}$, energy is ${{E}_{2}}=\dfrac{-13.6}{2{{a}_{o}}}$
Hence ${{E}_{2}}$>${{E}_{1}}$. That is the total energy of the electron is minimum when it is at a distance ${{a}_{0}}$ from the nucleus, not maximum. Hence the option (D) is incorrect.


Therefore the answer is option (D).


Note: It should be noted that , atomic orbitals can be uniquely defined by a set of integers known as quantum numbers and they are Principal Quantum Number(n) , the Azimuthal Quantum Number(l), the Magnetic Quantum Number (${{m}_{l}}$) and the Spin Quantum Number (${{m}_{s}}$).Also, nodes are those spaces where the probability of finding an electron is very low.