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Question: What is the correct relationship? (A) \({{E}_{1}}ofH=1/2{{E}_{2}}ofH{{e}^{+}}=1/3{{E}_{3}}ofL{{i}^...

What is the correct relationship?
(A) E1ofH=1/2E2ofHe+=1/3E3ofLi2+=1/4E4ofBe3+{{E}_{1}}ofH=1/2{{E}_{2}}ofH{{e}^{+}}=1/3{{E}_{3}}ofL{{i}^{2+}}=1/4{{E}_{4}}ofB{{e}^{3+}}
(B) E1(H)=E2(He+)=E3(Li2+)=E4(Be3+){{E}_{1}}(H)={{E}_{2}}(H{{e}^{+}})={{E}_{3}}(L{{i}^{2+}})={{E}_{4}}(B{{e}^{3+}})
(C) E1(H)=2E2(He+)=3E3(Li2+)=4E4(Be3+){{E}_{1}}(H)=2{{E}_{2}}(H{{e}^{+}})=3{{E}_{3}}(L{{i}^{2+}})=4{{E}_{4}}(B{{e}^{3+}})
(D) No relation

Explanation

Solution

Hydrogen atom is the simplest atom to apply Bohr model and Schrodinger equation. An atom in its normal state has Z protons and Z electrons in order to be neutral. Z is called the atomic number. From the Bohr’s equation, can calculate the energy of electron like single atoms H, He+2,Li+3and,Be+4H{{e}^{+2}},L{{i}^{+3}} and,B{{e}^{+4}} .

Complete answer:
In the equation given, En{{E}_{n}} = energy of nth orbit
We know, hydrogen like single atom, energy of the nth orbit, En=−13.6XZ2n2eV{{E}_{n}}=-13.6X\dfrac{{{Z}^{2}}}{{{n}^{2}}}eV -- (1)
Where, Z = atomic number of atoms, and n= number of orbits. This equation is known as energy of electrons in the Bohr orbit of the H atom.
Now, for H atom, the first energy level, E1{{E}_{1}} will be, (from equation-1)
E1=−13.6(1)2(1)2=−13.6eV{{E}_{1}}=-13.6\dfrac{{{(1)}^{2}}}{{{(1)}^{2}}}=-13.6eV--- (2), where Z= atomic number of H =1 and energy level n =1
For He+2H{{e}^{+2}} atom, E2{{E}_{2}} will be (from equation-1),
E2=−13.6(2)2(2)2=−13.6eV{{E}_{2}}=-13.6\dfrac{{{(2)}^{2}}}{{{(2)}^{2}}}=-13.6eV--- (3), where Z= 2 for He and n=2
For Li+3L{{i}^{+3}} , E3{{E}_{3}} will be (from equation-1),
E3=−13.6(3)2(3)2=−13.6eV{{E}_{3}}=-13.6\dfrac{{{(3)}^{2}}}{{{(3)}^{2}}}=-13.6eV -- (4), where Z=3 for Li and n=3
For Be+4B{{e}^{+4}} , E4{{E}_{4}} will be (from equation-1),
E4=−13.6(4)2(4)2=−13.6eV{{E}_{4}}=-13.6\dfrac{{{(4)}^{2}}}{{{(4)}^{2}}}=-13.6eV -- (5), where Z=4 for Be and n=3
From equation (2), (3), (4) and (5), the energy of electrons for a single electron atom is of equal value.
Hence, E1(H)=E2(He+)=E3(Li2+)=E4(Be3+){{E}_{1}}(H)={{E}_{2}}(H{{e}^{+}})={{E}_{3}}(L{{i}^{2+}})={{E}_{4}}(B{{e}^{3+}})

So the correct answer is option B

Note:
An electron in an orbit is to ignore the interactions between electrons with each other and assume each electron is moving under the action of the nucleus as a point charge +Ze. Each electron has an independent potential energy and wave function.