JEE Challenger
More from Structure of Atom
ChemMedAdv 2019 • Paper 2

Bohr Model Dependencies: Kinetic and Potential Energy Matching

Comprehension Passage

Answer Q.15 and Q.16 by appropriately matching the lists based on the information given in the paragraph.

Consider the Bohr's model of a one-electron atom where the electron moves around the nucleus. In the following, List-I contains some quantities for the nthn^{\text{th}} orbit of the atom and List-II contains options showing how they depend on nn.

List-I
List-II
(I)
Radius of the nthn^{\text{th}} orbit
(P)
∝n−2\propto n^{-2}
(II)
Angular momentum of the electron in the nthn^{\text{th}} orbit
(Q)
∝n−1\propto n^{-1}
(III)
Kinetic energy of the electron in the nthn^{\text{th}} orbit
(R)
∝n0\propto n^0
(IV)
Potential energy of the electron in the nthn^{\text{th}} orbit
(S)
∝n1\propto n^1
(T)
∝n2\propto n^2
(U)
∝n1/2\propto n^{1/2}

Which of the following options has the correct combination considering List-I and List-II?

Options

A

(III), (P)

Correct
B

(III), (S)

C

(IV), (Q)

D

(IV), (U)

Topics & Concepts

Structure of AtomBohr ModelQuantum NumbersElectronic Configuration

Step-by-Step Solution

According to Bohr's model for a one-electron hydrogen-like atom with nuclear charge ZZ:

  1. Orbital speed (vnv_n):
    vn=Ze22ϵ0h⋅1n  ⟹  vn∝n−1v_n = \frac{Z e^2}{2\epsilon_0 h} \cdot \frac{1}{n} \implies v_n \propto n^{-1}

  2. Kinetic Energy of the electron in the nthn^{\text{th}} orbit (KnK_n):
    Kn=12mvn2∝(n−1)2  ⟹  Kn∝n−2K_n = \frac{1}{2} m v_n^2 \propto (n^{-1})^2 \implies K_n \propto n^{-2}
    Thus, (III) corresponds to (P).

  3. Potential Energy of the electron in the nthn^{\text{th}} orbit (UnU_n):
    Un=−Ze24πϵ0rn∝−1rn∝−1n2  ⟹  ∣Un∣∝n−2U_n = -\frac{Z e^2}{4\pi\epsilon_0 r_n} \propto -\frac{1}{r_n} \propto -\frac{1}{n^2} \implies |U_n| \propto n^{-2}
    Thus, (IV) corresponds to (P).

Checking the given options for Question 16:

  • (A) (III), (P) →\rightarrow Correct (Kn∝n−2K_n \propto n^{-2})
  • (B) (III), (S) →\rightarrow Incorrect (Kn∝n−2K_n \propto n^{-2}, not n1n^1)
  • (C) (IV), (Q) →\rightarrow Incorrect (Un∝n−2U_n \propto n^{-2}, not n−1n^{-1})
  • (D) (IV), (U) →\rightarrow Incorrect (Un∝n−2U_n \propto n^{-2}, not n1/2n^{1/2})

Therefore, the correct option is A.

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