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Structure of Atom question

2023 · 30 Jan · Shift 2 · Q14
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Structure of Atom question

2023 · 30 Jan · Shift 2 · Q14

JEE MainChemistryStructure of AtomMCQ+4 / −1
The wave function (Ψ)(\Psi)(Ψ) of 2 s2 \mathrm{~s}2 s is given by Ψ2 s=122π(1a0)1/2(2−ra0)e−r/2a0\Psi_{2 \mathrm{~s}}=\frac{1}{2 \sqrt{2 \pi}}\left(\frac{1}{a_0}\right)^{1 / 2}\left(2-\frac{r}{a_0}\right) e^{-r / 2 a_0}Ψ2 s​=22π​1​(a0​1​)1/2(2−a0​r​)e−r/2a0​ At r=r0r=r_0r=r0​, radial node is formed. Thus, r0r_0r0​ in terms of a0a_0a0​
  1. A
    r0=a0r_0=a_0r0​=a0​
  2. B
    r0=4a0r_0=4 a_0r0​=4a0​
  3. C
    r0=2a0r_0=2 a_0r0​=2a0​
  4. D
    r0=a02r_0=\frac{a_0}{2}r0​=2a0​​
View written solutionFree

Correct answer: C

  1. Condition for a radial node

A radial node occurs where the wave function becomes zero for some finite value of rrr.

Given:

Ψ2s=122π(1a0)1/2(2−ra0)e−r/2a0\Psi_{2s}=\frac{1}{2\sqrt{2\pi}}\left(\frac{1}{a_0}\right)^{1/2}\left(2-\frac{r}{a_0}\right)e^{-r/2a_0}Ψ2s​=22π​1​(a0​1​)1/2(2−a0​r​)e−r/2a0​
  1. Set the wave function equal to zero

For Ψ2s=0\Psi_{2s}=0Ψ2s​=0, the constant prefactor cannot be zero, and the exponential term e−r/2a0e^{-r/2a_0}e−r/2a0​ is never zero.

So the only factor that can become zero is:

2−ra0=02-\frac{r}{a_0}=02−a0​r​=0
  1. Solve for rrr
2=ra02=\frac{r}{a_0}2=a0​r​ r=2a0r=2a_0r=2a0​

Hence,

r0=2a0r_0=2a_0r0​=2a0​
  1. Check options
  • A: a0a_0a0​ →\rightarrow→ incorrect
  • B: 4a04a_04a0​ →\rightarrow→ incorrect
  • C: 2a02a_02a0​ →\rightarrow→ correct
  • D: a02\dfrac{a_0}{2}2a0​​ →\rightarrow→ incorrect

Therefore, the correct option is C.

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