Atomic Structure Questions | Chemistry JEE Main Important Questions

Q 1 :
The de-Broglie's wavelength of an electron in the 4th orbit is _________ $π a_{0}$ . ( $a_{0}$ = Bohr's radius) [2024]

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(8) De-Broglie's wavelength ( $λ$ ) is related to mass (m) and velocity v of electron by:

$λ = \frac{h}{m v} - I,$ where $h$ is Planck's constant.

By quantization of angular momentum:

$m v r = \frac{n h}{2 π} - II$

Where $n$ is orbit number and $r$ is radius of $n^{t h}$ orbit.

Rearranging equation II

$2 π r = n \frac{h}{m v} - III$

From I and III

$2 π r = n λ - IV$

Radius ( $r$ ) of $n^{t h}$ orbit is related to radius of first orbit ( $a_{0}$ ) as:

$r = a_{0} \frac{n^{2}}{Z} - V$

From IV and V:

$2 π a_{0} \frac{n^{2}}{Z} = n λ$

$λ = 2 π a_{0} \frac{n}{Z}$

Taking atom to be H, Z = 1 and for fourth orbit, n = 4

$λ = 2 π a_{0} \frac{4}{1} = 8 π a_{0}$

Q 2 :
Frequency of the de-Broglie wave of electron in Bohr's first orbit of hydrogen atoms is ________ $\times 10^{13}$ Hz (nearest integer). (Given: $R_{H}$ (Rydberg constant) = $2.18 \times 10^{- 18}$ J, $h$ (Plank's constant) $= 6.6 \times 10^{- 34}$ J.s.) [2024]

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(656) Velocity ( $v$ ) of electron in $n^{t h}$ orbit of single electron specie with atomic number Z is given by:

$v = 2.18 \times 10^{6} \frac{Z}{n} m / s$ ...(i)

Radius ( $r$ ) of $n^{t h}$ orbit of single electron specie with atomic number Z is given by:

$r = 0.529 \times 10^{- 10} \frac{n^{2}}{Z} m$ ...(ii)

Circumference of $n^{t h}$ orbit is integral multiple of wavelength ( $λ$ ) of electron i.e.

$2 π r = n λ$

$λ = \frac{2 π r}{n}$ ...(iii)

Frequency (v) is related to velocity (v) and wavelength ( $λ$ ) by:

$ν = \frac{v}{λ}$ ...(iv)

Put $λ$ from (iii) in (iv)

$ν = \frac{v}{2 π r / n}$ ...(v)

Put $v$ and $r$ from (i) and (ii) in (iv)

$ν = \frac{2.18 \times 10^{6} \frac{Z}{n}}{(2 π \times 0.529 \times 10^{- 10} \frac{n^{2}}{Z}) / n} = \frac{2.18 \times 10^{16} Z^{2}}{2 π \times 0.529 n^{2}}$

For H, Z = 1 and for first orbit n = 1

$ν = \frac{2.18 \times 10^{16} \times {(1)}^{2}}{2 π \times 0.529 \times {(1)}^{2}} = 656 \times 10^{13} Hz$

Q 3 :
Based on Heisenberg's uncertainty principle, the uncertainty in the velocity of the electron to be found within an atomic nucleus of diameter $10^{- 15}$ m is ______ $\times 10^{9} {ms}^{- 1}$ (nearest integer) [Given: mass of electron = $9.1 \times 10^{- 31}$ kg, Plank's constant $(h) = 6.626 \times 10^{- 34}$ Js] (Value of $π = 3.14$ ) [2024]

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(58) $Δ x \times m Δ v \geq \frac{h}{4 π}$

$Δ x = 10^{- 15} m$

Substituting other values:

$10^{- 15} \times 9.1 \times 10^{- 31} \times Δ v \geq \frac{6.626 \times 10^{- 34}}{4 \times 3.14}$

$Δ v \geq 58 \times 10^{9} m / s$

Q 4 :
According to the wave-particle duality of matter by de-Broglie, which of the following graph plot presents most appropriate relationship between wavelength of electron ( $λ$ ) and momentum of electron (p)? [2024]

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(1)

According to de-Broglie hypothesis, wavelength ( $λ$ ) of a particle is related to its momentum (p) as:

$λ = \frac{h}{p}$ or $λ p = h$

Where $h$ is a constant called planck's constant.

Graph of $λ$ $v / s$ $p$ with $λ p$ = constant is a rectangular hyperbola.

Q 5 :
If $a_{0}$ is denoted as the Bohr radius of hydrogen atom, then what is the de-Broglie wavelength $(λ)$ of the electron present in the second orbit of hydrogen atom? [ $n$ : any integer] [2025]

$\frac{2 a_{0}}{n π}$
$\frac{4 π a_{0}}{n}$
$\frac{8 π a_{0}}{n}$
$\frac{4 n}{π a_{0}}$

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(3)

By de Broglie's equation:

$λ = \frac{h}{m v}$

$\Rightarrow m v = \frac{h}{λ} -I$

By quantization of angular momentum:

$m v r = n \frac{h}{2 π} -II$

$II / I$ gives:

$\frac{m v r}{m v} = \frac{n λ}{2 π}$

$2 π r = n λ$

$2 π \times a_{0} \frac{n^{2}}{Z} = n λ$

$2 π \times a_{0} \frac{n}{Z} = λ$

For second Bohr orbit, $n$ = 2 and for H atom Z = 1

$2 π \times a_{0} \frac{2}{1} = λ$

$λ = \frac{4 π a_{0}}{1}$

Putting $n$ = 2 in option (3) gives the correct answer.

Q 6 :
Given below are two statements:

Statement I: It is impossible to specify simultaneously with arbitrary precision, both the linear momentum and the position of a particle.

Statement II: If the uncertainty in the measurement of position and uncertainty in measurement of momentum are equal for an electron, then the uncertainty in the measurement of velocity is $\geq \sqrt{\frac{h}{π}} \times \frac{1}{2 m} .$

In the light of the above statements, choose the correct answer from the options given below: [2025]

Both Statement I and Statement II are true
Statement I is false but Statement II is true
Statement I is true but Statement II is false
Both Statement I and Statement II are false

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(1)

Statement I: This is Heisenberg's Uncertainty Principle.

Statement II: $Δ x Δ p \geq \frac{h}{4 π}, Δ p Δ p \geq \frac{h}{4 π},$

$Δ p \geq \sqrt{\frac{h}{4 π}}, Δ v \geq \frac{1}{m} \sqrt{\frac{h}{4 π}}$

Topic Question Set

Q 1 :
The de-Broglie's wavelength of an electron in the 4th orbit is _________ $π a_{0}$ . ( $a_{0}$ = Bohr's radius) [2024]

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Q 2 :
Frequency of the de-Broglie wave of electron in Bohr's first orbit of hydrogen atoms is ________ $\times 10^{13}$ Hz (nearest integer). (Given: $R_{H}$ (Rydberg constant) = $2.18 \times 10^{- 18}$ J, $h$ (Plank's constant) $= 6.6 \times 10^{- 34}$ J.s.) [2024]

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Q 4 :
According to the wave-particle duality of matter by de-Broglie, which of the following graph plot presents most appropriate relationship between wavelength of electron ( $λ$ ) and momentum of electron (p)? [2024]

Q 5 :
If $a_{0}$ is denoted as the Bohr radius of hydrogen atom, then what is the de-Broglie wavelength $(λ)$ of the electron present in the second orbit of hydrogen atom? [ $n$ : any integer] [2025]

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Topic Question Set

Q 1 : The de-Broglie's wavelength of an electron in the 4th orbit is _________ πa0. (a0 = Bohr's radius) [2024]

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Q 2 : Frequency of the de-Broglie wave of electron in Bohr's first orbit of hydrogen atoms is ________ ×1013 Hz (nearest integer). (Given: RH (Rydberg constant) = 2.18×10-18 J, h (Plank's constant) =6.6×10-34 J.s.) [2024]

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Q 3 : Based on Heisenberg's uncertainty principle, the uncertainty in the velocity of the electron to be found within an atomic nucleus of diameter 10-15 m is ______ ×109ms-1 (nearest integer) [Given: mass of electron = 9.1×10-31 kg, Plank's constant (h)=6.626×10-34 Js] (Value of π=3.14) [2024]

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Q 4 : According to the wave-particle duality of matter by de-Broglie, which of the following graph plot presents most appropriate relationship between wavelength of electron (λ) and momentum of electron (p)? [2024]

Q 5 : If a0 is denoted as the Bohr radius of hydrogen atom, then what is the de-Broglie wavelength (λ) of the electron present in the second orbit of hydrogen atom? [n : any integer] [2025]

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Q 1 :
The de-Broglie's wavelength of an electron in the 4th orbit is _________ $π a_{0}$ . ( $a_{0}$ = Bohr's radius) [2024]

Q 2 :
Frequency of the de-Broglie wave of electron in Bohr's first orbit of hydrogen atoms is ________ $\times 10^{13}$ Hz (nearest integer). (Given: $R_{H}$ (Rydberg constant) = $2.18 \times 10^{- 18}$ J, $h$ (Plank's constant) $= 6.6 \times 10^{- 34}$ J.s.) [2024]

Q 4 :
According to the wave-particle duality of matter by de-Broglie, which of the following graph plot presents most appropriate relationship between wavelength of electron ( $λ$ ) and momentum of electron (p)? [2024]

Q 5 :
If $a_{0}$ is denoted as the Bohr radius of hydrogen atom, then what is the de-Broglie wavelength $(λ)$ of the electron present in the second orbit of hydrogen atom? [ $n$ : any integer] [2025]