Atomic Structure | Physics JEE previous year questions with Solutions

Q 11 :
In an alpha particle scattering experiment distance of closest approach for the $α$ particle is $4.5 \times 10^{- 14} m$ . If target nucleus has atomic number 80, then maximum velocity of $α$ -particle is ________ $\times 10^{5}$ m/s approximately.

$(\frac{1}{4 π ε_{0}} = 9 \times 10^{9}$ SI unit, mass of $α$ particle = $6.72 \times 10^{- 27}$ kg $)$ [2024]

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(156) $\frac{K Q q}{r_{\min}} = \frac{1}{2} m v^{2} \Rightarrow v = \sqrt{\frac{2 K (Z e) \cdot (2 e) q}{m r_{\min}}} = \sqrt{\frac{4 K Z e^{2}}{m r_{\min}}}$

$\Rightarrow v = \sqrt{\frac{4 \times 9 \times 10^{9} \times 80 \times {(1.6 \times 10^{- 19})}^{2}}{6.72 \times 10^{- 27} \times 4.5 \times 10^{- 14}}}$

$= 1.56 \times 10^{7} m / s = 156 \times 10^{5} m / s$

Q 12 :
If Rydberg's constant is R, the longest wavelength of radiation in Paschen series will be $\frac{α}{7 R}$ , where $α$ = _______ . [2024]

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(144) Longest wavelength corresponds to transition between $n = 3$ and $n = 4$

$\frac{1}{λ} = R Z^{2} (\frac{1}{3^{2}} - \frac{1}{4^{2}}) = R Z^{2} (\frac{1}{9} - \frac{1}{16}) = \frac{7 R Z^{2}}{9 \times 16}$

$\Rightarrow λ = \frac{144}{7 R}$ for $Z = 1$

$∴ α = 144$

Q 13 :
A electron of hydrogen atom in an excited state is having energy $E_{n} = - 0.85$ $e V$ . The maximum number of allowed transitions to lower energy level is ______ . [2024]

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(6) $E_{n} = - \frac{13.6}{n^{2}} = - 0.85 \Rightarrow n = 4$

$Number of transitions = \frac{n (n - 1)}{2} = \frac{4 (4 - 1)}{2} = 6$

Q 14 :
A hydrogen atom changes its state from $n = 3 t o n = 2$ . Due to recoil, the percentage change in the wavelength of emitted light is approximately $1 \times 10^{- n}$ . The value of n is ____.

[Given $R h c = 13.6 eV, h c = 1242 eVnm, h = 6.6 \times 10^{- 34}$ mass of the hydrogen atom = $1.6 \times 10^{- 27} kg$ ] [2024]

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(7)

For $n = 3$ to $n = 2$

$Δ E = 13.6 (\frac{1}{2^{2}} - \frac{1}{3^{2}}) = 1.9 eV$

$Δ E = \frac{h c}{λ} \Rightarrow λ = \frac{h c}{Δ E}$

If recoil takes place

$P_{i} = P_{f} \Rightarrow 0 = - m v + \frac{h}{λ'} \Rightarrow v = \frac{h}{m λ'}$

$Δ E = \frac{1}{2} m v^{2} + \frac{h c}{λ'} = \frac{1}{2} m {(\frac{h}{m λ'})}^{2} + \frac{h c}{λ'}$

Now, $Δ E = \frac{h^{2}}{2 m λ'^{2}} + \frac{h c}{λ'}$

$λ'^{2} Δ E - h c λ' - \frac{h^{2}}{2 m} = 0$

$λ' = \frac{h c \pm \sqrt{h^{2} c^{2} + \frac{4 Δ E h^{2}}{2 m}}}{2 Δ E}$

$λ' = \frac{h c \pm h c \sqrt{1 + \frac{2 Δ E}{m c^{2}}}}{2 Δ E}$

$\frac{λ'}{λ} = \frac{1 + {(1 + \frac{2 Δ E}{m c^{2}})}^{\frac{1}{2}}}{2} = \frac{1 + 1 + \frac{Δ E}{m c^{2}}}{2}$

$\frac{λ'}{λ} = 1 + \frac{Δ E}{2 m c^{2}}$

$\frac{λ' - λ}{λ} = \frac{Δ E}{2 m c^{2}} = \frac{1.9 \times 1.6 \times 10^{- 19}}{2 \times 1.67 \times 10^{- 27} \times 9 \times 10^{16}} = 10^{- 9}$

$∴$ $% change \approx 10^{- 7}$

Q 15 :
When a hydrogen atom going from $n = 2 to n = 1$ emits a photon, its recoil speed is $x / 5$ m/s. Where $x =$ ______.

(Use: mass of hydrogen atom = $1.6 \times 10^{- 27} kg$ ) [2024]

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(17)

The hydrogen atom transitions from $n = 2$ to $n = 1,$

$Δ E = - 10.2 eV$

Recoil speed $v = \frac{Δ E}{m c}$

$= \frac{10.2 eV}{1.6 \times 10^{- 27} \times 3 \times 10^{8}} = \frac{10.2 \times 1.6 \times 10^{- 19}}{1.6 \times 10^{- 27} \times 3 \times 10^{8}}$

$v = 3.4 m/s = \frac{17}{5} m/s$

Therefore, $x = 17$

Topic Question Set

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Q 12 :
If Rydberg's constant is R, the longest wavelength of radiation in Paschen series will be $\frac{α}{7 R}$ , where $α$ = _______ . [2024]

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Q 13 :
A electron of hydrogen atom in an excited state is having energy $E_{n} = - 0.85$ $e V$ . The maximum number of allowed transitions to lower energy level is ______ . [2024]

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Q 15 :
When a hydrogen atom going from $n = 2 to n = 1$ emits a photon, its recoil speed is $x / 5$ m/s. Where $x =$ ______.

(Use: mass of hydrogen atom = $1.6 \times 10^{- 27} kg$ ) [2024]

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Topic Question Set

Q 11 : In an alpha particle scattering experiment distance of closest approach for the α particle is 4.5×10-14m. If target nucleus has atomic number 80, then maximum velocity of α-particle is ________ ×105m/s approximately. (14πε0=9×109 SI unit, mass of α particle = 6.72×10-27 kg) [2024]

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Q 12 : If Rydberg's constant is R, the longest wavelength of radiation in Paschen series will be α7R, where α = _______ . [2024]

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Q 13 : A electron of hydrogen atom in an excited state is having energy En=-0.85 eV. The maximum number of allowed transitions to lower energy level is ______ . [2024]

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Q 14 : A hydrogen atom changes its state from n=3 to n=2. Due to recoil, the percentage change in the wavelength of emitted light is approximately 1×10-n. The value of n is ____. [Given Rhc=13.6 eV, hc=1242 eVnm, h=6.6×10-34 mass of the hydrogen atom = 1.6×10-27 kg] [2024]

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Q 15 : When a hydrogen atom going from n=2 to n=1 emits a photon, its recoil speed is x/5 m/s. Where x= ______. (Use: mass of hydrogen atom = 1.6×10-27kg) [2024]

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Q 12 :
If Rydberg's constant is R, the longest wavelength of radiation in Paschen series will be $\frac{α}{7 R}$ , where $α$ = _______ . [2024]

Q 13 :
A electron of hydrogen atom in an excited state is having energy $E_{n} = - 0.85$ $e V$ . The maximum number of allowed transitions to lower energy level is ______ . [2024]

Q 15 :
When a hydrogen atom going from $n = 2 to n = 1$ emits a photon, its recoil speed is $x / 5$ m/s. Where $x =$ ______.

(Use: mass of hydrogen atom = $1.6 \times 10^{- 27} kg$ ) [2024]