10. Electron in hydrogen atom first jumps from third excited state to second excited state and then from second excited to the first excited state. The ratio of the wavelengths $λ$ ₁ : $λ$ ₂ emitted in the two cases is ⇒ (AIPMT 2012 Prelims)

A. $\frac{7}{5}$

B. $\frac{27}{20}$

C. $\frac{27}{5}$

D. $\frac{20}{7}$

The Correct Answer is Option (D)

In first case,

n₁ = 3 and n₂ = 4

$\frac{1}{λ_{1}} = R (\frac{1}{3^{2}} - \frac{1}{4^{2}})$ = $\frac{7 R}{144}$

In second case,

n₁ = 2 and n₂ = 3

$\frac{1}{λ_{2}} = R (\frac{1}{2^{2}} - \frac{1}{3^{2}})$ = $\frac{5 R}{36}$

$∴$ $\frac{λ_{1}}{λ_{2}}$ = $\frac{5}{36} \times \frac{144}{7}$ = $\frac{20}{7}$

11. An electron of a stationary hydrogen atom passes from the fifth energy level to the ground level. The velocity that the atom acquired as a result of photon emission will be ⇒ (AIPMT 2012 Prelims)

A. $\frac{24 h R}{25 m}$

B. $\frac{25 h R}{24 m}$

C. $\frac{25 m}{24 h R}$

D. $\frac{24 m}{25 h R}$

The Correct Answer is Option (A)

We know,

$\frac{1}{λ} = R Z^{2} (\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}})$

where R = Rydberg constant, Z = atomic number

Here, n₁ = 1, n₂ = 5

$∴$ $\frac{1}{λ} = R (\frac{1}{1^{2}} - \frac{1}{5^{2}})$ = $\frac{24 R}{25}$

According to conservation of linear momentum, we get

Momentum of photon = Momentum of atom

$\Rightarrow$ $\frac{h}{λ}$ = mv

$\Rightarrow$ v = $\frac{h}{m λ}$ = $\frac{h}{m} (\frac{24 R}{25})$ = $\frac{24 h R}{25 m}$

12. The wavelength of the first line of Lyman series for hydrogen atom is equal to that of the second line of Balmer series for a hydrogen like ion. The atomic number Z of hydrogen like ion is ⇒ (AIPMT 2011 Prelims)

A. 3

B. 4

C. 1

D. 2

The Correct Answer is Option (D)

The wavelength of the first line of lyman series for hydrogen atom is

$\frac{1}{λ} = R (\frac{1}{1^{2}} - \frac{1}{2^{2}})$

The wavelength of the second line of Balmer series for hydrogen like ion is

$\frac{1}{λ^{'}} = R Z^{2} (\frac{1}{2^{2}} - \frac{1}{4^{2}})$

According to question

$λ$ = $λ$ '

$R (\frac{1}{1^{2}} - \frac{1}{2^{2}})$ = $R Z^{2} (\frac{1}{2^{2}} - \frac{1}{4^{2}})$

$\Rightarrow$ $\frac{3}{4}$ = $\frac{3 Z^{2}}{16}$

$\Rightarrow$ Z = 2

13. The ionization energy of the electron in the hydrogen atom in its ground state is 13.6 eV. The atoms are excited to higher energy levels to emit radiations of 6 wavelengths, Maximum wavelength of emitted radiation corresponds to the transition between ⇒ (AIPMT 2009)

A. n = 3 to n = 1 states

B. n = 2 to n = 1 states

C. n = 4 to n = 3 states

D. n = 3 to n = 2 states

The Correct Answer is Option (C)

n(n – 1)/2 = 6

$\Rightarrow$ n² – n –12 = 0

$\Rightarrow$ (n – 4) (n + 3) = 0 or n = 4

So, n = 4 to n = 3 states.

14. Ionization potential of hydrogen atom is 13.6 eV. Hydrogen atoms in the ground state are excited by monochromatic radiation of photon energy 12.1 eV. According to Bohr's theory, the spectral lines emited by hydrogen will be ⇒ ( AIPMT 2006)

A. one

B. two

C. three

D. four

The Correct Answer is Option (C)

Energy required for exciting the hydrogen atom in the ground state to orbit n is given by

E = E_n - E₁

$\Rightarrow$ 12.1 = $- \frac{13.6}{n^{2}}$ - $(- \frac{13.6}{1^{2}})$ = 3

$\Rightarrow$ n² = $\frac{13.6}{1.5}$ = 9

$\Rightarrow$ n = 3

Number of spectral lines emitted

= $\frac{n (n - 1)}{2}$ = $\frac{3 \times 2}{2}$

15. Energy levels A, B and C of a certain atom corresponding to increasing values of energy i.e. E_A < E_B < E_C. If $λ$ ₁, $λ$ ₂ and $λ$ ₃ are wavelengths of radioations corresponding to transitions C to B, B to A and C to A respectively, which of the following relations is correct? ⇒ ( AIPMT 2005)

A. $λ_{3} = λ_{1} + λ_{2}$

B. $λ_{3} = \frac{λ_{1} λ_{2}}{λ_{1} + λ_{2}}$

C. $λ_{1} + λ_{2} + λ_{3} = 0$

D. ${λ_{3}}^{2} = {λ_{1}}^{2} + {λ_{2}}^{2}$

The Correct Answer is Option (B)

AIPMT 2005 Physics - Atoms and Nuclei Question 53 English Explanation

$∴$ (E_C – E_B) + (E_B – E_A) = (E_C – E_A)

$\Rightarrow$ $\frac{h c}{λ_{3}} = \frac{h c}{λ_{1}} + \frac{h c}{λ_{2}}$

$\Rightarrow$ $λ_{3} = \frac{λ_{1} λ_{2}}{λ_{1} + λ_{2}}$

16. Energy levels A, B and C of a certain atom corresponding to increasing values of energy i.e. E_A < E_B < E_C. If $λ$ ₁, $λ$ ₂ and $λ$ ₃ are wavelengths of radioations corresponding to transitions C to B, B to A and C to A respectively, which of the following relations is correct? ⇒ (AIPMT 2005)

A. $λ_{3} = λ_{1} + λ_{2}$

B. $λ_{3} = \frac{λ_{1} λ_{2}}{λ_{1} + λ_{2}}$

C. $λ_{1} + λ_{2} + λ_{3} = 0$

D. ${λ_{3}}^{2} = {λ_{1}}^{2} + {λ_{2}}^{2}$

The Correct Answer is Option (C)

AIPMT 2005 Physics - Atoms and Nuclei Question 53 English Explanation

$∴$ (E_C – E_B) + (E_B – E_A) = (E_C – E_A)

$\Rightarrow$ $\frac{h c}{λ_{3}} = \frac{h c}{λ_{1}} + \frac{h c}{λ_{2}}$

$\Rightarrow$ $λ_{3} = \frac{λ_{1} λ_{2}}{λ_{1} + λ_{2}}$

17. When an electron does transition from n = 4 to n = 2, then emitted line spectrum will be ⇒ (AIPMT 2000)

A. first line of Lyman series

B. second line of Balmer series

C. first line of Balmer series

D. second line of Paschen series.

The Correct Answer is Option (B)

Jump to second orbit leads to Balmer series. The jump from 4th orbit shall give rise to second line of Balmer series.

⇒ Atom

Topic 3 : Hydrogen Spectrum