Atoms Test

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Atoms Test - 32

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Weekly Quiz Competition

Question 1
1 / -0

If the radius of first Bohr orbit is $$r$$, then the radius of second orbit will be

A
2$$r$$

B
$$\dfrac{r}{2}$$

C
4$$r$$

D
$$\sqrt{2}{r}$$

Solution

The radius of an orbit of revolution of electron is:
$$r = \dfrac{n^2h^2\epsilon_0}{\pi m e^2}$$
where, variables have their usual meanings.
$$\Rightarrow r \propto n^2$$

Therefore, $$r_1 \propto n_1^2$$ and $$r_2 \propto n_2^2$$
$$\Rightarrow \dfrac{r_2}{r_1} = \dfrac{n_2^2}{n_1^2}$$

$$\therefore \dfrac{r_2}{r} = \dfrac{2_2^2}{1_1^2}$$

$$\therefore r_2 = 4r$$
Question 2
1 / -0

Whenever a hydrogen atom emits a photon in the Balmer series

A
It need not emit any more photon

B
It may emit another photon in the Paschen series

C
It must emit another photon in the Lyman series

D
It may emit another photon on the Balmer series

Solution

Balmer series, electron is still in excited state. ($$n=2$$), so it will definitely come to a lower energy state by transit from ($$n=2 to n=1$$)
Question 3
1 / -0

Two electrons in an atom are moving in orbit of radii $$R$$ and $$9R$$ respectively. The ratio of their frequencies will be

A
$$1:8$$

B
$$8:1$$

C
$$1:27$$

D
$$27:1$$

Solution

Radius is directly proportional to square of $$n$$. Therefore, ratio of principal quantum number is $$1:3$$
Hence, ratio of frequencies is $$27:1$$.
Question 4
1 / -0

The electron in a hydrogen atom jumps from the ground state to the higher energy where its velocity is reduced to one-third of its initial value. If the radius of the orbit in the ground state is $$r$$, the radius of new orbit will be

A
$$3r$$

B
$$9r$$

C
$$\displaystyle\frac{r}{3}$$

D
$$\displaystyle\frac{r}{9}$$

Solution

Velocity is inversely proportional to $$n$$, hence $${ n }_{ 2 }=3$$
Radius is directly proportional to square of $$n$$.
Therefore, $$ r_{ 3 }=9r$$
Question 5
1 / -0

The radius of the Bohr orbit in the ground state of hydrogen atom is $$0.5\mathring{A}$$. The radius of the orbit of the electron in the third excited state of $$He^+$$ will be

A
$$8\mathring{A}$$

B
$$4\mathring{A}$$

C
$$0.5\mathring{A}$$

D
$$0.25\mathring{A}$$

Solution

Radius, $$r\propto \dfrac{n^2}{Z}$$

Thus, $$\dfrac{r_2}{r_1}=\dfrac{n_2^2Z_1}{n^2_1Z_2}=\dfrac{4^2\times 1}{1^2\times 2}=8$$

$$\Rightarrow r_2=8r_1=8(0.5)=4 \mathring A$$
Question 6
1 / -0

In the above figure, $$E_1$$ to $$E_6$$ represent some of the energy levels of an electron in the hydrogen atom.
Which one of the following transitions produces a photon of wavelength in the ultra-violet region of the electromagnetic spectrum?

A
$$E_2-E_1$$

B
$$E_3-E_2$$

C
$$E_4-E_3$$

D
$$E_6-E_4$$

Solution

If a transition is occurred into $${ E }_{ 1 }$$, resulting radiation will be U.V
Question 7
1 / -0

The approximate value of quantum number $$'n\ '$$ for the circular orbit of hydrogen of $$0.0001\space mm$$ in diameter is

A
$$1000$$

B
$$60$$

C
$$10000$$

D
$$31$$

Solution

For $$d = 0.0001\ mm = 10^{-7} m = 1000\ A$$
$$ r = 0.529 \cfrac{n^2}{Z} $$
For hydrogen, $$Z =1$$
$$ 500 = 0.529 n^2$$
$$n = 30.74 $$
Question 8
1 / -0

How many revolutions does an electron complete in one second in the first orbit of hydrogen atom?

A
$$6.62\times10^{15}$$

B
$$6.62\times10^{5}$$

C
$$6.62\times10^{10}$$

D
$$1.62\times10^{15}$$

Solution

The radius of first orbit , $$r_1=0.53 \times 10^{-10} m$$

Thus, circumference of this orbit , $$p=2\pi r_1=2\pi\times (0.53\times 10^{-10})=3.33\times 10^{-10} m$$

The speed of first orbit is approximately , $$v_1=c/137=\dfrac{3\times 10^8}{137}=2.19\times 10^6 m/s$$

Time period, $$T=p/v_1=\dfrac{3.33\times 10^{-10}}{2.19\times 10^6}=1.52\times 10^{-16} s$$

The number of revolutions per sec is the frequency.
So, the frequency of the revolutions, $$f=\dfrac{1}{T}=\dfrac{1}{1.52\times 10^{-16}}=6.6\times 10^15$$
Question 9
1 / -0

Which of the following parameters are the same for all hydrogen-like atoms and ions in their ground states?

A
Radius of the orbit

B
Speed of the electron

C
Energy of the atom

D
Orbital angular momentum of the electron

Solution

According to Bohr's model of an atom, energy of nth orbit is given by: $$E_n = -13.6 \dfrac{Z^2}{n^2} eV$$
Radius of nth orbit, $$r_n = 0.529 \dfrac{n^2}{Z} A^o$$
Speed of an electron in nth orbit, $$v_n = 2.18 \times 10^6 \dfrac{Z}{n} m/s$$
And orbital angular momentum, $$L_n =\dfrac{nh}{2\pi}$$
$$\implies$$ Radius of the orbit, speed of the electron, and energy of an atom depends on $$Z$$ whereas orbital angular momentum is independent of $$Z$$.
Hence, orbital angular momentum remains the same for all hydrogen like atoms and ions in ground state $$(n=1)$$.
Question 10
1 / -0

The electric potential energy between a proton and an electron is given by $$U=U_0ln\dfrac{r}{r_0}$$, where $$r_0$$ is a constant. Assuming Bohr's model to be applicable, write the variation of $$'r\ '$$ with $$n$$, $$n$$ being the principal quantum number.

A
$$r_n \propto n$$

B
$$r_n \propto 1/n$$

C
$$r_n \propto n^2$$

D
$$r_n \propto 1/n^2$$

Solution

The electrostatic interaction force between electron and proton is $$F=\dfrac{dU}{dr}=\dfrac{U_0}{rr_0}$$
According Bohr's first postulate, $$\dfrac{mv^2}{r}=F=\dfrac{U_0}{rr_0}$$
or $$v^2=\dfrac{U_0}{mr_0} \rightarrow constant$$
According Bohr's second postulate, $$mvr_n=\dfrac{nh}{2\pi}$$
As m,v and h are constant so we can write, $$r_n\propto n$$

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Atoms Test - 32

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Atoms Test - 32

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Question 1

2$$r$$

$$\dfrac{r}{2}$$

4$$r$$

$$\sqrt{2}{r}$$

Solution

Question 2

It need not emit any more photon

It may emit another photon in the Paschen series

It must emit another photon in the Lyman series

It may emit another photon on the Balmer series

Solution

Question 3

$$1:8$$

$$8:1$$

$$1:27$$

$$27:1$$

Solution

Question 4

$$3r$$

$$9r$$

$$\displaystyle\frac{r}{3}$$

$$\displaystyle\frac{r}{9}$$

Solution

Question 5

$$8\mathring{A}$$

$$4\mathring{A}$$

$$0.5\mathring{A}$$

$$0.25\mathring{A}$$

Solution

Question 6

$$E_2-E_1$$

$$E_3-E_2$$

$$E_4-E_3$$

$$E_6-E_4$$

Solution

Question 7

$$1000$$

$$60$$

$$10000$$

$$31$$

Solution

Question 8

$$6.62\times10^{15}$$

$$6.62\times10^{5}$$

$$6.62\times10^{10}$$

$$1.62\times10^{15}$$

Solution

Question 9

Radius of the orbit

Speed of the electron

Energy of the atom

Orbital angular momentum of the electron

Solution

Question 10

$$r_n \propto n$$

$$r_n \propto 1/n$$

$$r_n \propto n^2$$

$$r_n \propto 1/n^2$$

Solution

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