Atoms and Nuclei

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Questions:

Multiple Choice Question

3 x 1 = 3

Two H atoms in the ground state collide inelastically. The maximum amount by which their combined kinetic energy is reduced is

(a)

10.20 eV

(b)

20.40 eV

(c)

13.6 eV

(d)

27.2 eV

The Bohr model for the spectre of a H-atom

(a)

will not be applicable to hydrogen in the molecular form

(b)

will not be applicable as it is for a He-atom

(c)

is valid only at room temperature

(d)

predicts continuous as well as discrete spectral lines

Taking the Bohr radius as a₀ = 53pm, the radius of Li⁺⁺ ion its ground state, on the basis of Bohr's model, will be about

(a)

53pm

(b)

27pm

(c)

18pm

(d)

13pm

Generic 5

4 x 5 = 20

Marie Curie and her teacher turned husband Pierre Curie worked hard to extract radium chloride (R_aCl₂) from uranium ore. They succeeded in 192 after a long struggle. About 0.19 g of R_aCl₂ was extracted and its radioactivity was studied. They were awarded by the noble prize, which they shared it which they shared it which they shared it with Henri Becquerel.
Read the above passage and answer following questions:
(i) What are the values shown by Marie Curie and her husband?
(ii) What do you understand by radioactivity? How the half-life period is related to the disintegration constant?
(iii) How is average-life of radioactive element related to half-life?

According to Bohr's theory of hydrogen atom, total energy of electron in a stationary orbit is \(E=-\frac { 13.6 }{ { n }^{ 2 } } eV,\) where n is the number of orbit. Clearly, total energy of electron in a stationary orbit is negative, which means the electron is bound to the nucleus and is not free to leave it. An n increases, value of negative energy decreases, i.e., energy is progressively larger in the outer orbits.
Read the above passage and answer the following questions :
(i) What is total energy of electron in ground state of hydrogen atom? What does it imply?
(ii) Energy required to remove an electron is smaller when atoms is in any one excited state. Comment.
(iii) How is this concept translated in day to day life?

Einstein was the first to establish the equivalence between mass energy. According to him, whenever a certain mass \(\left( \Delta m \right) \) disappears in some process, the amount of energy released is \(E=\left( \Delta m \right) { c }^{ 2 },\) where c is velocity of light vacuum \(\left( =3\times { 10 }^{ 8 }m/s \right) .\) The reverse is also true, i.e., whenever energy E disappears, an equivalent mass \(\left( \Delta m \right) ={ E/c }^{ 2 }\) appears.
Read the above passage and answer the following questions :
(i) What is the energy released when 1 a.m.u. of mass disappears in a nuclear reaction?
(ii) Do you know any phenomenon in which energy materialises?
(iii) What values of life do you learn from this famous relation?

In the Auger process an atom makes a transition to a lower state without emitting a photon.the excess energy is transferred to an outer electron which may be ejected by the atom.(This is called an Auger electron).Assuming the nucleus to be massive, calculate the kinetic energy of an n = 4 Auger electron emitted by Chromium by absorbing the energy from a n = 2 to n = 1 transition.

Generic 2

5 x 2 = 10

Calculate the energy equivalent of 1 a.m.u. in MeV.

What is the effect on neutron to proton ratio in a nucleus when (i) an electron, (ii) a positron is emitted?

10.

Assuming the nuclei to be spherical in shape, how does the surface area of a nucleus of mass number \({ A }_{ 1 }\) compare with that of a nucleus of mass number \({ A }_{ 2 }\)?

11.

Define ionisation energy. How would the ionisation energy change when electron in hydrogen atom is replaced by a particle 200 times heavier than electron, but having the same charge?

12.

In a hydrogen atom, if the electron is replaced by a particle which is 200 times heavier but has the same charge, how would its radius change?

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Answers Key:

Multiple Choice Question

3 x 1 = 3

(a)
10.20 eV

(b)
will not be applicable as it is for a He-atom

Generic 5

4 x 5 = 20

(i) Values shown here are dedication, hard working nature, brilliance, true justification of the help when helped by another, honesty and gratitude.
(ii) The phenomenon of spontaneous emission of \(\alpha \) , \(\beta \) and \(\gamma \)-particles by the nuclide is known as radioactivity. Half-life period is related with the disintegration constant as,
\({ T }_{ 1/2 }=\frac { 0.693 }{ \lambda } \)
(iii) Average life of radioactive element,
\(\tau ={ 1.44T }_{ 1/2 }\)

(i) For ground state, n = 1
\(\therefore E=\frac { -13.6 }{ { n }^{ 2 } } eV= \ \frac { -13.6 }{ { I }^{ 2 } } eV= \ -13.6 \ eV\)
It implies that 13.6 eV energy is required to remove an electron from hydrogen atom in its ground state.
(ii) In first excited state, n = 2,
\(\therefore \ E=-\frac { 13.6 }{ { 2 }^{ 2 } } eV \ = \ -3.4eV\)
It means that energy required to remove an electron from hydrogen atom in first excited state is 3.4 eV, which is less than 13.6 eV. Therefore, the statement is true.
(iii) Negative energy of electron indicates that it is bound to the nucleus and cannot leave it until energy equal to its negative energy is supplied from outside. The same is true in day to day life. A person intending to leave the country has to show that nothing is pending against him in a court of law, and he has cleared income tax/sales tax payments due from him. A person from whom any type of payment is due, is not free to leave the country. He is bound till he clears all his dues.

(i) Here, \(\Delta m=1\quad a.m.u.=1.66\times { 10 }^{ -27 }kg\)
\(E=\left( \Delta m \right) { c }^{ 2 }=1.66\times { 10 }^{ -27 }{ \left( 3\times { 10 }^{ 8 } \right) }^{ 2 }=1.49\times { 10 }^{ -10 }J\)
(ii) Yes, in the phenomenon of pair production. Under suitable conditions, a photon materialises into an electron and a position : \(\gamma ={ e }^{ -1 }+{ e }^{ +1 }\)
(iii) Einstein's relation, \(E=\left( \Delta m \right) { c }^{ 2 }\) emphasis that when certain mass disappears, an equivalent amount of energy appears. The reverse is also true. It Implies that to gain something, you have to lose another in equivalent amount. No one can have all gains together or all losses together. It also implies that nothing come for free. You have to pay the price in one form and acquire something in the desired form.

As the nucleus is massive, recoil momentum of the atom may be neglected and the entire energy of the transition may be considered transferred to the Auger electron. As there is single valence electron in Cr, the energy states may be thought of as given by the Bohr model.
The energy of the nth state
\(E_n=Z^2R{1\over n^2}\) where R is the Rydberg constant and Z = 24
In transition from n = 2 to n = 1,
Energy released
\(\Delta E=-RZ^2\left[{1\over4}-1\right]={3\over4}Z^2R\)
The energy required to eject a n = 4 electron is
\(E_4=Z^2R{1\over16}={Z^2R\over16}\)
So K.E. of Auger electron is given by
K.E. \(=Z^2R\left({3\over4}-{1\over10}\right)\)
\(={11\over16}\times24\times24\times13.6eV\)
= 5385.6eV

Generic 2

5 x 2 = 10

From \(E=m{ c }^{ 2 }\)
when m=1 a.m.u = \(1.66\times { 10 }^{ -27 }kg\)
\(E=(1.66\times { 10 }^{ -27 }){ \left( 3\times { 10 }^{ 8 } \right) }^{ 2 }joule\)
\(=\frac { 1.66\times { 9\times 10 }^{ -11 } }{ 1.6\times { 10 }^{ -13 } } MeV=933.75MeV\)

In emission of an electron, a neutron is converted into a proton.Therefore, number of neutrons decreases and the number of protons increases.The neutron to proton ratio decreases. In the emission of a positron, a proton is converted into a neutron. Hence the ratio increases.

10.

\(\frac { { A }_{ 1 } }{ { A }_{ 2 } } ={ \left( \frac { { R }_{ 1 } }{ { R }_{ 2 } } \right) }^{ 2 }={ \left[ { \left( \frac { { A }_{ 1 } }{ { A }_{ 2 } } \right) }^{ { 1 }/{ 3 } } \right] }^{ 2 }={ \left( \frac { { A }_{ 1 } }{ { A }_{ 2 } } \right) }^{ 2/{ 3 } }\)

11.

Ionisation energy is the minimum energy required to knock out an electron from an atom.Its value will be different for different atoms. Ionisation energy will also depend on the orbit from which electron is to be removed.
When an electron in hydrogen atom is replaced by a particle 200 times heavier than electron but having the same charge, ionisation energy will not change, as it depends only on charge and not on mass of particle.

12.

As radius, \(r\propto \frac { 1 }{ m } \)
\(\therefore \) When electron is replaced by a particle 200 times heavier, the radius would decrease to \(\frac { 1 }{ 200 } \) time the original radius.