Chapter # 43
Bohr Model and Physics of the Atom
SOLVED EXAMPLES
43.1
Calculate the energy of a He+ ion in its first excited state.
Sol.
The energy is En =
43.2
Calculate the wavelength of radiation emitted when He+ makes a transition from the state n = 3 to the state n = 2. The wavelength is given by
Sol.
RhcZ 2 2
=–
(13.6 eV )Z 2
n n2 For a He+ ion, Z = 2 and for the first excited state, n = 2 so that the energy of He ion in the first excited state is – 13.6 eV.
1 1 1 RZ 2 2 2 m n 1 1 5 = 4R R 4 9 9
or, 43.3 Sol.
=
9 9 = = 164.0 nm. 5R 5 1.00737 107 m 1
The excitation energy of hydrogen-like ion in its first excited state in 40.8 eV. Find the energy needed to remove the electron from the ion. The excitation energy in the first excited state is 1 1 E = RhcZ2 2 2 2 1
3 4 Equating this to 40.8 eV, we get Z = 2. So, the ion in question is He+. The energy of the ion in the ground state is = (13.6 eV) × Z2 ×
E=–
RhcZ 2
= – 4 × (13.6 eV) 12 = – 54.4 eV. Thus 54.4 eV is required to remove the electron from the ion.
QUESTIONS
FOR
SHORT
ANSWER
1.
How many wavelengths are emitted by atomic hydrogen in visible range (380 nm – 780 nm)? In the range 50 nm to 100 nm ?
2.
The first excited energy of a He+ ion is the same as the ground state energy of hydrogen. Is it always true that one of the energies of any hydrogen-like ion will be the same as the ground state energy of a hydrogen atom ?
3.
Which wavelengths will be emitted by a sample of atomic hydrogen gas (in ground state) if electrons of energy 12.2 eV collide with the atoms of the gas ?
4.
When white radiation is passed through a sample of hydrogen gas at room temperature, absorption lines are observed in Lyman series only. Explain.
5.
Balmer series was observed and analysed before the other series. Can you suggest a reason for such an order?
6.
What will be the energy corresponding to the first excited state of a hydrogen atom if the potential energy of the atom is taken to be 10 eV when the electron is widely separated from the proton? Can we still write widely separated from the proton? Can we still write En = E1/n2? rn = a0 n2?
7.
The difference in the frequencies of series limit of Lyman series and Balmer series is equal to the frequency manishkumarphysics.in
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Chapter # 43 Bohr Model and Physics of the Atom of the first line of the Lyman series.. Explain. 8.
The numerical value of ionization energy in eV equals the ionization potential in volts. Does the equality hold if these quantities are measured in some other units?
9.
We have stimulated emission and spotaneous emission. Do we also have stimulated absorption and spontaneous absorption?
10.
An atom is in its excited state. Does the probability of its coming to ground state depend on whether the radiation is already present or not? If yes, does it also depend on the wavelength of the radiation present?
Objective - I 1.
The minimum orbital angular momentum of the electron in a hydrogen atom is gkbMªkstu ijek.kq ls bysDVªkWu dk U;wure d{kh; dks.kh; laoxs gS (A) h (B) h/2 (C*) h/2 (D) h/
2.
Three photons coming from excite3d atomic-hydrogen sample are picked up. Their energies are 12.1 V, sample 10.2eV and 1.9eV. These photons must come from (A) a single atom (B) two atoms (C) three atom (D*) either two atoms or three atoms mÙksftr ijekf.od&gkbMªkt s u ds izfrn'kZ ls vkus okys rhu QksVkWuksa dk p;u fd;k tkrk gSA budh ÅtkZ,a 12.1 eV, 10.2eV vkSj 1.9eV gSA ;s QksVkWu vk jgs gS]a fuf'pr :i ls (A) ,d vdsys ijek.kq ls (B) nks ijek.kqvksa ls (C) rhu ijek.kqvksa ls (D*) ;k rks nks ijek.kqvksa ls ;k rhu ijek.kqvksa ls
3.
Suppose, the electron in a hydrogen atom makes transition from n = 3 to n = 2 in 10–8 s. The order of the torque acting on the electron in this period, using the relation between troque and angular momentum as discussed in the chapter on rotational mechanics is ekuk fd gkbMªkstu ijek.kq esa n = 3 ls n = 2 rd bysDVªkWu dk laØe.k 10–8 lsd.M esa gksrk gSA bl dky esa bysDVªkWu ij yx jgs] cy vk?kw.kZ dh dksfV dk eku] ?kw.kZu ;kaf=kdh ds v/;k; esa of.kZr cyk?kw.kZ ,oa dks.kh; laoxs ds vk/kkj ij gksxk (A) 10-34 N-m (B*) 10-24 N-m (C) 10-42 N-m (D) 10-8 N-m
4.
In which of the following transitions will the wavelength be minimum ? (A) n = 5 to n = 4 (B) n = 4 to n = 3 (C) n = 3 to n = 2 (D*) n = 2 to n = 1 fuEu laØe.kksa esa ls fdlesa rjaxnS/;Z U;wure gksxh (A) n = 5 ls n = 4 (B) n = 4 ls n = 3 (C) n = 3 ls n = 2 (D*) n = 2 ls n = 1
5.
In which of the following systems will the radius of the first orbit (n=1) be minimum ? (A) hydrogen atom (B) deuterium atom (C) singly ionized helium (D*) doubly ionized lithium fuEu esa ls fdlesa izFke d{kk (n=1) dh f=kT;k U;wure gS (A) gkbMªkstu ijek.kq (B) M~;wfVfj;e ijek.kq (C) ,d/kk vk;fur ghfy;e (D*) f}vk;fur fyfFk;e
6.
In which of the following systems will the wavelength corresponding to n=2 to n=1 be minimum ? (A) hydrogen atom (B) deuterium atom (C) singly ionized helium (D*) doubly ionized lithium fuEu esa ls fdlds fy;s n=2 ls n=1 ds laxr rjaxnS/;Z U;wure gS (A) gkbMªkstu ijek.kq (B) M~;wfVfj;e ijek.kq (C) ,d/kk vk;fur ghfy;e (D*) f}vk;fur fyfFk;e
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Chapter # 43 Bohr Model and Physics of the Atom 7. Which of the following curves may represent the speed of the electron in a hydrogen atom as a function of the principal quantum number ? fuEu oØksa esa ls dkSulk gkbMªkt s u ijek.kq dh bysDVªkWu dh pky dks eq[; Dok.Ve la[;k n ds Qyu :i esa O;Dr dj ldrk gS -
(c*)
8.
As one considers orbits with higher values of n in a hudrogen atom, the electric potential energy of the atom (A) decreases (B*) increases (C) remains the same (D) does not increase ;fn dksbZ gkbMªkstu ijek.kq esa n ds mPp ekuksa ds fy;sd{kkvksa ij fopkj djrk gS] rks ijek.kq dh oS|qr fLFkfrt ÅtkZ (A) de gksrh gSA (B*) c<+rh gSA (C) leku jgrh gSA (D) c<+rh ugha gSA
9.
The energy of an atom (or ion) in its ground state is - 54.4eV. It may be (A) hydrogen (B) deuterium (C*) He (D) Li fdlh ijek.kq ¼;k vk;u½ dh ewy voLFkk esa ÅtkZ - 54.4eV gSA ;g gks ldrk gS (A) gkbMªkstu (B) M~;wfVfj;e (C*) He (D) Li
10.
The radius of the shortest orbit in a one-electron system is 18 pm. It may be (A) hydrogen (B) deuterium (C) He (D*) Li ,dy bysDVªkWu fudk; esa lcls NksVh d{kk dh f=kT;k 18 pm ¼fidks ehVj½ gSA ;g gks ldrk (A) gkbMªkstu (B) M~;wfVfj;e (C) He (D*) Li
gS -
11.
A hydrogen atom in ground state absorbs 10.2eV of energy. The orbital angular momentum of the electron is increased by gkbMªkt s u ijek.kq ewy voLFkk esa 10.2eV ÅtkZ vo'kksf"kr djrk gSA bysDVªkWu ds d{kh; dks.kh; laoxs esa o`f) gksrh gS (A*) 1.05 x 10 -34 J-s (B) 2.11 x 10 -34 J-s (C) 3.16 x 10 -34 J-s (D) 4.22 x 10 -34 J-s
12.
Which of the following parameters are the same for all hydrogen-like atoms and ions in their ground states ? (A) radius of the orbitq (B) speed of the electron (C) energy of the atom (D*) orbital angular momentum of the electron gkbMªkt s u tSls leLr ijek.kqvksa vkSj vk;uksa dh ewy voLFkkvksa ds fy;s fuEu jkf'k;ksa esa ls dkSulh ,d leku gksrh gS (A) d{kk dh f=kT;k (B) bysDVªkWu dh pky (C) ijek.kq dh ÅtkZ (D*) bysDVªkWu dk d{kh; dks.kh; laox s
13.
In a laser tube, all the photons (A) have same wavelength (C) move in same direction ,d yslj uyh esa] lkjs QksVkWuksa (A) dh rjaxnS/;Z leku gksrh gSA (C) xfr dh fn'kk leku gksrh gSA
(B) have same energy (D*) move with same speed (B) dh ÅtkZ leku gksrh gSA (D*) pky ,d leku gksrh gSA
Objective - II 1.
In a laboratory experiment on emission from atomic hydrogen in a discharge tube, only a small number of lines are abserved whereas a large number of lines are present in the hydrogen spectrum of a star. This is because in a laboratory (A) the amount of hydrogen taken in much smaller than that present in the star. (B*) the temperature of hydrogen is much smaller than that of the star (C) the pressure of hydrogen is much smaller than that of the star (D) the gravitational pull is much smaller than that in the star.
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Chapter # 43
Bohr Model and Physics of the Atom
iz;ksx'kkyk esa foltZu ufydk ds iz;ksx esa ijekf.od gkbMªkstu ds mRltZu esa] dsoy dqN gh la[;k esa js[kk;sa izsf{kr gksrh gS] tcfd rkjksa ds gkbMªkt s u LisDVªe esa js[kkvksa dh la[;k cgqr vf/kd gksrh gSA bldk dkj.k ;g gS fd cgqr vf/kd gksrh gSA bldk dkj.k ;g gS fd iz;ksx'kkyk esa (A) yh x;h gkbMªkt s u dh ek=kk] rkjksa dh rqyuk esa cgqr de gksrh gSA (B*) gkbMªkstu dk rki] rkjksa dh rqyuk esa cgqr de gksrk gSA (C) gkbMªkstu dk nkc] rkjksa dh rqyuk esa cgqr de gksrk gSA (D) xq:Roh; vkd"kZ.k rkjksa dh rqyuk esa cgqr de gksrk gSA 2.
An electron with kinetic energy 5 eV is incident on a hydrogen atom in its ground state. The collision (A*) must be elastic (B) may be partially elastic (C) must be completely inelastic (D) may be completely inelastic gkbMªkstu ijek.kq dh ewy voLFkk esa bl ij 5 eV xfrt ÅtkZ dk ,d bysDVªkWu vkifrr gksrk gSA VDdj (A*) fuf'pr :i ls izR;kLFk gksxhA (B) vkaf'kd :i ls izR;kLFk gks ldrh gSA (C) iw.kZr;k vizR;kLFk gks ldrh gSA (D) iw.kZr;k vizR;kLFk gks ldrh gSA
3.
Which of the following products in a hydrogen atom are independent of the principal quantum n ? The symbols have their usual meanings gkbMªkstu ijek.kq ds fy;s fuEu xq.kuQyksa esa ls dkSuls eq[; Dok.Ve la[;k n ij fuHkZj ugha djrs gSa\ ladsrksa dk lkekU; vFkZ gS (A*) n (B*) Er (C) En (D) r
4.
Let An be the area enclosed by the nth orbit in a hydrogen atom. The graph of n (An / A1) against n (n) (A*) will pass through the origin (B*) will be a straight line with slope 4 (C) will be a monotonically increasing nonlinear curve (D) will be a circle ekuk fd gkbMªkstu ijek.kq esa nth d{kk dk {ks=kQy n (An / A1) against n(n) (A*) ewy fcUnq ls xqtjsxk (B*) ,d ljy js[kk gksxk] ftldk
Sol.
rn2 An ln A = ln 2 1 r1
= ln n4 = 4 ln (n) ln (An/A1)
5.
Ionization energy of a hydrogen-like ion B. Let r, u, E and L represent the radius of the orbit, speed of the electron, energy of the atom and orbital angular momentum of the electron respectively. In ground state gkbMªkt s u tSls vk;u A dh vk;fudj.k ÅtkZ] ,d vU; gkbMªkstu tSls vk;u B dh ÅtkZ ls vf/kd gSA ekuk fd d{kk dh f=kT;k] bysDVªkWu dk d{kh; dks.kh; laoxs Øe'k% r, u, E vkSj L }kjk O;Dr fd;s tkrs gSAa ewy voLFkk esa (A) rA > rB (B*) uA > uB (C) EA > EB (D) LA > LB
6.
When a photon stimulates the emission of another photon, the two photons have (A*) same energy (B*) same direction (C*) same phase (D*) same wavelength tc QksVkWu fdlh vU; QksVkWu ds mRltZu dks míhIr djrk gS] rks nksuksa QksVkWu dh (A*) ÅtkZ leku gksrh gSA (B*) fn'kk leku gksrh gSA (C*) dyk leku gksrh gSA (D*) rjaxnS/;Z
leku gksrh gSA
WORKED OUT EXAMPLES 1. Sol.
Find the radius of Li++ ions in its ground state assuming Bohr’s model to be valid. cksgj ds ekWMy dks lR; ekurs gq, Li++ vk;u dh f=kT;k dh x.kuk dhft,A HCV_Ch-43_WOE_1 For hydrogen -like ions, the radius of the nth orbit is
n2a0 Z For Li++, Z = 3 and in ground state n = 1. The radius is an =
a1 = 2.
53 pm 18 pm. 3
A particular hydrogen-like ion emits radiation of frequency 2.467 × 1015 Hz when it makes transition from n = 2 to n = 1. What will be the frequency of the radiation emitted in a transition from n = 3 to n = 1? manishkumarphysics.in
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Chapter # 43 Bohr Model and Physics of the Atom Sol. The frequency of radiation emitted is given by v=
c =K
1 1 2 2 n 1 n2
1 1 Thus, 2.467 × 10 15 Hz = K 2 2 2 1 4 × 2.467 × 10–15 Hz. 3 The frequency of the radiation emitted in the transition n = 3 to n = 1 is
or,
K=
1 1 v’ = K 2 2 3 1 8 8 4 K= × × 2.467 × 10 15 Hz 9 9 3 = 2.92 × 10 15 Hz
=
3. Sol.
Calculate the two highest wavelengths of the radiation emitted when hydrogen atoms make transitions from higher states to n = 2 states. The highest wavelength corresponds to the lowest energy of transition. This will be the case for the transition n = 3 to n = 2. The second highest wavelength corresponds to the transition n = 4 to n = 2. E1
The energy of the state n is En = Thus,
and
n2
.
E2 = –
13.6 eV = – 3.4 eV 4
E3 = –
13.6 eV = – 1.5 eV 9
E4 = –
13.6 eV = – 0.85 eV.. 16
The highest wavelength is 1 = =
hc E
1242 eV nm = 654 nm. (3.4 eV 1.5 eV )
The second highest wavelength is 2 = 4. Sol.
1242 eV nm = 487 nm. (3.4 eV 0.85 eV )
What is the wavelength of the radiation emitted when the electron in a hydrogen atoms jumps from n = to n = 2? The energy of n = 2 state is
13.6 eV = – 3.4 eV.. 4 The energy of n = state is zero. The wavelength emitted in the given transition is E2 =
l= 5. Sol.
hc E
=
1242 eV nm = 365 nm. 3.4 eV
(a) Find the wavelength of the radiation required to excite the electron in Li++ from the first to the third Bohr orbit. (b) How many spectral linea are observed in the emission spectrum of the above excited system? (a) The energy in the first orbit = E1 = Z2 E0 where E0 = – 13.6 eV is the energy of a hydrogen atom in ground state thus for Li++, E1 = 9E0 = 9 × (– 13.6 eV). The energy in the third orbit is manishkumarphysics.in
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Chapter # 43
Bohr Model and Physics of the Atom E1
E 1 = 13.6 eV.. 9 n2 Thus, E3 – E1 = 8 × 13.6 eV = 108.8 eV. The wavelength of radiation required to excite Li++ from the first orbit to the third orbit is given by
E3 =
E1 = – 13.6 eV.. 9 n Thus, E3 – E1 = 8 × 13.6 eV = 108.8 eV. The wavelength of radiation required to excite Li++ from the first orbit to the third orbit is given by E3
E1 2
hc E 3 E1
or,
=
hc E 3 E1
1242 eV nm 11.4 nm 108.8 eV
(b) The spectral lines emitted are due to the transitions n = 3 n = 2, n = 3 n = 1 and n = 2 n = 1. Thus, there will be three spectral lines in the spectrum. 6. Sol.
Find the wavelengths present in the radiation emitted when hydrogen atoms excited to n = 3 states return to their ground states. A hydrogen atom may return directly to the ground state or it may go to n = 2 and from there to the ground state. Thus, wavelengths corresponding to n = 3 n = 1, n = 3 n = 2 are present in the radiation. The energies in n = 1, 2 and 3 states are E1 = – 13.6 eV E2 = –
13.6 eV = – 3.4 eV 4
13.6 eV = – 1.5 eV 9 The wavelength emitted in the transition n = 3 to the ground state is
and
E3 = –
=
hc E
1242 eV nm = 13.6 eV 1.5 eV = 103 nm. Similarly, the wavelength emitted in the transition n = 3 to n = 2 is 654 nm and that emitted in the transition n = 2 to n = 1 is 122 nm. The wavelengths present in the radiation are, therefore, 103 nm, 122 nm and 654 nm. 7. Sol.
How may different wavelengths may be observed in the spectrum from a hydrogen sample if the atoms are excited to states with principal quantum number n ? From the nth state, the atom may go to (n – 1)th state, ...., 2nd state or 1st state. So there are (n – 1) possible transitions starting from the nth state. The atoms reaching (n – 1)th state may make (n – 2) different transitions. Similarly for other lower states. The total number of possible transitions is (n – 1) + (n – 2) + (n – 3) +............2 + 1
n(n 1) 2 Monochromatic radiation of wavelength is incident on a hydrogen sample in ground state. Hydrogen atoms absorb a fraction of light and subsequently emit radiation of six different wavelengths. Find the value As the hydrogen atoms emit radiation of six different wavelengths, some of them must have been excited to n = 4. The energy in n = 4 state is =
8. Sol.
E1
13.6 eV = –0.85 eV 16 4 The energy needed to take a hydrogen atom from its ground state to n = 4 is 13.6 eV – 0.85 eV = 12.75 eV The photons of the incident radiation should have 12.75 eV of energy. So,
E4 =
2
=
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Chapter # 43
Bohr Model and Physics of the Atom hc = 12.75 eV
or, 9.
Sol.
=
hc 12.75 eV
=
1242 eV - nm = 97.5 nm. 12.75 eV
The energy needed to detach the electron of a hydrogen–like ion in ground state is 4 rydberg. (a) What is the wavelength of the radiation emitted when the electron jumps from the first excited state to the ground state ? (b) What is the radius of the first orbit for this atom ? (a) In energy units, 1 rydberg = 13.6 eV. The energy needed to detach the electron is 4 × 13.6 eV. The energy in the ground state is, therefore, E1 = –4 × 13.6 eV. The energy of the first excited state (n = 2) is E2 =
E1 4
= 13.6 eV = 40.8 eV. The wavelength of the radiation emitted is
hc E (b) The energy of a hydrogen-like ion in ground state is E = Z2E0 where Z = atomic number and E0 = – 13.6 =
eV. Thus, Z = 2. The radius of the first orbit is r= 10.
Sol.
11.
Sol.
a0 where a0 = 53 pm. Thus, Z
53 pm = 26.5 pm 2
A hydrogen sample is prepared in a particular excited state A. Photons of energy 2.55 eV get absorbed into the sample to take some of the electrons to a further excited state B. Find the quantum numbers of the states A and B. The allowed energies of hydrogen atoms are E1 = – 13.6 eV E2 = –3.4 eV E3 = –1.5 eV E4 = –0.85 eV E5 = – 0.54 eV We see that a different of 2.55 eV can only be absorbed in transition n = 2 to n = 4. So the state A has quantum number 2 and the state B has quantum number 4. (a) Find the maximum wavelength 0 of light which can ionize a hydrogen atom in its ground state. (b) Light of wavelength 0 is incident on a hydrogen atom which is in its first excited state. Find the kinetic energy of the electron coming out. (a) To ionize a hydrogen atom in ground state, a minimum of 13.6 eV energy should be given to it. A photon of light should have this much of energy in order to ionize a hydrogen atom. Thus,
hc 0 = 13.6 eV or,
l0 =
1242 eV - nm = 91.3 nm, 13.6 eV
13.6 eV = – 3.4 eV. Thus, 4 3.4 eV of energy is needed to take the electron out of the atom. The energy of a photon of the light of wavelength 0 is 13.6 eV. Thus, the electron coming out will have a kinetic energy 13.6 eV – 3.4 eV = 10.2 eV. (b) The energy of the hydrogen atom in its first excited state is
12.
Sol.
Derive an expression for the magnetic field at the site of the nucleus in a hydrogen atom due to the circular motion of the electron. Assume that the atom is in its ground state and give the answer in terms of fundamental constants. We have
e2 mv 2 = 4 0r 2 r
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Chapter # 43 or
Bohr Model and Physics of the Atom v2r =
e2 4 0m
............(i)
From Bohr’s quantization rule, in ground rule, in ground state, h 2m From (i) and (ii),
and
vr =
.........(ii)
e2 v= 2 0h
..........(iii)
r=
0h 2
...........(iv)
me 2
As the electron moves along a circle, it crosses any point on the circle crossing per unit time, that is the current, is i =
v times per unit time. The charge 2r
eV . The magnetic field at the centre due to this circular 2r
current is B=
0i 0 ev 2r 4v 2
From (iii) and (iv),
0e e2 2m 2 e 4 B = 4 2 h 2 4 0 h 0 0 e 7 m 2
=
13.
Sol.
4 03h5
A lithium atom has three electrons, Assume the following simple picture of the atom. Two electrons move close to the nucleus making up a spherical cloud around it and the third moves outside this cloud in a circular orbit. Bohr’s model can be used for the motion of this third electron but n = 1 states ar not available to it. Calculate the ionization energy of lithium in ground state using the above picture. In this picture, the third electron moves in the field of a total charge + 3e – 2e = + e. Thus, the energies are the same as that of hydrogen atoms. The lowest energy is :
E1 13.6 eV = = – 3.4 eV 4 4 Thus, the ionization energy of the atom in this picture is 3.4 eV. E2 =
14.
Sol.
A particle known as -meason, has a charge equal to that of an electron and mass 208 times the mass of the electron. It moves in a circular orbit around a nucleus of charge +3e. Take the mass of the nucleus to be infinite. Assuming that the Bohr’s model is applicable to this system, (a) derive an expression for the radius of the nth Bohr orbit, (b) find the value of n for which the radius of the orbit is approximately the same as that of the first Bohr orbit for a hydrogen atom and (c) find the wavelength of the radiation emitted when the – meson jumps from the third orbit to the first orbit. (a) We have,
mv 2 Ze 2 r 4 0 r 2 or,
v 2r
Ze 2 4 0m
The quantization rule is vr =
The radius is r =
( vr )2 2
v r
=
...(i)
nh 2m
4 0m Ze 2 manishkumarphysics.in
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Chapter # 43
Bohr Model and Physics of the Atom =
n2h2 0
....(ii)
Zme 2 For the given system, Z = 3 and m = 208 me.
n 2h 2 0
r
Thus
624 m e e 2
(b) From (ii), the radius of the first Bohr orbit for the hydrogen atom is
h2 0
rh
m e e 2
n 2h 2 0
For r = rh,
624 m e e 2
=
h20 m e e 2
or, n2 . = 624 or, n = 25 (c) From (i), the kinetic energy of the atom is
Ze 2 mv 2 = 8 0r 2 Ze 2 and the potential energy is – 4 0r Ze 2 8 0r
The total energy is En = Using (ii),
Z 2 me 4 En = –
8 02n 2h 2
=–
4 1872 m e e = 2 2 n 2 8 0 h
me e 4 But 2 2 8 0 h
9 208m e
4
8 02n 2h 2
is the ground state energy of hydrogen atom and hence is equal to – 13.6 eV..
From (iii), En = –
1872 n
2
× 13.6 eV =
Thus, E1 = – 25459.2 eV and E3 =
25459 .2 eV n2
E1 = – 2828.8 eV. The energy difference is E3 – E1 = 22630.4eV.. 9
The wavelength emitted is = 15. Sol.
hc E
1242 eV nm = 22630 .4 eV = 55 pm.
Find the wavelengths in a hydrogen spectrum between the range 500 nm to 700 nm. The energy of a photon of wavelength 500 nm is
1242 eV nm hc = = 1.77 eV 500 nm The energy difference between the states involved in the transition should, therefore, be between 1.77 eV and 2.44 eV.
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Chapter # 43
Bohr Model and Physics of the Atom n = 4, E = –0.85 eV n = 3, E = –1.5 eV n = 2, E = –3.4 eV n = 1, E = –13.6 eV
Figure shows same of the energies of hydrogen states. It is clear that only those transitions which end at n = 2 may emit photons of energy between 1.77 eV the proper range. The energy of the photon emitted in the transition n = 3 to n = 2 is DE = (3.4 – 1.5) eV = 1.9 eV. The wavelength is =
16.
Sol.
hc E
=
1242 eV nm = 654 nm. 1.9 eV
A beam of ultraviolet radiation having wavelength between 100 nm and 200 nm is incident on a sample of atomic hydrogen gas. Assuming that the atoms are in ground state, which wavelength will have low intensity in the transmitted beam? If the energy of a photon is equal to the difference between the energies of an excited being absorbed by an atom in the ground state. The energy of a photon corresponding to = 100 nm is
1242 eV nm = 12.42 eV 100 nm and that corresponding to l = 200 nm is 6.21 eV. The energy needed to take the atom from the ground state to the first excited state is E2 – E1 = 13.6 eV – 3.4 eV = 10.2 eV, to the second excited state is E3 – E1 = 13.6 eV – 1.5 eV = 12.1 eV. to the third excited state is E4 – E1 = 13.6 eV – 0.85 eV = 12.75 eV etc. Thus, 10.2 eV photons and 12.1 eV photons have large probability of being absorbed from the given range 6.21 eV to 12.42 eV. The corresponding wavelengths are
and
1 =
1242 eV nm = 122 nm. 10.1eV
1 =
1242 eV nm = 103 nm. 12.1eV
These wavelengths will have low intensity in the transmitted beam. 17.
Sol.
A neutron moving with speed v makes a head-on collision with a hydrogen atom in ground state kept at rest. Find the minimum kinetic energy of the neutron for which inelastic (completely or partially) collision may take place. The mass of neutron = mass of hydrogen = 1.67 × 10–27 kg. Suppose the neutron and the hydrogen atom move at speed v1 and v2 after the collision. The collision will be inelastic if a part of the kinetic energy is used to excite the atom. Suppose an energy E is used in this way. Using conservation of linear momentum and energy. mv = mv1 + mv2 ....(i)
From (i),
1 1 1 mv2 = mv12 + mv22 + E 2 2 2 v2 = v12 + v22 + 2v1v2 ,
From (ii),
v2 = v12 + v22 +
Thus,
2v1v2 =
and
....(ii)
2E m
2E m
Hence, (v1 – v2)2 – 4v1v2 = v2 –
4E m
As v1 – v2 must be real, v2 –
4E 0 m manishkumarphysics.in
Page # 10
Chapter # 43
Bohr Model and Physics of the Atom
1 mv2 > 2E. 2 The minimum energy that can be absorbed by the hydrogen atom in ground state to go in an excited state is 10.2 eV. Thus, the minimum kinetic energy of the neutron needed for an inelastic collision is or,
1 2 mv min 2 10.2 eV 20.4 eV 2 18. Sol.
Light corresponding to the transition n = 4 to n = 2 in hydrogen atoms falls on cesium metal (work function = 1.9 eV). Find the maximum kinetic energy of the photoelectrons emitted. The energy of the photons emitted in transition n = 4 to n = 2 is 1 1 hv = 13.6 eV 2 2 = 2.55 eV.. 4 2 The maximum kinetic energy of the photoelectrons is = 2.55 eV – 1.9 eV = 0.65 eV.
1 2 2 2 m r where is a 2 constant and r is the distance of the particle from the origin. Assuming Bohr’s model of quantization of
19.
A small particle of mass m moves in such a way that the potential energy U =
Sol.
angular momentum and circular orbits, show that radius of the nth allowed orbit is proportional to The force at a distance r is
n.
dU = – m2r.. ....(i) dr Suppose the particle moves along a circle of radius r. The net force on it should be mv2/r along the radius. Comparing with (i),
F=–
mv 2 = m2 r r or, v = r The quantization of angular momentum gives mvr =
or,
v=
....(ii)
nh 2
nh 2mr
...(iii)
Thus, the radius of the nth orbit is proportional to
n.
EXERCISE 1.
The Bohr radius is given by ao =
cksgj f=kT;k ao =
oh 2 me 2
oh 2 me 2
Verify that the RHS has dimensions of length.
}kjk O;Dr dh tkrh gSA tkap dhft;s fd nka;h vksj dh jkf'k;ksa dh foek,a yEckbZ dh gSAa
Ans : 2.
Ans:
Find the wavelength of the radiation emitted by hydrogen in the transitions (a) n = 3 to n = 2, (b)n = 5 to n = 4 and (c) n = 10 to n = 9. gkbMªkstu ds }kjk fuEu laØe.kksa esa mRlftZr fofdj.kksa dh rjaxnS/;Z Kkr dhft;s % (a) n = 3 ls n = 2, (b) n = 5 ls n = 4 vkSj (c) n = 10 ls n = 9 (a) 654 nm (b) 4050 nm (c) 38860
3.
Calculate the smallest wavelength of radiation that may be emitted by (a) hydrogen, (b) He + and
manishkumarphysics.in
Page # 11
Chapter # 43 (c) Li
Bohr Model and Physics of the Atom
++
fuEu ds }kjk mRlftZr gks ldus okys fofdj.kksa dh U;wure rjanS/;Z dh x.kuk dhft;s % (a) gkbMªkstu , (b) He+ vkSj Ans:
(c) Li++ (a) 91 nm (b) 23 nm (c) 10 nm
4.
Evaluate Rydberg constant by putting the values of the fundamental constants in its expression.
fjMcxZ fu;rkad ds O;atd ds ewyHkwr fu;rkadksa dk eku izfrLFkkfir djds] bldk eku Kkr dhft;sA –1
Ans :
1.097 × 10 7 m
5
Find the binding energy of a hydrogen atom in the state n = 2. n = 2 voLFkk esa gkbMªkstu ijek.kq dh ca/ku&ÅtkZ Kkr dhft;sA 3.4 eV
Ans : 6. Ans : 7.
Find the radius and energy of a He+ ion in the states (a) n = 1, (b) n = 4 and (c) n = 10. He+ vk;u dh fuEu voLFkkvksa esa f=kT;k vkSj ÅtkZ Kkr dhft;sA (a) n = 1, (b) n = 4 and (c) n = 10. (a) 0.265 A, – 54.4 eV (b) 4.24 A ,– 3.4 eV A hydrogen atom emits ultraviolet radiation of wavelength 102.5 nm. What are the quantum numbers of the states involved in the transition ? ,d gkbMªkstu ijek.kq 102.5 nm uSuksehVj rjaxnS/;Z ds ijkcSaxuh fofdj.k mRlftZr djrk gSA laØe.k esa Hkkx ysus okys
voLFkkvksa dh Dok.Ve la[;k,¡ D;k gS\ Ans :
1 and 3
8.
(a) Find the first excitation potential of He+ ion (b) Find the ionization potential of Li++ ion. (a) He+ vk;u dk izFke mÙkstu foHko Kkr dhft;sA (b) Li++ vk;u dk vk;uu foHko Kkr dhft;sA (a) 40.8 V (b) 122.4 V
Ans : 9.
A group of hydrogen atoms are prepared in n = 4 states. List the wavelength that are emitted as the atoms make transitions and return to n = 2 states . gkbMªkt s u ijek.kqvksa dk ,d lewg n = 4 voLFkk esa rS;kj fd;k tkrk gSA ijek.kqvksa esa laØe.k ls vkSj n = 2 voLFkkvksa esa
ykSVus ds dkj.k mRlftZr rjaxnS/;ks± dh lwph cukb;sA Ans :
487 nm, 654 nm 1910 nm
10.
A positive ion hydrogen just one electron ejects it if a photon of wavelength 228 Å of less is absorbed by it identify the ion. ,d /kuk;u ds ikl dsoy ,d bysDVªkuW gS] ;fn ;g 228 Å ;k blls de rjaxnS/;Z dk QksVkWu vo'kksf"kr djrk gS] rks bysDVªkuW
mRlftZr dj nsrk gSA vk;u dks igpkfu;sA +
Ans :
He
11.
Find the maximum coulomb force that can act on the electron due to the nucleus in a hydrogen atom.
gkbMªkstu ijek.kqvksa esa ukfHkd ds dkj.k bysDVªkWu ij yxus okyk vf/kdre dwykWeh; cy Kkr dhft;sA Ans:
8.2 × 10 – 8 N
12.
A hydrogen atom in a state having a binding energy of 0.85 eV makes transition to a state with excitation energy 10.2 eV (a) identify the quantum numbers n of the upper and the lower energy states involved in the transition. (b) Find the wavelength of the emitted radiation. 0.85 eV ca/ku&ÅtkZ voLFkk dk gkbMªkt s u ijek.kq mÙkstu ÅtkZ 10.2 eV okyh voLFkk esa laØe.k djrk gS % (a) bl laØe.k esa 'kkfey Åij okyh vkSj uhps okyh ÅtkZ voLFkkvksa dh Dok.Ve la[;k,sa crkb;sA (b) mRlftZr fofdj.k dh rjaxnS/;Z Kkr
dhft;sA Ans:
(a) 4, 2 (b) 487 nm
13.
Whenever a photon is emitted by hydrogen in Balmer series it is followed by another photon in Lyman series. What wavelength does this latter photon correspond to ?
tc Hkh gkbMªkt s u ds }kjk ckej Js.kh esa QksVkWu mRlftZr fd;k tkrk gS] ykbeu Js.kh dk ,d vU; QksVkWu Hkh izkIr gksrk gSA ckn okys QksVkWu ds laxr rjaxnS/;Z D;k gS\ Ans:
122 nm
14.
A hydrogen atom in stage n = 6 makes two successive transitions and reaches the ground state. In the first transition a photon of 1.13 eV is emitted in the second transition (b) What is the value of n in the intermediate state? n = 6 voLFkk dk gkbMªkt s u ijek.kq nks Øekxr laØe.k djds ewy voLFkk esa igqp a rk gSA izFke laØe.k esa 1.13 eV dk QksVkWu manishkumarphysics.in
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Chapter # 43
Bohr Model and Physics of the Atom
mRlftZr gksrk gSA (a) f}rh; laØe.k esa mRlftZr QksVkWu dh ÅtkZ Kkr dhft;sA (b) chp dh voLFkk ds fy;s n dk eku fdruk gS\ Ans:
121.eV , 3
15.
What is the energy of a hydrogen atom in the first excited state if the potential energy is taken to be zero in the ground state ?
;fn ewy voLFkk esa fLFkfrt ÅtkZ 'kwU; eku yh tk;s rks gkbMªkt s u ijek.kq dh izFke mÙksftr voLFkk esa ÅtkZ fdruh gksxh\ Ans:
23.8 eV
15.
What is the energy radiation of wavelengths 46.0 nm, 82.8 nm and 103.5 nm only. Assume that the atoms have only two excited states and the difference between consecutive energy levels decreases as energy is increased. Taking the energy of the highest energy state to be zero, find the energies of the ground state and the first excited state.
;fn ewy voLFkk esa fLFkfrt ÅtkZ 'kwU; eku yh tk;s rks gkbMªkt s u ijek.kq dh izFke mÙksftr voLFkk esa ÅtkZ c<+us ds lkFk nks Øekxr ÅtkZ Lrjksa dk varj de gksrk gSA mPpre ÅtkZ voLFkk dh ÅtkZ 'kwU; ekurs gq, ewy voLFkk ,oa izFke mÙksftr voLFkk dh ÅtkZ,¡ Kkr dhft;sA Ans. 23.8 eV 16.
A hot gas emits radiation of wavelengths 46.0 nm 82.8 nm and 103.5 nm only. Assume that the atoms have only two excited states and the difference between consecutive energy levels decreases as energy is increased. Taking the energy of the ground state and the first excited state. ,d xeZ xSl ls dsoy 46.0 nm , 82.8 nm vkSj 103.5 nm rjaxnSs/;ks± ds fofdj.k mRlftZr gksrs gSaA eku yhft;s fd
ijek.kqvksa esa dsoy nks gh mÙksftr voLFkk,a gSa vkSj ÅtkZ c<+us ds lkFk nks Øekxr ÅtkZ] Lrjksa dk varj de gksrk gSA mPpre ÅtkZ voLFkk dh ÅtkZ 'kwU; ekurs gq, ewy voLFkk ,oa izFke mÙksftr voLFkk dh ÅtkZ,¡ Kkr dhft;sA Ans:
– 27 eV , – 12 eV
17.
A gas of hydrogen like ions is prepared in a particular excited state A it emits photons having wavelength equal to the wavelength of the first line of the lyman series together with photons of five other wavelength identify the gas and find the principal quantum number of the state A. gkbMªkt s u tSls ijek.kqvksa dh ,d xSl ,d fof'k"V mÙksftr voLFkk A esa rS;kj dh tkrh gSA ;g ykbeu Js.kh dh izFke js[kk ds rqY; rjaxnS/;Z ds QksVkWuksa ds lkFk ikap vU; rjaxnS/;k± ds QksVkWu mRlftZr djrh gSA xSl dks igpkfu;s vkSj voLFkk A
dh eq[; Dok.Ve la[;k Kkr dhft;sA Ans:
He + 4,
18.
Find the maximum angular speed of the electrons of a hydrogen atom in a stationary orbit.
gkbMªkstu ijek.kq dh LFkk;h d{kk esa bysDVªkWu dh vf/kdre dks.kh; pky Kkr dhft;sA Ans:
4.1 × 10 16 rad/s
19.
A spectroscopic instrument can resolve two nearby wavelength and + id / is smaller than 8000 .This is used to study the spectral lines of the Balmer series of hydrogen Approximately how many lines will be resolved by the instrument ? ,d LisDVªkWLdkWfid midj.k nks lehi okyh rjaxnS/;ks± vkSj + dks foHksfnr dj ldrk gS] ;fn / dk eku 8000
ls de gksA bldks gkbMªkt s u dh ckej Js.kh dh LisDVªeh js[kkvksa ds v/;;u ds fy;s iz;D q r fd;k tkrk gSA bl midj.k ds }kjk lfUudVr% fdruh js[kk,¡ foHksfnr dh tk ldsxh\ Ans:
38
20.
Suppose in certain conditions only those transitions are allowed to hydrogen atoms in which the principal quantum number n change by 2 (a) Find the smallest wavelength emitted by hydrogen (b) List the wavelengths emitted by hydrogen in the visible range (380 nm to 780)
eku yhft;s fd] fdUgha fof'k"V ifjfLFkfr;ksa esa gkbMªkt s u ijek.kq ds dsoy ogh laØe.k laHko gS]a ftuesa eq[; Dok.Ve la[;k n esa 2 dk ifjorZu gksrk gSA (a) gkbMªkstu }kjk mrlftZr U;wure rjaxnS/;Z Kkr dhft;sA (b) gkbMªkstu }kjk n`'; ijkl (380 nm ls 780nm) esa mRlftZr rjaxnS/;ks± dh lwph cukb;sA Ans:
(a) 103 nm (b) 487 nm
21.
According to Maxwell’s theory of electrodynamics an electrons going in a circle should emit radiation of frequency equal to its frequency of revolution .What should be the wavelengths of the radiation emdiation by a hydrogen atom in ground state if this rule is followed ? oS|rq xfrdh (electrodynamics) ds fy;s eSDlosy ds fl)kUr ds vuqlkj o`Ùkkdkj iFk ij xeu djus okyk bysDVªkuW ] ?kw.kZu
dh vko`fÙk ds rqY; vko`fÙk ds fofdj.k mRlftZr djsxkA ;fn bl fu;e dk ikyu gksrk gS] rks ewy voLFkk ds gkbMªkt s u ijek.kq ls mRlftZr fofdj.k dh rjaxnS/;Z fdruh gksxh\ manishkumarphysics.in
Page # 13
Chapter # 43 Ans: 45.7nm 22.
Bohr Model and Physics of the Atom
The average lometic energy of molecules in a gas at temperature T is .Find the temperature at which the average kinetic energy of the molecules of hydrogen equals the binding energy of its atoms will hydrogen remain in molecular form at this temperature ?Take = 8.62 × 10– 5 eV/K fdlh xSl ds v.kqvksa dh T rki ij vkSlr xfrt ÅtkZ 1.5 kT gksrh gSA og rki Kkr dhft;s ftl ij gkbMªkt s u v.kqvksa dh
vkSlr xfrt ÅtkZ dk eku blds ijek.kqvksa dh ca/ku ÅtkZ ds cjkcj gks tk;sA D;k bl rki ij gkbMªkt s u vkf.od :i esa –5 jgsxh\ Take = 8.62 × 10 eV/K Ans:
1.05 × 10 5 K
23.
Find the temperature at which the average thermal kinetic energy is equal to the energy needed to take a hydrogen atom from its ground state to n = 3 state. Hydrogen can now emit red light of wavelengths 653.1nm Because of Maxwellian distributions of speeds, a hydrogen sample emits red light at temperature much lower than that obtained from this problem. Assume that hydrogen molecules dissociate into atoms. og rki Kkr dhft;s ftl ij vkSlr Å"eh; xfrt ÅtkZ dk eku] gkbMªkt s u ijek.kq dks ewy voLFkk ls n = 3 voLFkk rdys tkus ds fy;s vko';d ÅtkZ ds rqY; gks tk;sA vc gkbMªkt s u 653.1 uSuksehVj rjaxnS/;Z dk yky izdk'k mRlftZr dj ldrh
gSA eSDlosy ds osx forj.k fu;e ds vuqlkj] izkIr fd;s x;s mDr rki ls cgqr de rki ij gh gkbMªkt s u v.kq] ijek.kqvksea as foHkDr gks tkrs gSAa Ans:
9.4 × 10 4 K
24.
Average lifetime of a hydrogen atom excited n = 2 state is 10– 8 s. Find the number of revolutions made by the electrons on the average before it jumps to ground state. n = 2 voLFkk ds mÙkstu ds fy;s gkbMªkt s u ijek.kq dk vkSlr vk;qdky 10– 8 lsd.M gSA bysDVªkWu }kjk ewy voLFkk esa dwnus
ls igys yxk;s x;s pDdjksa dh vkSlr la[;k Kkr dhft;sA 6
Ans:
8.2 × 10
25.
Calculate the magnetic dipole moment corresponding to the motion of the electrons in the ground state of a hydrogen atom.
gkbMªkt s u ijek.kq dh ewy&voLFkk esa bysDVªkWu dh xfr ds laxr pqEcdh; f}/kzoq vk?kw.kZ dh x.kuk dhft;sA Ans:
9.2 × 10
26.
Show that the ratio of the magnetic dipole moment to the angular momentum ( = mnr) is universal constant for hydrogen like atoms and ions. Find its value. O;Dr dhft;s fd gkbMªkt s u tSls ijek.kqvksa ,oa vk;uksa ds fy;s pqEcdh; f}&/kzoq vk?kw.kZ vkSj dks.kh; laoxs ( = mnr) dk vuqikr
,d lkoZf=kd fu;rkad gSA bldk eku Kkr dhft;sA Ans: 27.
e 8.8 1010 C / kg 2m A beam of light having wavelengths distributed uniformly between 450 nm to 550 passes through a sample of hydrogen gas. Which wavelength will have the least intensity in the transmitted beam ? gkbMªkt s u xSl ls ,d izdk'k iqt a xqtjrk gSA ftlesa 450 nm ls 550nm rd rjaxnS/;Z ,d leku forj.k gSA ikjxfer iqt a
esafdl rjaxnS/;Z dh rhozrk U;wure gksxh\ Ans: 28.
487 nm Radiation coming from transitions n = 2 to n = 1 of hydrogen atoms falls on helium ions in n = 1 and n =2 states. What are the possible transitions of helium ions as they absorb energy from the radiation? gkbMªkt s u ijek.kq esa n = 2 ls n = 1 laØe.k dkj.k vkus okys fofdj.k n = 1 vkSj n =2 voLFkkvksa okys ghfy;e vk;uksa ij
vkifrr gksrs gSAa ;fn ghfy;e vk;u bu fofdj.kksa ls ÅtkZ vo'kksf"kr djrs gS]a rks buesa laHko laØe.k D;k gks ldrs gS\a Ans:
n = 2 to n = 3 and n = 2 to n = 4
29.
A hydrogen atom in ground state absorbs a photon of ultraviolet radiation of wavelength 50 nm Assuming that the entire photon energy is taken up by the electron with what kinetic energy will the electrons be ejected ? ,d gkbMªkt s u ijek.kq ewy voLFkk esa 50 nm rjaxnS/;Z ds ijkcSxa uh fofdj.k dk QksVkWu vo'kksf"kr djrk gSA eku yhft;s fd
QksVkWu dh lEiw.kZ ÅtkZ bysDVªkWu }kjk xzg.k dj yh tkrh gS] bysDVªkWu fdruh xfrt ÅtkZ ds lkFk mRlftZr gksxk\ Ans :
11.24 eV manishkumarphysics.in
Page # 14
Chapter # 43 Bohr Model and Physics of the Atom 30. A parallel beam of light of wavelength 100nm passes through a sample of atomic hydrogen gas in ground state (a) Assume that when a photon supplies some of its energy to a hydrogen atom the rest of the energy by the excited hydrogen atoms in the direction of the incident beam what wavelengths may be observed in the transmitted beam ? (b) A radiation detector is placed neat the gas to detect radiation coming perpendicular to the incident beam .Find the wavelengths of radiation that may be detected by the detector. 100nm rjaxnS/;Zdk izdk'k iqt a ewy voLFkk dh ijekf.od gkbMªkt s u xSl ls xqtjrk gSA (a) eku yhft;s fd tc ,d QksVkWu
viuh dqN ÅtkZ gkbMªkstu ijek.kq dks iznku djrk gS] 'ks"k ÅtkZ ,d vU; QksVkWu dh fn'kk esa gh xfr djrk gSA mÙksftr ijek.kq }kjk vkifrr iqt a dh fn'kk esa mRlftZr izdk'k dks ux.; ekurs gq, ikjxfer iqt a esa dkSulh rjaxnS/;ks± dks izfs {kr fd;k tk;sxk\ (b) vkifrr iqat ds yEcor~ fn'kk esa vkus okys fofdj.kksa dks lalwfpr djus ds fy;s ,d fofdj.k lalwpd dks xSl ds lehi j[kk tkrk gSA lalwpd }kjk lalwfpr fofdj.kksa dh rjaxnS/;ks± dk eku Kkr dhft;sA
Ans:
(a) 100 nm , 560 nm 3880 (b) 103 nm, 121 nm 654 nm
31.
A beam of monochromatic light of wavelength ejects photoelectrons from a cesium surface ( = 1.9 eV) These photoelectrons are made to collide with hydrogen atoms in ground state .Find the maximum value of for which (a) hydrogen atoms may be ionised (b) hydrogen atoms may get excited from the ground state to the first excited state and (c) the excited hydrogen atoms may emit visible light. rjaxnS/;Z dk ,d o.khZ izdk'k iqat lhft+;e ( = 1.9 eV) dh lrg ls QksVks&bysDVªkWuksa dks mRlftZr djrk gSA bu QksVks&bysDVªkuW ksa dh ewy voLFkk okys gkbMªkt s u ijek.kqvksa ls VDdj djok;h tkrh gSA dk og vf/kdre eku Kkr dhft;s] ftlds fy;s (a) gkbMªkt s u ijek.kq vk;fur gks ldrk gS] (b) gkbMªkstu ijek.kq ewy voLFkk ls izFke mÙksftr voLFkk rd mÙksftr gks ldrs gSa rFkk (c) mÙksftr gkbMªkstu ijek.kq n`'; izdk'k mRlftZr dj ldrs gSaA (a) 80 nm( (b) 102 nm (c) 89 nm
Ans: 32.
Ans: 33.
Electrons are emitted from an electron gun at almost zero velocity and accelerated by an electric field E through a distance of 1.0 m The electrons are now scattered by an atomic hydrogen sample in ground state. What should be the minimum value of E so that red light of wavelength 656 .3 may be emitted by the hydrogen ? ,d bysDVªkWu&xu ls bysDVªkWu yxHkx 'kwU; osx ls mRlftZr gksrs gSa] vkSj 1.0 m nwjh rd fo|qr {ks=k E ls Rofjr gksrs gSAa vc Hkhk bysDVªkuW ewy voLFkk ds ijekf.od gkbMªkt s u izfrn'kZ ls izdhf.kZr gksrs gSAa E dk U;wure eku fdruk gks fd gkbMªkt s u ls 656.3nm rjaxnS/;Z dk yky izdk'k mRlftZr gks\ 12.1 V/m A neutron having kinetic energy 12.5 eV collides with a hydrogen atom at rest Neglect the difference in mass between the neutron and the hydrogen atom and assume that the neutron does not leaves its line of motion Find the possible kinetic energies of the neutron after the event. 12.5 eV xfrt ÅtkZ dk U;wVkª uW ] fojkekoLFkk esa fLFkr ,d gkbMªkt s u ds nzO;eku vUrj dks ux.; eku yhft;s vkSj ;g eku
yhft;s fd U;wVkª Wu bldh xeu js[kk dks ugha NksM+rk gSA bl ?kVuk ds i'pkr~ U;wVªkWu dh laHko xfrt ÅtkZ,a Kkr dhft;sA Ans:
zero
34.
A hydrogen atom moving at speed colloids with another hydrogen atom kept at rest Find the minimum value of for which one of the atoms may get ionized the mass of a hydrogen atom = 1.67 × 10–27 kg. pky ls xfr'khy ,d gkbMªkt s u ijek.kq fojkekoLFkk esa fLFkr ,d vU; gkbMªkt s u ijek.kq ls Vdjkrk gSA dk og U;wure eku Kkr dhft;s] ftlds fy;s gkbMªkstu ijek.kqvksa esa ls ,d vk;fur gks ldsA gkbMªkstu ijek.kq dk nzO;eku = 1.67 × 10–27 fdxzkA 7.2 × 10 4 m/s
Ans: 35.
A neutron moving with a speed v strikes a hydrogen atom in ground state moving towards it with the same speed .Find the minimum speed of the neutron for which inelastic (completely of partially) collision may get ionized The mass of neutron = mass of hydrogen = 1.67 × 10– 27kg v pky ls xfr'khy ,d U;wVkWu bldh vksj leku pky ls xfr'khy ewy voLFkk okys gkbMªkt s u ijek.kq ls Vdjkrk gSA U;wVkª Wu
dh og U;wure pky Kkr dhft;s] ftlds fy;s vizR;kLFk ¼iw.kZr;k ;k vkaf'kd½ VDdj gks ldsA U;wVkª Wu dh og U;wure pky Kkr dhft;s] ftlds fy;s vizR;kLFk ¼iw.kZr;k ;k vkaf'kd½ VDdj gks ldsA U;wVkª Wu dk nzO;eku = gkbMªkt s u dk nzO;eku = –27 1.67 × 10 fdxzk Ans:
3.13 × 104 m/s
manishkumarphysics.in
Page # 15
Chapter # 43 Bohr Model and Physics of the Atom 36. When a photons is emitted by a hydrogen atom, the photon carries a momentum with it(a) Calculate the momentum carried by the photon when a hydrogen atom emits light of wavelength 656.3nm (b) With what speed does the atom recoil during this transition ?Take the mass of the hydrogen atom =1.67 × 10 – 27 kg (c) Find the kinetic energy of recoil of the atom tc fdlh gkbMªkt s u ijek.kq ls QksVkWu mRlftZr gksrk gS] rks QksVkWu vius lkFk laoxs ys tkrk gSA (a) tc gkbMªkt s u ijek.kq 656.3 nm rjaxnS/;Z dk izdk'k mRlftZr djrk gS] rks QksVkWu }kjk ys tk;sx;slaosx dh x.kuk dhft;sA (b) bl laØe.k esa ijek.kq esa ijek.kq dh izfr{ksi pky D;k gksxh\ eku yhft;s fd gkbMªkt s u ijek.kq dk nzO;eku 1.67 × 10 – 27 fdxzk gSA (c)
izfr{ksfir ijek.kq dh xfrt ÅtkZ Kkr dhft;sA
Ans :
(a) 1.0 × 10 27 kg m/s (b) 0.6 m/s
37.
When a photon is emitted from an atom the atom recoils. The kinetic energy of recoil and the energy of the photon come from the difference in energies between the states involved in the energies between the states involved transition suppose a hydrogen atom changes its state from n = 3 to n = 2 Calculate the fractional change in the wavelength of light emitted, due to the recoil
tc fdlh ijek.kq ls QksVkWu mRlftZr gksrk gS] rks ijek.kq izfr{ksfir gksrk gSA izfr{ksi ds fy;s xfrt ÅtkZ vkSj QksVkWu dh ÅtkZ laØe.k esa Hkkx ysus okyh ÅtkZ voLFkkvksa dh ÅtkZvksa ds vUrj ls izkIr gksrh gSA eku yhft;s fd ,d gkbMªkt s u ijek.kq bldh voLFkk n = 3 ls n = 2 rd ifjofrZr djrk gSA izfr{ksi ds dkj.k mRlftZr izdk'k dh rjaxnS/;Z esa fdl xq.kkad ls ifjorZu gksxk] x.kuk dhft;sA Ans:
10 – 9
38.
The light emitted in the transition n = 3 to n = 2 in hydrogen is called H light. Find the maximum work function a metal can have so that H light can emit photoelectrons from it. gkbMªkstu ijek.kq esa n = 3 ls n = 2 laØe.k ls mRlftZr izdk'k H izdk'k blls QksVks&bysDVªkWu mRlftZr dj ldsA 1.9 eV
Ans: 39.
Light from Balmer series of hydrogen is able to eject photoelectrons from a metal .What can be the maximum work function of the metal?
gkbMªkt s u dh ckej Js.kh dk izdk'k fdlh /kkrq ls QksVks bysDVªkWuksa dks mRlftZr djus ds fy;s l{ke gSA /kkrq dk vf/kdre dk;ZQyu fdruk gks ldrk gS\ Ans:
3.4 eV
40.
Radiation from hydrogen discharge tube falls on a cesium plate. Find the maximum possible kinetic energy of the photoelectrons .Work function of cesium is 1.9 eV.
gkbMªkt s u foltZu ufydk esa mRlftZr fofdj.k lhft;e IysV ij vkifrr gksrs gSAa QksVks&bysDVªkuW ksa dh vf/kdre laHko xfrt ÅtkZ Kkr dhft;sA lhft+;edk dk;ZQyu 1.9 eV gSA Ans: 41.
11.7 eV A filter transmits only the radiation of wavelength greater than 440 nm. Radiation from a hydrogen discharge tube goes through such a filter and is incident on a metal of work function 2.0 eV Find the stopping potential which can stop the photoelectrons. fdlh fQYVj 440 nm ls vf/kd rjaxnS/;Z okys fofdj.k ikjxfer gksrs gSaA gkbMªkt s u foltZu ufydk ls mRlftZr fofdj.k ,sls gh fQYVj ls xqtjus ds i'pkr~ 2.0 eV dk;ZQyu okyh /kkrq dh lrg ij vkifrr gksrs gSAa fujks/kd foHko dk og eku
Kkr dhft;s] tks QksVks&bysDVªkWuksa dks jksd ldsA Ans: 42.
0.55 The earth revolves round the sun due to gravitational attraction Suppose that the sun and the earth are point particles with their existing masses and that Bohar ‘s quantization rule for angular momentum is valid in the case of gravitation (a) Calculate the minimum radius the earth can have for its orbit (b) What is the value of the principal quantum number for the present radius ? Mass of the earth = 6.0 ×1024 kg mass of the sun 2.0 × 1030 kg earth sun distance = 1.5 × 1011m.
i`Foh xq:Rokd"kZ.k cy ds dkj.k lw;Z ds pkjksa vksj ifjØek djrh gSA eku yhft;s fd i`Foh ,oa lw;Z buds nzO;ekuksa okys fcUnqor~ d.k gS rFkk cksgj dk dks.kh; laoxs ds Dokf.Vdj.k dk fu;e xq:Rokd"kZ.k ds fy;s Hkh ykxw gksrk gSA (a) i`Foh dh d{kk ds fy;s laHko U;wure f=kT;k dh x.kuk dhft;sA (b) f=kT;k ds vHkh okys eku ds fy;s eq[; Dok.Ve la[;k dk eku fdruk gksxk\ i`Foh dk nzO;eku = 6.0 ×1024 kg , lw;Z dk nzO;eku = 2.0 × 1030 kg , i`Foh&lw;Z dh nwjh = 1.5 × 1011m
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Chapter # 43 Bohr Model and Physics of the Atom Ans: (a) 2.3 × 10 – 138 m (b) 2.5 × 10 74 43. Consider a neutron and an electrons bound to each other due to gravitational force .Assuming Bohar s quantization rule for angular momentum to be valied in this case derive an expression for the energy of the neutron - electron system.
eku yhft;s fd ,d bysDVªkWu ,oa ,d U;wVkª Wu xq:Rokd"kZ.k cy ds dkj.k ijLij ca/ks gq, gSaA eku yhft;s fd bl fLFkfr esa cksgj dk dks.kh; laoxs Dokf.Vdj.k fu;e ykxw gksrk gS] U;wVkª Wu&bysDVªkWu fudk; ds fy;s ÅtkZ dk O;atd fuxfer dhft;sA Ans:
2 2 G mn2m 3e 2h 2 n 2
44.
A uniform magnetic field B exists in a region An electron projected perpendicular to the field goes in a circle Assuming Bohar’s quantization rule for angular momentum calculate (a) the smallest possible radius of the electron (b) the radius of the nth orbit and (c) the minimum possible speed of the electron. fdlh LFkku ij le:i pqEcdh; {ks=k B fo|eku gSA {ks=k ds yEcor~ iz{ksfir ,d bysDVªkWu o`Ùkkdkj iFk ij xeu djrk gSA cksgj dk dks.kh; laoxs Dokf.Vdj.k fu;e ekurs gq,] x.kuk dhft;s : (a) bysDVªkuW dh laHko U;wure f=kT;k (b) noha d{kk dh f=kT;k] rFkk (c) bysDVªkWu dh laHko U;wure pky
Ans:
(a)
45.
Suppose in an imaginary world the angular momentum is quantized to be even integral multiples of h/ 2p What is the longest possible wavelength emitted by hydrogen atoms in visible range in such a world according to Bohr’s model? eku yhft;s fd fdlh dkYifud nqfu;k¡ esa dks.kh; laoxs h/2 ds le xq.ktksa ds fy;s Dokf.VÑr gSA ,slh nqfu;k¡ esa cksgj izfr:i
h nh heB (b ) (c ) 2eB 2eB 2m 2
ds vuqlkj gkbMªkstu ijek.kqvksa }kjk n`'; ijkl esa mRlftZr rjaxnS/;Z dk vf/kdre laHko eku fdruk gks ldrk gS\ Ans: 46.
487 nm Consider an excited hydrogen atom in state n moving with a velocity ( << c ).It emits a photons in the direction of its motion and changes its state to a lower state m Apply momentum and energy conservation principles to calculate the frequency of the emitted radiation compare this wite the frequency vo emitted if the atom were at rest. ( << c ) osx ls xfr'khy ,oa n voLFkk esa mÙksftr gkbMªkt s u ijek.kq ij fopkj dhft;sA ;g bldh xfr dh fn'kk esa ,d QksVkWu mRlftZr djrk gS] vkSj bldh voLFkk ,d fuEu voLFkk m rd ifjofrZr dj ysrk gSA mRlftZr fofdj.k dh vko`fÙk dh x.kuk djus ds fy;s laoxs rFkk ÅtkZ ds laj{k.k fu;eksa dks ykxw dhft;sA ;fn ijek.kq fLFkj gksrk rks mRlftZr vko`fÙk vo
ds lkFk bldh rqyuk dhft;sA Ans:
v v = vo 1 c
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