q. what is the de broglie wavelength of an electron that has a kinetic energy of 100 ev? after an...
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Q. What is the de Broglie wavelength of an electron that has a kinetic energy of 100 eV?
After an electron is accelerated in 100 V potential difference, its kinetic energy is 100 eV.eV unit has to be converted into SI unit, Joule.
1 eV = 1.6 x 10-19 J
Ek = (1/2)mov2 = 1.6 x 10-17 J
v2 = 2Ek/mo = 2(1.6 x 10-17 J)/(9.1 x 10-31 kg) = 3.52 x 1013 m2/s2
v = 5.93 x 106 m/s low speed: no need to use relativistic
= h/p = h/mov = (6.6 x 10-34 Js)/(9.1 x 10-31 kg x 5.93 x 106 m/s) = 1.23 x 10-10 m = 0.123 nm
The Atomic Nucleus
Composed of nucleons: proton (+e) and neutrons (neutral)mp = 1.673 x 10-27 kg = 938.28 MeVmn = 1.675 x 10-27 kg = 939.4 MeVme = 9.1 x 10-31 kg = 0.511 MeV
Positively charged (# of protons = # of electrons)Heavy object: most of the atomic mass is concentrated in the nucleusSize ≈ 10-14 m = 10 femtometer (10-15 = femto)
Considering the size of an atom ≈ 10-10 m, an atom is mostlyempty.
The force binding nucleons together should overcome Coulomb repulsion!Nuclear force is very strong and attractive.
Each element has different # of protons periodic table is arranged by the # of protons.
E = mc2
atomic number: Z
Z= # of electrons: chemical property
Missing information about the # of neutrons?
8 neutrons
146 neutrons
“I began to look about and write down the elements with their atomic weights and typical properties, analogous elements and like atomic weights on separate cards, and this soon convinced me that the properties of elements are in periodic dependence upon their atomic weights.”
--Principles of Chemistry, 1905, Vol. II
Dmitri Ivanovich Mendeleev (1834-1907)
atomic mass number (A): total # of protons and neutronsA = Z + N (integer)
XA
Zelement
ElementElement SymbolSymbol ZZ NN Mass #Mass #
HydrogenHydrogen 11 00 11HeliumHelium 22 22 44LithiumLithium 33 44 77UraniumUranium 9292 146146 238238
H11He42
U23892
Li73
Approximate radius, R, for the nucleusR = (1.2 x 10-15)A1/3 m
The actual mass of atom or nucleus is expressed in atomic mass units (U).
1 u = 1.6607 x 10-27 kg= 931 MeV
Exactly equal to one-twelfth of the mass of the most abundant form of the carbon atom
mp = 1.673 x 10-27 kg = 1.007276 umn = 1.675 x 10-27 kg = 1.008665 ume = 9.1 x 10-31 kg = 0.00055 u
The average atomic mass of mercury (Hg) is 200.59 u. What is the average nuclear mass for Hg atom?
Z = 80 for Hg there are 80 electrons.
Mass of electrons in Hg atom = 80 x 0.00055 u = 0.044 u
Nuclear mass = atomic mass – electron mass = 200.59 u – 0.044 u = 200.55 u
include everything (p, n, and e)
Isotopes
Atoms that have the same atomic number (Z) but different mass numbers (A).
XA
Z
Li7
3Li
6
3
Natural abundance: 92% 8%
Co58
27Co
60
27
Helium Nucleus
He4
2
atomic mass = 4.002602 u2 electrons: electron mass = 2 x 0.00055 u
= 0.0011 unucleus mass = 4.002602 u – 0.0011 u
= 4.001502 u
We know He nucleus is composed of 2 protons and 2 neutrons:
2 proton mass = 2 x 1.007276 u2 neutron mass = 2 x 1.008665 utotal mass = 4.031882 u > He nucleus mass
m = 4.031882 u – 4.001502 u = 0.030376 u
All nuclei have a mass that is smaller than the sum of the masses of its separated nucleons. This difference in mass is called the mass defect.
E = mc2E = mc2
binding energy of the nucleons in the nucleus
m for He = 0.030376 u = 0.030376 x (1.66 x 10-27 kg) = 5.0 x 10-29 kg
E = mc2 = (5.0 x 10-29 kg)(3 x 108 m/s)2
= 4.5 x 10-12 J = 28 MeV
1 u = 1.6607 x 10-27 kg = 931 MeV
= 0.030376 x 931 MeV= 28.2 MeV
binding energy per nucleon in He = 7 MeV
np
He nucleus
28 MeV
Ex. 31.4 How much energy is released as 1 kg of neutrons and 1 kg of protons are changed into 2 kg of helium 4 nuclei?
# of He nuclei in 2 kg = 2 kg/(4 x 1.66 x 10-27 kg) = 3.0 x 1026
28 MeV is released per nucleus:total energy released = 28 x 3.0 x 1026 MeV
= 8.4 x 1027 MeV = 1.5 x 1015 J
New York city needs 1010 W of electricity. This amount can run NY city for 2 days!!
Radioactivity
Sometimes particles or photons are emitted from the nuclei of atoms: unstable nuclei
radioactive
unstable nuclei stable nuclei
all naturally occurring elements with Z > 83
Three major forms of radioactive emission:
Alpha particle (): nucleus of a He atom
Beta particles (): beta minus particle (electron)
beta plus particle (positron)
Gamma rays (): high energy photons, short wavelength x-ray
420101
XA
ZX
A
Z+
XA
ZX
A-4
Z-242+
XA
ZX
A
Z+1+ 01
A and Z are conserved in nuclear reaction!!
T1/2: half-life
N(t) = Noe-pt
# of unstable nuclei at time t
Initial # of unstable nuclei at time t = 0
At T1/2, N(T1/2) = (0.5)No
692.02ln
2
1lnln
2
1)(
2/1
2/1
2/1
2/1
pT
e
eN
TN
pT
pT
o
If you know T1/2, you can get p from this equation. Then, at any time you can calculate # of unstable nuclei. Or, with the knowledge of # of unstable nuclei, we can calculate how much time has passed. Carbon dating
decay constant
Carbon Dating
By measuring the ratio of 14C isotope to 12C in a biological sampleone can calculate the age of the object from which the sample came with the knowledge of the initial ratio when the biological activity ceased.
should be dead!
half-life: 5730 yrs?
bbc.co.uk
A mummy found in Egypt
147N + 1
0n 146C + ? 1
1p
Ratio of 14C to 12C?
Before 1900 (atomic bomb testing)1.5 x 10-12
146C 14
7N + ? 0-1
beta decay
bbc.co.uk
A mummy found in Egypt
Measured the ratio of 14C to 12Cr(t) = 1.05 x 10-12
ro = 1.5 x 10-12
T1/2 = 5730 yrs
Let’s calculate decay const, pp = 0.693/T1/2
= 0.693/5730= 1.21 x 10-4 (yr-1)
Total # of 12C = N12
# of 14C = N14 = r(t)N12
r(t)N12 = roN12e-pt
r(T)/ro = (1.05/1.5) = e-pT
ln(0.7) = -pTT = -ln(0.7)/(1.21 x 10-4) = 2848 yr
Ex. 31.10 Cobalt-60 has a half-life of 5.25 yrs (1.66 x 108 s). Given 1 mg of the substance, find (a) the number of atoms in the sample, (b) the decay constant, (c) the time taken for 0.01 mg to decay.
mass of Co-60 = 59.93381 u = 60 u= 60 x 1.66 x 10-27 kg= 1.0 x 10-25 kg
No = (1 x 10-6 kg)/(1 x 10-25 kg) = 1 x 1019 atoms
p = 0.693/T1/2 = 0.693/(1.66 x 108 s) = 4.2 x 10-9 s-1
How long does it take to reach N/No = 0.99?ln(N/No) = -pt
ln(0.99) = -(4.2 x 10-9)tt = 2.4 x 106 s about a month
1 yr = x 107 s
Which of the following nuclear reaction is(are) impossible?Which of the following nuclear reaction is(are) impossible?
(a) (a) 2312319191Pa Pa + + 227227
8989AcAc (b) (b) 1414
66C C 141477N + N + +1+1
(c) (c) 2402409292U U -1-1 + + 240240
9393NpNp
45
20% 20% 20%20%20%
1.1. (a)(a)
2.2. (b)(b)
3.3. (c)(c)
4.4. AllAll
5.5. NoneNone
Nuclear Fission
In a chemical reaction, changes in atoms are in their electron configuration. However, in a nuclear reaction, the configuration of nucleus changes by emitting or absorbing typically high energy photons, nucleons, alpha particles, and/or light charged particles. In this process, charge, nucleons, and mass-energy are conserved.
Nuclear fission: Unstable heavy nucleus multiple intermediate mass nuclei
+ large energy
U235
92
n
stable isotopeT1/2 ≈ 109 yrs
U236
92
unstable
Kr95
36
Ba138
56
n
n
n
200 MeV per fission
In 1938, Otto Hahn, Lise Meitner, and Fritz Strassmann discovered:
235U 236U 92Kr141Ba
n
Chain Reaction
Through the release of atomic energy, our generation has brought into the world the most revolutionary force since prehistoric man's discovery of fire. This basic force of the universe cannot be fitted into the outmoded concept of narrow nationalisms. For there is no secret and there is no defense; there is no possibility of control except through the aroused understanding and insistence of the peoples of the world. We scientists recognise our inescapable responsibility to carry to our fellow citizens an understanding of atomic energy and its implication for society. In this lies our only security and our only hope - we believe that an informed citizenry will act for life and not for death.
A. Einstein, 1947
"When you sit with a nice girl for two hours, it seems like two minutes. When you sit on a hot stove for two minutes, it seems like two hours that's relativity."
"Gravitation is not responsible for people falling in love."