50Radioactivity

📚
RADIOACTIVITY
▢ Radioactivity:
❖ Definition: Spontaneous disintegration of unstable nucleus with emission of radioactive radiations
❖ Nature: Nuclear process, not atomic process
❖ Order: First order reaction
❖ Important Points:
  • •
    Discovered by Becquerel
  • •
    Radioactivity is spontaneous and random
  • •
    Independent of external physical and chemical conditions
  • •
    Fusion is not mode of radioactive decay
  • •
    Radioactive substance decays exponentially
  • •
    Lead is heaviest stable nucleus
▢ Types of Radioactive Radiations:

Table 1: Alpha, Beta and Gamma Rays

Feature
Alpha \((\alpha)\)
Beta \((\beta)\)
Gamma \((\gamma)\)
Nature
Helium nucleus
Fast electron / positron
Electromagnetic photon
Symbol
\(*2He^4\)
\(*{-1}e^0\) or \(*{+1}e^0\)
\(\gamma\)
Charge
+2e
-e / +e
0
Mass
4 amu
Nearly zero compared to alpha
0
Ionising power
Maximum
Medium
Minimum
Penetrating power
Minimum
Medium
Maximum
Magnetic deflection
Least
Maximum
No deflection
Spectrum
Line spectrum
Continuous spectrum
Line spectrum
❖ Order:
◉ Ionising Power: \(\alpha>\beta>\gamma\)
◉ Penetrating Power: \(\gamma>\beta>\alpha\)
◉ Magnetic Deflection: \(\beta>\alpha>\gamma\)
▢ Radioactive Displacement Law:
❖ Discovered By: Soddy and Fajans, 1913

Table 1: Displacement Laws

Emission
Mass number change
Atomic number change
Periodic table shift
\(\alpha\)-emission
\(A\to A-4\)
\(Z\to Z-2\)
Two places left
\(\beta^-\)-emission
No change
\(Z\to Z+1\)
One place right
\(\gamma\)-emission
No change
No change
No displacement
❖ Equations:
  • •
    \(*Z^AX\to *{Z-2}^{A-4}Y+*2^4He\)
  • •
    \(*Z^AX\to *{Z+1}^{A}Y+*{-1}^{0}e\)
  • •
    \(*Z^AX^*\to *Z^AX+\gamma\)
▢ Radioactive Series:

Table 1: Natural Radioactive Series

Series
Mass number type
Parent nucleus
\(\alpha\) particles
\(\beta\) particles
Stable end product
Thorium
\(4n\)
\(*{90}^{232}Th\)
6
4
\(*{82}^{208}Pb\)
Neptunium
\(4n+1\)
\(*{93}^{237}Np\)
7
3
\(*{83}^{209}Bi\)
Uranium
\(4n+2\)
\(*{92}^{238}U\)
8
6
\(*{82}^{206}Pb\)
Actinium
\(4n+3\)
\(*{92}^{235}U\)
7
4
\(*{82}^{207}Pb\)
▢ Decay Law:
❖ Statement: Rate of disintegration at any instant is directly proportional to number of undecayed nuclei present

Table 1: Radioactive Decay Formulae

Quantity
Formula
Decay rate
\(-\frac{dN}{dt}\propto N\)
Activity
\(R=-\frac{dN}{dt}=\lambda N\)
Number left
\(N=N_0e^{-\lambda t}\)
Activity after time \(t\)
\(R=R_0e^{-\lambda t}\)
Mass after time \(t\)
\(M=M_0e^{-\lambda t}\)
Fraction left
\(\frac{N}{N_0}=e^{-\lambda t}\)
Fraction decayed
\(1-\frac{N}{N_0}=1-e^{-\lambda t}\)
❖ Graph: Decay curve is exponential
▢ Decay Constant:
❖ Symbol: \(\lambda\)
❖ Meaning: Probability of decay per unit time

Table 1: Decay Constant Relations

Relation
Formula
Activity
\(R=\lambda N\)
Half-life relation
\(\lambda=\frac{0.693}{T*{1/2}}\)
Mean life relation
\(\lambda=\frac{1}{T*{av}}\)
Half-life
\(T*{1/2}=\frac{0.693}{\lambda}\)
Mean life
\(T*{av}=\frac{1}{\lambda}\)
❖ Depends On: Nature of radioactive substance
❖ Independent Of: Temperature, pressure, physical state and chemical state
▢ Half Life and Mean Life:

Table 1: Half Life and Mean Life

Quantity
Meaning / Formula
Half-life \((T*{1/2})\)
Time in which 50% of original atoms decay
Mean life \((T*{av})\)
Average life of radioactive atoms
Mean life relation
\(T*{av}=1.44T*{1/2}\)
Mean life comparison
Mean life > half-life
After \(n\) half-lives
\(\frac{N}{N_0}=\left(\frac{1}{2}\right)^n\)
Number of half-lives
\(n=\frac{t}{T*{1/2}}\)
Mass left
\(\frac{M}{M_0}=\left(\frac{1}{2}\right)^n\)
Activity left
\(\frac{R}{R_0}=\left(\frac{1}{2}\right)^n\)
▢ Radioactive Equilibrium:
❖ Condition: Activity of parent nuclide equals activity of daughter nuclide
❖ Formula: \(R_1=R_2\)
❖ Relation: \(N_1\lambda_1=N_2\lambda_2\)
❖ Equivalent: \(\frac{N_1}{T*{1/2,1}}=\frac{N_2}{T*{1/2,2}}\)
▢ Simultaneous Decay:
❖ Condition: Radioactive substance decays by two modes with half-lives \(T_1\) and \(T_2\)

Table 1: Effective Decay Constant and Half-Life

Quantity
Formula
Effective decay constant
\(\lambda*{eff}=\lambda_1+\lambda_2\)
Using half-life
\(\frac{0.693}{T*{eff}}=\frac{0.693}{T_1}+\frac{0.693}{T_2}\)
Effective half-life
\(\frac{1}{T*{eff}}=\frac{1}{T_1}+\frac{1}{T_2}\)
Result
\(T*{eff}=\frac{T_1T_2}{T_1+T_2}\)
▢ Activity Units:

Table 1: Radioactivity Units

Unit
Value
Becquerel
1 disintegration per second
Curie
\(1Ci=3.7\times10^{10}\ dps\)
Rutherford
\(1Rd=10^6\ dps\)
1 Curie
\(3.7\times10^4\ Rutherford\)
▢ Uses of Radioisotopes:

Table 1: Radioisotope Uses

Radioisotope
Use
\(C-14\)
Determination of age of fossils and plants
\(Co-60\)
Treatment of cancer
\(I-131\)
Treatment of thyroid gland
\(Na-24\)
Detection/remedy of blood circulation defects
▢ Nuclear Stability:

Table 1: Stability Rules

Condition
Result
Stable light nuclei
\(\frac{n}{p}=1\)
Radioactive / unstable nuclei
\(\frac{n}{p}>1\) or \(\frac{n}{p}<1\)
\(\frac{n}{p}>1\)
\(\beta^-\)-decay tendency
\(\frac{n}{p}<1\)
\(\alpha\)-decay / positron-related tendency
Most stable elements
Even protons and even neutrons
❖ Lead: Heaviest stable nucleus
▢ Alpha Decay:
❖ Definition: Emission of helium nucleus from radioactive nucleus
❖ Equation: \(_Z^AX\to _{Z-2}^{A-4}Y+_2^4He+Q\)

Table 1: Alpha Decay Effects

Quantity
Change
Mass number
Decreases by 4
Atomic number
Decreases by 2
Periodic table
Two places left
Product type
Isodiaphers
Explanation
Tunnel effect
❖ Energy Distribution:
◉ Daughter Energy: \(E_1=\frac{m_2}{m_1+m_2}E\)
◉ Alpha Particle Energy: \(E_2=\frac{m_1}{m_1+m_2}E\)
▢ Beta Decay:
❖ Beta Minus: \(_Z^AX\to *{Z+1}^{A}Y+*{-1}^{0}e+\bar\nu\)
❖ Beta Plus: \(_Z^AX\to *{Z-1}^{A}Y+*{+1}^{0}e+\nu\)

Table 1: Beta Decay Effects

Emission
Mass number
Atomic number
Product
\(\beta^-\)
No change
Increases by 1
Isobar
\(\beta^+\)
No change
Decreases by 1
Isobar
❖ Important Points:
  • •
    \(\beta\)-particles produce continuous spectrum
  • •
    \(\beta\)-particles are most deflected by magnetic field
  • •
    \(\beta\)-particles are fast-moving electrons having one unit negative charge
▢ Gamma Decay:
❖ Definition: Emission of high-energy electromagnetic photon from excited nucleus

Table 1: Gamma Decay

Feature
Answer
Mass number
No change
Atomic number
No change
Energy of nucleus
Decreases
Product
Isomer
Direct production
Generally not direct; follows alpha or beta decay
Spectrum
Line spectrum
▢ Geiger Muller Counter:
❖ Also Called: GM counter
❖ Use: Measures/detects radioactivity
❖ Detects:
  • •
    \(\alpha\)-particles
  • •
    \(\beta\)-particles
▢ Cloud Chamber:
❖ Use: Detecting radioactive radiation and determining path, range and energy
▢ Penetrating and Ionising Power:

Table 1: Power Relations

Quantity
Relation
Ionising power
Directly proportional to mass and energy
Penetrating power
\(\propto\frac{1}{mass}\)
Penetrating order
\(\gamma>\beta>\alpha\)
Ionising order
\(\alpha>\beta>\gamma\)
▢ Nuclear Reactions and Fission:
❖ Fission: Splitting of heavy nucleus into lighter nuclei with release of large energy

Table 1: Fission Facts

Fact
Answer
Energy from 1 atom of \(U^{235}\)
About 200 MeV
Best nuclear fuel
\(U^{235}\), followed by \(Pu^{239}\)
Mass-energy relation
\(E=\Delta mc^2\)
If \(\Delta m\) in amu
\(E=931\Delta m\ MeV\)
▢ Read and Digest:
❖ _*table:
    ▢ Objective Answer Key:
    ❖ _*table:
      ▢ High-Yield Recall:

      Table 1: Radioactivity One-Liners

      Fact
      Answer
      Radioactivity
      Spontaneous nuclear disintegration
      Radioactivity order
      First order
      Decay law
      \(N=N_0e^{-\lambda t}\)
      Activity
      \(R=\lambda N\)
      Half-life
      \(T*{1/2}=\frac{0.693}{\lambda}\)
      Mean life
      \(T*{av}=\frac{1}{\lambda}=1.44T*{1/2}\)
      After \(n\) half-lives
      \(N=N_0\left(\frac{1}{2}\right)^n\)
      Curie
      \(3.7\times10^{10}\ dps\)
      Rutherford
      \(10^6\ dps\)
      Alpha particle
      \(_2^4He\)
      Alpha decay
      \(A-4,Z-2\)
      Beta minus decay
      \(A\ same,Z+1\)
      Gamma decay
      \(A,Z\ same\)
      Penetrating power
      \(\gamma>\beta>\alpha\)
      Ionising power
      \(\alpha>\beta>\gamma\)
      Most deflected
      \(\beta\)-rays
      Least penetrating
      \(\alpha\)-rays
      Most penetrating
      \(\gamma\)-rays
      GM counter
      Measures radioactivity
      Cloud chamber
      Detects path/range/energy of radiations
      C-14
      Age of fossils and plants
      Co-60
      Cancer treatment
      I-131
      Thyroid treatment
      Na-24
      Blood circulation defect
      Alpha decay product
      Isodiapher
      Beta decay product
      Isobar
      Gamma decay product
      Isomer
      Nuclear fission of \(U^{235}\)
      About 200 MeV per atom
      Mass-energy
      \(E=\Delta mc^2=931\Delta m\ MeV\)
      Q1.
      The percentage of mass which changes into energy fission is in the order of:
      📅IOM 2011
      Q2.
      Activity of 1.85×1011 dps is nearly equivalent to:
      📅IOM 2011
      Q3.
      Thorium (Atomic mass 232 and atomic no-90) is converted to lead (Atomic mass-208 & atomic no-82) then no. of α & β particles emitted is
      📅IOM 2009
      Q4.
      75% of substance decays in 32 min, then 50% decays in
      📅IOM 2009
      Q5.
      The half life of a radioactive substance is 2 months then the amount of substance left after 1 year is
      📅IOM 2009
      Q6.
      Two proton and a deuteron fuse to form a nucleus with a positron. The product nucleus may be
      📅MOE 2013
      Q7.
      1n + 92U235 → 56Ba141 + 36Kr92 + 31n + 0.1824 represents a nuclear fission reaction. The energy released in this reaction is
      📅MOE 2012
      Q8.
      Energy released by the fission of 1kg of U-235 is approx
      📅MOE 2012
      Q9.
      A nucleus with mass number 220 initially at rest emits an α-particle. If Q-value of the reaction is 5.5 MeV, K.E. of an α particle is
      📅MOE 2012
      Q10.
      The disintegration constant of radioactive sample is 3×10-6 s-1. Its half life is
      📅MOE 2063
      Q11.
      Law of radioactivity is generally expressed as N = N0e-λt when symbols have usual meanings. A plot of N against time t will be
      📅MOE 2010
      Q12.
      Initial mass of a radioactive sample of half-life 6 hours is 0.8kg. The amount of sample left after 1 day (24 hrs) is
      📅MOE 2009
      Q13.
      If the half life of radioactive radium is 1600yrs. Then find the radium left after 4800 yrs
      📅KU 2013/2016
      Q14.
      The half life of a radioactive substance is 20 min. Difference between the points of time when it is 33% disintegrated and 67% disintegrated is approximately
      📅KU 2012
      Q15.
      A radioactive substance has a half life of one hour. The fraction of substance that would remain un-decayed in three hours will be:
      📅KU 2011
      Q16.
      The rate of disintegration of radioactive substance at a certain instant of time is directly proportional to
      📅KU 2010
      Q17.
      Half life of radioactive substance is 10yrs and time taken for 99.9% decay is
      📅BP 2011
      Q18.
      What fraction of a radioactive substance of initial mass is left after 4 days if its half life time is 8 days?
      📅BP 2012
      Q19.
      1 curie of radioactivity is equal to how many Rutherford's
      📅BP 2013/2016
      Q20.
      A radioactive substance decays to 1/16th of initial activity in 2 hours. The half-life of the radioactive substance expressed in minute is
      📅BP 2013
      Q21.
      The half life of radium is 1600 years. When will the 100gm of the substance reduces to 25gm?
      📅I.E. 2012
      Q22.
      13Al27 is a stable isotope. 13Al24 is expected to disintegrate by
      📅I.E. 2010
      Q23.
      A radioactive substance decays as:
      n → p + e- + X
      The substance 'X' is
      📅I.E. 2011
      Q24.
      A radioactive substance has a half life of 5 minutes. The amount of substance decayed in 20 minutes is
      📅J.E. 2011
      Q25.
      The no. of atoms initially is 106. If the half life of materials is 2 days then after 5 days the activity in Becquerel is
      📅MOE 2014
      Q26.
      Correct decay for an emission of β-particle from a radioactive nucleus ZXA may be
      📅MOE
      Q27.
      The activity of a sample of radioactive material is A1 at time t1 and A2 at time t2 (t2 > t1), its mean life in time T is such that
      Q28.
      The fraction of the radioactive substances decayed in a time equal to the average life is
      Q29.
      What percentage of original radioactive atoms is left after four half lives
      Q30.
      Half life of a radioactive substance is 2 min, then time between 33% decay and 67% decay will be
      Q31.
      The mass number of a nucleus is
      Q32.
      The activity of a radioactive sample falls from 600 s-1 to 500 s-1 in 40 minutes. Calculate its half life
      Q33.
      75% of substance decay in 32 min, then 50% decay in
      📅IOM
      Q34.
      One Curie is the activity of 1 gram of
      Q35.
      Two radioactive materials X1 and X2 have decay constants 10λ and λ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of X1 to that of X2 will be 1/e after time
      Q36.
      Activity of 1.85 × 1011 dps is nearly equivalent to
      📅IOM
      Q37.
      If Nt = N0e-λt then the number of atoms decayed during time interval from t1 to t2 (t2 > t1) is
      Q38.
      A radioactive elements has half life of 7 yr. What fraction will remain after 150 yr.
      📅BPKIHS
      Q39.
      A radioactive source had decayed to 1/64 of its initial activity in 60 days. The half life of source is
      Q40.
      N-atoms of a radioactive substance emits n α-particles/sec on decaying. The half life of the radioactive substance is
      Q41.
      If one end A of a wire is irradiated with α-rays and the other end B is irradiated with β-rays then:
      Q42.
      A radioactive source has a half life of 3hr. A freshly prepared sample of the same emits radiation 16 times the permissible safe value. The minimum time after which it would be possible work safely with source is
      Q43.
      A radioactive isotope has a half life of T yr. After how much time is its activity reduced to 6.25% of its original activity?
      Q44.
      The (e/m) of β-particle emitted in radioactivity nuclide in comparison to the value of (e/m) for photoelectron is
      Q45.
      There are two radioactive nuclei A and B. A is an alpha emitter and B is a beta emitter. Their disintegration constant is in the ratio of 1:2. What should be the ratio of number of atoms of A and B at any time so that probabilities of getting alpha and beta particle are same at that instance.
      Q46.
      A sample contains 16 gm of a radioactive material, the half life of which is two days. After 32 days, the amount of radioactive material left in the sample is
      Q47.
      At any instant, the ratio of the amount of radioactive substances is 2:1. If their half lives be respectively 12 and 16 hr, then after 48 days, what will be the ratio of the substances?
      Q48.
      The half life of a radioactive substance is 6 h. After 24 h, its activity is 0.01 μCi. What was its initial activity?
      Q49.
      In a sample of radioactive material, what fraction of initial number of active nuclei will remain undisintegrated after half of a half life of the sample
      Q50.
      Two radioactive sources A and B of half live 1 hr and 2 hr respectively initially contain the same number of radioactive atoms. At the end of two hr, their rates of disintegration are in the ratio of
      Q51.
      The half life of a radioactive substance is 30 days what is the time taken for 3/4 of original mass to disintegrate
      Q52.
      Initial mass of a radioactive sample of half-life 6 hours is 0.8kg. The amount of the sample left after 1 day (24 hours) is
      📅MOE 2009
      Q53.
      If the half life of a radioactive specimen is 1600 years. Then find the specimen left after 6400 years.
      📅KU 2010
      Q54.
      A radioactive element ZXA emits an α particle, and changed into Z-2YA-4. The other particle emitted is
      📅Bangladesh 09
      Q55.
      The mass of a radioactive salt of half life 2 days is 10 gm. What amount of the salt will be left after 10 days?
      📅IOM 05
      Q56.
      A radioactive sample has half-life of 5 days. What time is taken by 7/8 of the sample to decay.
      📅MOE 066
      Q57.
      Which of the following does not belong to the electromagnetic spectrum
      📅MOE 2063
      Q58.
      50% of a radioactive substance decomposes in 5 years. What is the time for the 99.9% decomposition?
      📅MOE 2062/10M
      Q59.
      In most stable elements the number of proton and neutron is
      📅MOE 2062
      Q60.
      What is the average life of a radioactive substance having half-life period of 1600 years
      📅MOE 2053
      Q61.
      The % of radioactivity remained after 5 half-life time periods is:
      📅MOE
      Q62.
      The number of radiation of a radioactive sample is 1,28,000 counts min-1. After 2 min, it is reduced to 8,000 counts min-1. Find its half life period.
      📅IE-01
      Q63.
      A nuclei having maximum number of neutrons emits
      📅IE-02
      Q64.
      Cu29(z = 29) is converted to Cu(z = 30) by the emission of
      📅IE-06
      Q65.
      The particle(?) in the reaction
      197Au + ? → 197Hg + 0e-
      📅IE-07
      Q66.
      A nucleus with an excess of neutrons may decay radioactivity with the emission of:
      📅IE-07
      Q67.
      Naturally occurring radioactive atoms can spontaneously emit:
      📅IE-07
      Q68.
      A radioactive nucleus is being produced at a constant rate α per second. Its decay const. is λ. If N0 are the number of nuclei at time t = 0, the number at time t = t will be
      📅BPKIHS-09
      Q69.
      92U238 → 90Th234 + ... The products are
      📅BPKIHS 02
      Q70.
      Which of the following is the best nuclear fuel?
      📅BPKIHS-07
      Q71.
      A radioactive source has decayed to 1/64th after 60 days. The half life of the source is:
      📅BPKIHS 05
      Q72.
      One sixteenth of the initial amount of radioactive isotope remains undecayed after two hours. The half life of isotope is
      📅BPKIHS-06
      Q73.
      Which of the following has maximum energy?
      📅BPKIHS-06
      Q74.
      The rate of decay of a radioactive element
      📅BPKIHS-97
      Q75.
      Maximum ionization is exhibited by which rays
      📅BPKIHS 98/IOM
      Q76.
      In the Uranium radioactivity series, the initial nucleus is 92U238 and the final nucleus is 82Pb206. When Uranium decays into lead, the number of α and β particles emitted are
      Q77.
      A deutron is bombarded on 8O16 nucleus and an α-particle is emitted. The product nucleus is
      Q78.
      A nucleus ZXA emits one alpha particle and two β-particles. The resulting nucleus is
      Q79.
      In the disintegration series 92U238 → ZXA + 3α + 2β, the value of Z and A respectively will be
      Q80.
      A nucleus ZXA emits an α-particle. The resultant nucleus emits a β+ particle. The respective atomic and mass numbers of the final nucleus will be
      Q81.
      In the nuclear reaction 6C11 → 5B11 + β+ + X, what does X stand for?
      Q82.
      When 88Ra decays in a series by emission of 3 alpha particles and one beta particle, isotope X formed is
      Q83.
      When a β- particle is emitted from a radioactive nucleus, the ratio of the number of neutrons to the number of protons in the resulting nucleus
      Q84.
      At time t = 0, activity of radioactive substance is 1600 Bq and becomes 100 Bq at 8 sec. Then the activity of the substance at 2 sec is
      Q85.
      At time t = 0, number of atoms of radioactive substance is 100 and becomes 90 at 1s. Then number of atom left at 2s is
      Q86.
      If 200 MeV of energy is released in disintegration (fission) of 1 nucleus of Uranium. Then, how many nuclei must be disintegrated per sec to produce a power of 1KW.
      Q87.
      Percentage of atom disintegrated in 5 days is 10%. Then, the percentage of atom left after 20 days will be
      📅BPKIHS/IOM/MOE
      Q88.
      A sample with half life 2hr is 64 times stronger than safer amount. After how much time, it's safe to work with the sample?
      📅KU 2016