  ## Nuclei

- Understand the topics that are frequently asked in JEE exam
1
JEE Mains

## Binding energy and binding energy per nucleon

If a certain number of neutrons and protons are brought together to form a nucleus of a certain charge and mass, an energy will be released in the process. The energy is called the binding energy of the nucleus.

The ratio of the binding energy of a nucleus to the number of the nucleons, A, in that nucleus is called the binding energy per nucleon, .
Consider the nuclear fission,
Given that the binding energy/nucleon of , and are, respectively, 8.03 MeV, 7.07 MeV and 7.86 MeV, identify the correct statement:
A
8.3 MeV energy will be released
B
energy of 12.4 MeV will be supplied
C
energy of 3.6 MeV has to be supplied
D
none of the above
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## Decay constant

If is the number of nuclei in the sample and undergo decay in time then

or
where is called the radioactive decay constant or disintegration constant.
The activity of a nuclide is millicurie. If its decay constant is sec, the number of atoms present in it is
A
11.1 x 10
B
1 x 10
C
111 x 10
D
1.1 x 10
3

## Activity of a radioactive substance

The total decay rate R of a sample of one or more radionuclides is called the activity of that sample.

Here is the radioactive decay rate at time t = 0, and R is the rate at any subsequent time t.

A certain radioactive material is known to decay at a rate proportional to the amount present. Initially there is of the material present and after two hours it is observed that the material has lost 10%of its original mass, then the
A
mass of the material after four hours is
B
mass of the material after four hours is
C
time at which the material has decayed to half of its initial mass (in hours) is
D
time at which the material has decayed to half of its initial mass (in hours) is
The activity of a radioactive sample is measured as N counts per minute at t = 0 and N/e counts per minute at t = 5 minutes. The time (in minutes) at which the activity reduces to half its value is
A
log2/5
B
C
5 log2
D
5 log2
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## Half life of a radioactive substance

The half-life of a radio nuclide is the time at which both N and R have been reduced to one-half their initial values.
Putting and in the law of radioactive decay, we get:

The initial number of atoms of a radioactive element with half life days, is . The number of atoms remaining undecayed after days, will be
A
B
C
D
5

## Fraction of substance decayed in a period of time

Tritium has a half-life of 12.5 y undergoing beta decay.What fraction of a sample of pure tritium will remain undecayed after 25 y.
Solution
By definition of half-life, half of the initial sample will remain undecayed after 12.5 y. In the next 12.5 y, one-half of these nuclei would have decayed. Hence, one fourth of the sample of the initial pure tritium will remain undecayed.

A piece of bone from an archaeological site is found to give a count rate of counts per minute. A similar sample of fresh bone gives a count rate of counts per minute. Calculate the age of the specimen. Given
6

## Decay constant

If is the number of nuclei in the sample and undergo decay in time then

or
where is called the radioactive decay constant or disintegration constant.
In an experiment the initial number of radioactive nuclei is . It is found that nuclei decayed in the . For up to first power in . The error , in the determination of the decay constant , is , is
A
B
C
D
decays to with half-life year. The resulting is in excited state and hence, emits a further a gamma ray to come to the ground state, with half-life . A sample of emits gamma rays per second. In what time, the emission rate will drop to gamma ray per second ?
A
B
C
year
D
year
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## Activity of a radioactive substance

The total decay rate R of a sample of one or more radionuclides is called the activity of that sample.

Here is the radioactive decay rate at time t = 0, and R is the rate at any subsequent time t.

A small amount of solution containing radionuclide with activity disintegration per second was injected into the bloodstream of a man. The activity of 1 of the blood sample taken t= 5.0 hours later turned out to be A'= 16 disintegration per minute per . The half-life of the radionuclide is T= 15 hours. Find the volume of the man's blood.
A human body excretes ( removes by waste discharge sweating etc.) certain materials by a law similar to radioactivity. if technitium is injected in some form in a human  body.the body exerts half - the amount in 24hours . A patient is given an injection containing this isotope is radioactive with a half-life of 6 hours.the activity frim the body just after the injection is How much time will elapse before the activity falls to ?
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## Half life of a radioactive substance

The half-life of a radio nuclide is the time at which both N and R have been reduced to one-half their initial values.
Putting and in the law of radioactive decay, we get:

The radionuclide has a half-life of 2.58 h and is produced in a cyclotron by bombarding a manganese target with deuterons. The target contains only the stable manganese isotope , and the manganesedeuteron reaction that produces is

If the bombardment lasts much longer than the half-life of , the activity of the produced in the target reaches a final value of . (a) At what rate is being produced? (b) How many Mn nuclei are then in the target? (c) What is their total mass?
A source contains two phosphorous radio nuclides
and . Initially, % of the decays come from . How long one must wait until % do so?
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## Changes within the nucleus in alpha, beta and gamma emission

Alpha emission: If the nucleus of a radioactive element X of mass number A and atomic number Z emits an particle, a new element Y (daughter nucleus) is formed which has mass number equal to (A-4) and atomic number equal to (Z-2). Thus due to emission of an alpha particle, atomic number Z decreases by two units and mass number decreases by 4 units.
Beta emission: In emitting a beta particle the number of nucleons in the nucleus (i.e. protons and neutrons) remain same, but the number of nuetrons is decreased by one and the number of protons is increased by one.
Gamma emission: In emitting gamma particle there is no change in mass number A and atomic number Z of the nucleus.
In a radioactive decay chain, nucleus decays to nucleus. Let and be the number of and particles respectively, emitted in this decay process. Which of the following statements is (are) true?
This question has multiple correct options
A
B
C
D