Notes for Nucleus chapter of class 12 physics. Dronstudy provides free comprehensive chapterwise class 12 physics notes with proper images & diagram.

**Radioactivity**

Radioactive decay is a random process. Each decay is an independent event, and one cannot tell when a particular nucleus will decay. When a given nucleus decays, it is transformed another nuclide, which may or may not be radioactive. When there is a very large number of nuclei in a sample, the rate of decay is proportional to the number of nuclei *N *that are present

Where is called the ** decay constant**. This equation may be expressed in the form Â and integrated

to yield

where *N _{o} *is the initial number of parent nuclei at

This function is plotted in the following figure.The *time required *for the number of parent nuclei to fall to 50% is called the ** half-life**,

we have . Therefore

It takes one half-life to drop to 50% of any starting value. The half-life for the decay of the free neutron is 12.8 min. Other half-lives range from about 10^{-20} s to 10^{16 }years.

Since the number of atoms is not directly measurable, we measure the ** decay rate** or

. On taking the derivative of equation we findÂ Â Â Â Â Â Â Â

where Â is the initial activity. The SI unit for the activity is the

1 becquerel (Bq) Â Â Â Â Â Â Â Â Â Â Â Â Â Â = Â Â Â Â Â Â Â Â 1 disintegration per second (dps)

1 curie Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = Â Â Â Â Â Â Â Â 3.7 x 10^{10} dps

1 rutherford Â Â Â Â Â Â Â Â Â Â Â = Â Â Â Â Â 10^{6} dps

Mean life of a radioactive sample is defined as the average of the lives of all nuclei.

i.e. Â

*Example 1
*

*Solution:*

The activity is proportional to the number of undecayed atoms.

In each half-life, half the remaining sample decays.

SinceÂ Â , therefore, three half-lives or 15.75 years are required for the sample to decay to Â itsÂ original strength.

*Example 2
*

*Solution:*

Initial velocity Â Â Â Â (i)

Final velocity Â Â Â Â (ii)

Dividing (i) by (ii), we get

Â Â (iii)

The decay constant is given by

Â Â or Â Â Â

Half life of the sample is

*Example 3
*

*Solution:*

When an substance decays by a and b emission simultaneously, the average disintegration constant Â is given by

where Â = disintegration constant for - emission only

Â Â Â Â Â = disintegration constant for - emission only

Mean life is given byÂ Â

Â or Â

*Example 4
*

*Solution:*

Number of atoms in 1 g sample is

The decay constant is

Taking one year

Now, Â

ThusÂ Â Â Â nuclei decay in one second.

(1)** Â **Â

(2)

Both processes are accompanied by -radiation, i.e. the flow of photons having a very small wavelength, and hence a very high energy. Like other electromagnetic waves, -rays propagate at a velocity of light. The penetrability of -rays is 1-100 times higher than the penetrability of -rays and 1000-10000 times higher than the penetrability of -rays. It also exceeds the penetrability of *X*-rays.

In a magnetic field, a beam of -, -, and -rays splits into three parts as shown in the figure.

Nuclei possessing the ** artificial radioactivity **are obtained by bombarding stable nuclei of heavy elements by

Radioactive processes occur in accordance with the laws of *conservation of energy*, *momentum*, *angular momentum*, *electric charge*, and *mass number *(amount of nucleons).

In ** -decay**, the mass number of the nucleus decreases by four and the charge decreases by two units, as a result of which two electrons are removed from the atomic shell. The element transforms into another element with the atomic number which is two units lower.

In

The atomic nucleus consists of two types of elementary particles, viz. *protons* and *neutrons.* These particles are called *nucleons*.

The **proton** (denoted by *p*) has a charge +*e* and a mass , which is approximately 1840 times larger than the electron mass. The proton is the nucleus of the simplest atom with ** Z **= 1, viz. the hydrogen atom.

The **neutron** (denoted by *n*) is an electrically neutral particle (its charge is zero). The neutron mass . The fact that the neutron mass exceeds the proton mass by about 2.5 electron masses is of essential importance. It follows from this that the neutron in free state (outside the nucleus) is unstable (radioactive). During the time equal on the average to 12 min, the neutron spontaneously transforms to the proton by emitting an electron (e^{-}) and a particle called the antineutrino (). This process can be schematically written as follows:

The most important characteristics of the nucleus are the *charge number **Z* (coinciding with the atomic number of the element) and the *mass number **A*. The charge number *Z* is equal to the number of protons in the nucleus, and hence it determines the nuclear charge equal to *Ze*. The mass number *A* is equal to the number of nucleons in the nucleus (i.e. to the total number of protons and neutrons).

Nuclei are symbolically designated as

Â orÂ Â

where *X *stands for the symbol of a chemical element. For example, the nucleus of the oxygen atom is symbolically written as Â or .

Most of the chemical element have several types of atoms differing in the number of neutrons in their nuclei. These varieties are called *isotopes*. For example, oxygen has three stable isotopes: , Â and . In addition to stable isotopes, there also exist unstable (radioactive) isotopes.

Atomic masses are specified in terms of the atomic mass unit or unified mass unit (u). The mass of a neutral atom of the carbon isotope _{6}C^{12} is defined to be exactly 12 u.

*Example 5
*

*Solution:*

(a)Â One mole of C^{12} has a mass of 12 g and contains Avogadroâ€™s number, *N _{A}*, of atoms.

By definition, each C

Thus, 12 g corresponds to 12

or Â

(b)Â From Einstein relationÂ ^{
}â¸« Â Â Â

Since Â

â¸« Â Â Â Â

Hence

The shape of nucleus is approximately spherical and its radius is approximately related to the mass number by

Â Â Â Â Â whereÂ 1 fermi (fm) = 10^{-15} m**Â **

*Example 6
*

*Solution:*

Volume Â Â Â

Mass of oxygen atoms (*A *= 16) is approximately 16 u.

Therefore, density is

or Â Â Â Â

This is 10^{14} times the density of water.