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Question: Radius of nucleus varies as \(R = {R_0}{A^{1/3}}\), where \({R_0}\)=1.3 Fermi. What is the volume of...

Radius of nucleus varies as R=R0A1/3R = {R_0}{A^{1/3}}, where R0{R_0}=1.3 Fermi. What is the volume of Be8B{e^8} nucleus (approx) [A = atomic mass]?

Explanation

Solution

In this question, we will use the direct formula of radius of nucleus or atom. Now, when we substitute the given values in the equation, it will give us the required result. Also, we will see the basics of an atom and nuclei. Further, we will also discuss the types of radioactive radiations and the de-Broglie wave-matter concept, for our better understanding.

Formula used:
R=R0A1/3R = {R_0}{A^{1/3}}

Complete answer:
As we know, an atom is a smallest part of an ordinary matter. An atom consists of a small positive portion, called nucleus. Nucleus has protons and neutrons, together called nucleons.
Radius of the nucleus is given by:
R=R0A1/3R = {R_0}{A^{1/3}}
Now, by substituting the given values in above equation, we get:
R=1.3×(8)1/3FermiR = 1.3 \times {(8)^{1/3}}Fermi
\eqalign{ & \Rightarrow R = 2.6Fermi \cr & \Rightarrow R = 2.6 \times {10^{ - 13}}cm \cr}
As we know, the nucleus is sphere, so the volume is given by:
V=43πR3V = \dfrac{4}{3}\pi {R^3}
Now, by substituting the values in above equation, we get:
V=4.1866×(2.6×10−13)3V = 4.1866 \times {\left( {2.6 \times {{10}^{ - 13}}} \right)^3}
∴V=7.36×10−38cc\therefore V = 7.36 \times {10^{ - 38}}cc
Therefore, we get the required result of the volume of the given nucleus.

Additional information:
Wave- particle duality is a concept of quantum mechanics that according to this every particle may be described as either a particle or a wave. It expresses the inability of the classical concepts of particle and wave, to fully describe the behavior of quantum scale objects.
In general, an electron in a metal has a de-Broglie wavelength in order of ~10nm. So, we observe quantum-mechanical effects in the properties of a metal when the width of the sample is around this value. The S.I unit of this wavelength is meter (m).
De-Broglie won the Nobel Prize for physics in 1929, after the wave- like behavior of matter was experimentally demonstrated in 1927.
We also know that radioactive material is defined as any material containing unstable atoms that emit ionizing radiation as it decays.
Here, we know that the ability of any radiation to damage molecules is defined as ionizing power. Also, when these particles interact with atoms, this interaction can cause the atom to lose its electrons and become ionized.
Penetration power is defined as the ability of radiation to pass through the matter is expressed. Here, we can also say that the more a radiation can pass through a material, the greater the penetration power and so, the more dangerous these radiations are.

Note:
We should remember that, there are many different models proposed for an atom, according to Bohr, electrons revolve around the nucleus and according to Rutherford- an atom consists of small dense positively charged nucleus.