Obtain approximately the ratio of the nuclear radii of the gold isotope \(^{197}_{ 79} Au \) and the silver isotope \(^{107}_{ 47} Ag\).
Obtain approximately the ratio of the nuclear radii of the gold isotope \(^{197}_{ 79} Au \) and the silver isotope \(^{107}_{ 47} Ag\).
Solution and Explanation
Nuclear radius of the gold isotope \(_{ 79} Au ^{197}\) = RAu
Nuclear radius of the silver isotope \( _{47} Ag^{107}\) = RAg
Mass number of gold, AAu = 197
Mass number of silver, AAg = 107
The ratio of the radii of the two nuclei is related with their mass numbers as:
\(\frac{R_{Au}}{R_{Ag}}\) = \((\frac{RAU}{RAG})^{\frac{1}{3}}\)
= \((\frac{197}{107})^{\frac{1}{3}}\)
= 1.2256
Hence, the ratio of the nuclear radii of the gold and silver isotopes is about 1.23
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Concepts Used:
Nuclear Physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies the atom as a whole, including its electrons
Radius of Nucleus
‘R’ represents the radius of the nucleus. R = RoA1/3
Where,
- Ro is the proportionality constant
- A is the mass number of the element
Total Number of Protons and Neutrons in a Nucleus
The mass number (A), also known as the nucleon number, is the total number of neutrons and protons in a nucleus.
A = Z + N
Where, N is the neutron number, A is the mass number, Z is the proton number
Mass Defect
Mass defect is the difference between the sum of masses of the nucleons (neutrons + protons) constituting a nucleus and the rest mass of the nucleus and is given as:
Δm = Zmp + (A - Z) mn - M
Where Z = atomic number, A = mass number, mp = mass of 1 proton, mn = mass of 1 neutron and M = mass of nucleus.