Obtain approximately the ratio of the nuclear radii of the gold isotope \({ 79^{ Au } }^{ 197 }\) and the silver isotope \({ 79^{ Au } }^{ 197 }\)

1

(a)

Radius of nuclei,\(R={ R }_{ 0 }{ A }^{ 1/3 }\)
where A is the mass number of the nucleus and \({ R }_{ 0 }\) is an empirical constant.
\(\therefore \ R\alpha { A }^{ 1/3 }\)
\(\Rightarrow \ \cfrac { R \ gold }{ R \ silver } =(\cfrac { A \ gold }{ A \ silver } { ) }^{ 1/3 }\)
\(=(\frac { 197 }{ 107 } { ) }^{ 1/3 }=1.225=1.23\)

Draw a graph showing the variation of decay rate with number of active nuclei.

1

(a)

We know that, rate of decay  
\(=\frac { -dN }{ dt } =\lambda N\)  
Where, decay constant\(\lambda\) is constant for a give radioactive material.Therefore, graph between N and\(\frac { dN }{ dt } \) is a straight line as shown in the figure.

\(_{ 2 }^{ 3 }{ He }\) and \(​​​​_{ 1 }^{ 3 }{ He }\)  nuclei have the same mass number. Do they have same binding energy?

1

(a)

Since the repulsive force between protons is missing in \(_{ 1 }^{ 3 }{ He }\), so the building energy of \(_{ 1 }^{ 3 }{ He }\) is greater than that of \(_{ 2 }^{ 3 }{ He }\)

Two nuclei have mass numbers in the ratio of 27:512. What is the ratio of their nuclear radii?

1

(a)

3 : 8

 \({ }_{2}^{3} \mathrm{He}\) and \({ }_{1}^{3} \mathrm{He}\) nuclei have the same mass number. Do they have same binding energy?

1

(a)

Since, the repulsive force between protons is missing in \({ }_{1}^{3} \mathrm{He}\), so the binding energy of \({ }_{1}^{3} \mathrm{He}\) is greater than that of  \({ }_{2}^{3} \mathrm{He}\).