Question

Two nuclei have mass numbers in the ratio $27:125$ , what is the ratio of their nuclear radii?

Answer 1

This question is related to nuclear chemistry and physics. The formula that will be applied here is the formulaAlkanesAlkanesAlkanes of the average radius of the nucleus with a nucleon. In the question mass number values are given in ratio just have to substitute the values $27:125$ and by solving within 2-3 steps you will get the required answer.

Complete answer:
Given above is the mass number of two nuclei in ratio $27:125$ we need to find the ratio of their nuclear radii. The formula that will be used here is the formula of average radius of the nucleus given as:
$R{\text{ }} = {\text{ }}{R_0}{A^{1/3}}\;\;\;$
And since we are given here with two nuclei we will write the formula for two nuclei as
${R_1}{\text{ }} = {\text{ }}{R_0}{A_1}^{1/3}\;\;\;$ And ${R_2}{\text{ }} = {\text{ }}{R_0}{A_2}^{1/3}\;\;\;$
When we take the ratio of the two it can be written as
$\dfrac{{{R_1}}}{{{R_2}}} = \dfrac{{{R_0}}}{{{R_0}}} \times {\left( {\dfrac{{{A_1}}}{{{A_2}}}} \right)^{\dfrac{1}{3}}}$
From the above equation ${A_1}{\text{ }}and{\text{ }}{A_2}$ are given in ratio as $27:125$ and we need to find $R_1{\text{ }}and{\text{ }}R_2.$ R0 being common in both can be cancelled out.
So after substituting the value the equation become:
$\dfrac{{{R_1}}}{{{R_2}}} = {\left[ {{{\left( {\dfrac{3}{5}} \right)}^3}} \right]^{\dfrac{1}{3}}}$
(In this step we have just taken the cube root of $27{\text{ }}and{\text{ }}125$ )
Next just cancel out $3$ from and we get the final answer.
$\dfrac{{{R_1}}}{{{R_2}}} = {\left[ {{{\left( {\dfrac{3}{5}} \right)}^{{3}}}} \right]^{\dfrac{1}{{{3}}}}}$
$\dfrac{{{R_1}}}{{{R_2}}} = {\left[ {\left( {\dfrac{3}{5}} \right)} \right]^1}$
This can be written as
$\dfrac{{{R_1}}}{{{R_2}}} = \dfrac{3}{5}$
Thus we get $R_1{\text{ }}and{\text{ }}R_2.$ as $3{\text{ }}and{\text{ }}5$ . Therefore the ratio of their nuclear radii $R_1:R_2$ is $3:5$ .

Note:
The equation used here is $R{\text{ }} = {\text{ }}{R_0}{A^{1/3}}\;\;\;$ , (where R is the radius, A is the mass number of the nuclei and R0 is the constant whose value is given as $1.2{\text{ }} \times {\text{ }}{10^{ - 15}}$ meter or $1.2$ fm where fm is fermi). The mass number is the total number of protons and neutrons in the nucleus.