Q.Obtain approximately the ratio of the nuclear radii of the gold isotope $^{197}_{79}\text{Au}$ and the silver isotope $^{107}_{47}\text{Ag}$.

The key idea is that nuclear radius follows the empirical formula $R = R_0 A^{1/3}$, where $A$ is the mass number and $R_0$ is a constant (~1.2 fm). This means the ratio of radii depends only on the cube root of the mass numbers.

Step 1: Write the radius for each nucleus:

$R_{\text{Au}} = R_0 (197)^{1/3}$, $R_{\text{Ag}} = R_0 (107)^{1/3}$.

Step 2: Take the ratio: …

The key idea is that nuclear density is roughly constant for all nuclei — a fact known from scattering experiments. Since mass number $A$ is proportional to volume, and volume scales as $R^3$, we get $R \propto A^{1/3}$. This is one of the most reliable scaling laws in nuclear physics.

Let’s work through it.

1. The nuclear radius formula For any nucleus, the radius is given by

where $R_0 \approx 1.2 \times 10^{-15}\,\text{m}$ (a constant). The exact value of $R_0$ cancels out when we take a ratio, so we don’t need it here.

2. Write the ratio For gold: $R_{\text{Au}} = R_0 (197)^{1/3}$ For silver: $R_{\text{Ag}} = R_0 (107)^{1/3}$ Dividing:

3. Approximate the fraction $197/107 \approx 1.841$. Now take the cube root. A quick mental check: $1.2^3 = 1.728$, $1.25^3 = 1.953$. So the cube root lies between 1.2 and 1.25. More precisely, $1.23^3 = 1.23 \times 1.23 \times 1.23 = 1.5129 \times 1.23 \approx 1.861$ — very close to 1.841. So $\left(\frac{197}{107}\right)^{1/3} \approx 1.23$ …

Method: Using the Radius–Mass-Number Relation

The key idea is that nuclear radius depends only on the mass number $A$, not on the atomic number $Z$. The standard empirical formula is:

where $R_0 \approx 1.2 \times 10^{-15}\ \text{m}$ is a constant (the same for all nuclei). This means the ratio of two nuclear radii depends only on the ratio of their mass numbers.

Steps

1. Write the radius for each nucleus

   For gold: $R_{\text{Au}} = R_0 (197)^{1/3}$

   For silver: $R_{\text{Ag}} = R_0 (107)^{1/3}$

2. Take the ratio — the $R_0$ cancels immediately:

3. Simplify the fraction

   $\frac{197}{107} \approx 1.841$

4. Take the cube root …

Common Mistakes in Nuclear Radius Ratio Problems

Students typically make the same few errors when calculating the ratio of nuclear radii from mass numbers. Here is what goes wrong and how to fix each.

Mistake 1: Forgetting the Cube Root

The most frequent error is treating the radius as directly proportional to the mass number $A$, rather than to $A^{1/3}$.

The nuclear radius formula is:

where $R_0$ is a constant (about $1.2 \times 10^{-15}$ m). So the ratio of radii is:

Students often write $\frac{R_1}{R_2} = \frac{A_1}{A_2}$ instead, which gives a wildly wrong answer.

How to avoid: Every time you see "nuclear radius" and "mass number" together, immediately write $R = R_0 A^{1/3}$ before doing anything else. Then take the ratio — the $R_0$ cancels, and you are left with the cube root of the mass number ratio.

Mistake 2: Confusing Mass Number with Atomic Number

The formula uses $A$ (mass number = protons + neutrons), not $Z$ (atomic number = protons only). For gold, $A = 197$; for silver, $A = 107$. The atomic numbers (79 and 47) are irrelevant here.

How to avoid: Circle the mass numbers in the isotope notation. The superscript (top-left) is $A$; the subscript (bottom-left) is $Z$. Use only the superscript.

Mistake 3: Inverting the Ratio

Students sometimes write $\frac{R_{\text{Au}}}{R_{\text{Ag}}} = \left(\frac{107}{197}\right)^{1/3}$ instead of $\left(\frac{197}{107}\right)^{1/3}$. Gold has more nucleons, so its nucleus is larger — the ratio must be greater than 1.

How to avoid: Check your common sense. Gold ($A=197$) is heavier than silver ($A=107$), so its radius should be larger. If your ratio comes out less than 1, you have inverted it.

Mistake 4: Not Simplifying the Cube Root

Students often leave the answer as $\left(\frac{197}{107}\right)^{1/3}$ without approximating. The question asks for an approximate ratio — you need a numerical value. …