Calculating the Ratio of Nuclear Radii

The radius of a nucleus is related to its atomic mass number. The relationship is given by a specific formula in nuclear physics. Understanding this formula is key to solving problems involving nuclear sizes.

Relationship Between Nuclear Radius and Mass Number

The radius of a nucleus, denoted by \(R\), is approximately proportional to the cube root of its atomic mass number, \(A\). The formula is:

\(R = R_0 A^{1/3}\)

Where:

• \(R\) is the radius of the nucleus.
• \(R_0\) is a constant, approximately \(1.2 \times 10^{-15}\) m (or 1.2 fm), known as the nuclear radius constant.
• \(A\) is the atomic mass number (total number of protons and neutrons).

This formula tells us that as the atomic mass number increases, the nuclear radius also increases, but not linearly; it increases with the cube root of the mass number.

Calculating the Ratio of Radii

We are given two nuclei with atomic mass numbers \(A_1 = 27\) and \(A_2 = 8\). Let their radii be \(R_1\) and \(R_2\), respectively. Using the formula \(R = R_0 A^{1/3}\), we can write the radii for the two nuclei as:

\(R_1 = R_0 A_1^{1/3}\)

\(R_2 = R_0 A_2^{1/3}\)

To find the ratio of their radii, \(\frac{R_1}{R_2}\), we divide the first equation by the second:

\(\frac{R_1}{R_2} = \frac{R_0 A_1^{1/3}}{R_0 A_2^{1/3}}\)

The constant \(R_0\) cancels out:

\(\frac{R_1}{R_2} = \frac{A_1^{1/3}}{A_2^{1/3}}\)

Using the property of exponents \(\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n\), we get:

\(\frac{R_1}{R_2} = \left(\frac{A_1}{A_2}\right)^{1/3}\)

Substituting Given Values

Now, substitute the given atomic mass numbers \(A_1 = 27\) and \(A_2 = 8\) into the ratio formula:

\(\frac{R_1}{R_2} = \left(\frac{27}{8}\right)^{1/3}\)

Evaluating the Cube Root

To evaluate \(\left(\frac{27}{8}\right)^{1/3}\), we find the cube root of the numerator and the cube root of the denominator separately:

\(\left(\frac{27}{8}\right)^{1/3} = \frac{27^{1/3}}{8^{1/3}}\)

The cube root of 27 is 3, because \(3 \times 3 \times 3 = 27\).

The cube root of 8 is 2, because \(2 \times 2 \times 2 = 8\).

So, substituting these values:

\(\frac{R_1}{R_2} = \frac{3}{2}\)

Conclusion on Nuclear Radii Ratio

The ratio of the radii of the two nuclei having atomic mass numbers 27 and 8 is \(\frac{3}{2}\).

Let's check the given options:

1. \(\frac{R_1}{R_2}=\frac{3}{2}\)
2. \(\frac{R_1}{R_2}=\frac{4}{2}\)
3. \(\frac{R_1}{R_2}=\frac{6}{4}\)
4. \(\frac{R_1}{R_2}=\frac{\sqrt3}{2}\)

Our calculated ratio \(\frac{3}{2}\) matches Option 1.

Revision Table: Nuclear Radii Calculation

Additional Information: Nuclear Structure and Size

The atomic nucleus is the dense, central part of an atom. It is composed of protons and neutrons, collectively called nucleons. The total number of nucleons is the atomic mass number \(A\).

• Nuclear Density: The density of nuclear matter is remarkably constant across most nuclei, around \(2.3 \times 10^{17} \, \text{kg/m}^3\). This high density implies that nucleons are packed very closely together. The fact that density is roughly constant leads to the relationship \(R \propto A^{1/3}\), because the volume of a sphere is proportional to \(R^3\), and if density is constant, mass (proportional to A) is proportional to volume. So, \(A \propto R^3\), which means \(R \propto A^{1/3}\).
• Nuclear Force: The nucleons are held together by the strong nuclear force, which is much stronger than the electromagnetic repulsion between protons at short distances but drops off rapidly at larger distances.
• Isotopes: Nuclei with the same number of protons but different numbers of neutrons are isotopes. They have the same atomic number (\(Z\)) but different mass numbers (\(A\)). Their radii will be slightly different based on their mass numbers.
• Measurement of Nuclear Radius: Nuclear radii can be measured experimentally using techniques like electron scattering or studying characteristic X-rays from muonic atoms. These experiments confirm the \(R \propto A^{1/3}\) relationship.

Understanding the \(R \propto A^{1/3}\) relationship is fundamental in nuclear physics when discussing properties like nuclear volume, density, and reaction cross-sections.