Question

It the radius of \({ }^{27}{ }_{13} \mathrm{~A} \ell\) nucleus is \(3.6 \mathrm{fm}\) the radius of \({ }^{125}{ }_{52} \mathrm{Te}\) nucleus is nearly equal to (A) \(8 \mathrm{fm}\) (B) \(6 \mathrm{fm}\) (C) \(4 \mathrm{fm}\) (D) \(5 \mathrm{fm}\)

Short answer

The radius of the \({ }^{125}{ }_{52} \mathrm{Te}\) nucleus is approximately \(6 \mathrm{fm}\).

Step-by-step solution

Understand the formula for Nuclear Radius

The formula for nuclear radius is given by: \[R = R_0 A^{1/3}\] where R is the nuclear radius, R_0 is the constant (approximately equal to 1.2 \(\mathrm{fm}\)), and A is the mass number of the nucleus.

Identify the given values

The problem gives us the radius of \({ }^{27}{ }_{13} \mathrm{~A} \ell\) nucleus as \(3.6 \mathrm{fm}\). So, in this case, A = 27 and R = 3.6 \(\mathrm{fm}\).

Calculate the value of R_0

To find the value of R_0, we need to rearrange the nuclear radius formula and substitute the values of A and R: \[R_0 = \frac{R}{A^{1/3}}\] Plug in the given values: \[R_0 = \frac{3.6}{27^{1/3}}\] \[R_0 \approx 1.2 \mathrm{fm}\]

Calculate the radius of \({ }^{125}{ }_{52} \mathrm{Te}\) nucleus

To find the radius of \({ }^{125}{ }_{52} \mathrm{Te}\) nucleus, use the nuclear radius formula. Assign the values A = 125, and R_0 = 1.2 \(\mathrm{fm}\). \[R = R_0 A^{1/3}\] \[R = 1.2 (125)^{1/3}\] \[R \approx 6 \mathrm{fm}\]

Identify the correct option

After calculating the radius of \({ }^{125}{ }_{52} \mathrm{Te}\) nucleus, compare it with the given options. Since the radius is approximately \(6 \mathrm{fm}\), the correct answer is (B) \(6 \mathrm{fm}\).

Key concepts

Nuclear Physics

Nuclear physics is the branch of physics that studies the components and interactions of atomic nuclei. Within this field, a key concept is understanding the structure and properties of the nucleus, such as nuclear radius, binding energy, and stability. The nuclear radius is a measure of the size of an atomic nucleus and is typically expressed in femtometers (fm), whereby one femtometer equals 10^-15 meters.

• The nuclear radius gives insights into how the protons and neutrons (collectively known as nucleons) are distributed within the nucleus.
• Understanding nuclear dimensions is fundamental in predicting nuclear reactions and stability.

The nuclear radius is influenced by the mass number, as more nucleons generally result in a larger nuclear size. This affects how the nucleus interacts with incoming particles, essential in various applications like nuclear energy and medical imaging.

Atomic Structure

The atomic structure consists of a central nucleus surrounded by a cloud of electrons. The nucleus itself is composed of protons and neutrons, collectively known as nucleons. Each element has a unique identity determined by the number of protons in its nucleus, termed the atomic number. However, the size of the nucleus does not correlate directly with the number of protons or electrons, but rather with the mass number which includes neutrons.

• Electrons arrange themselves in energy levels or shells around the nucleus, while the nucleus remains at the core.
• The nucleus is held together by the strong nuclear force, which overcomes the repulsion between the positively charged protons.

Atomic structure is key not only in identifying elements but also in understanding physical and chemical properties influenced by electron configurations and nuclear compositions.

Mass Number

The mass number (A) of an atom is the total count of protons and neutrons in its nucleus. Unlike the atomic number, which only considers protons, the mass number gives a snapshot of the entire nucleon content, essential for calculating properties like the nuclear radius.

• For the element symbolized \(^{A}_{Z}X\), \(A\) is the mass number, \(Z\) is the atomic number, and \(X\) is the element symbol.
• The mass number directly influences the nuclear radius, as seen in calculations using the formula \(R = R_0 A^{1/3}\).

The mass number is crucial in nuclear physics for predicting nuclear size and behavior. It also helps distinguish isotopes of the same element, which may have similar chemical properties but varying nuclear characteristics.