Question

The radius of Ge nucleus is measured to be twice the radius of \({ }^{9}{ }_{4}\) Be. The number of nucleons in Ge are (A) 72 (B) 78 (C) 65 (D) 80

Short answer

Using the nuclear radius formula, we find that the mass number of Germanium (Ge) is $A_{Ge} = 72$. Therefore, the number of nucleons in Ge is 72. The correct option is (A).

Step-by-step solution

Write down the formula for the nuclear radius

The formula for the nuclear radius (R) in terms of the mass number (A) is given by: \[R = R_0 A^{1/3}\] where \(R_0\) is a constant, approximately equal to 1.2 fm (fermi, a unit of length used in nuclear physics, 1 fermi = \(10^{-15}\)m).

Write down the given information

We are given that the radius of Ge nucleus (R_Ge) is twice the radius of \({ }^{9}_{4}\) Be nucleus (R_Be): \[R_{Ge} = 2 R_{Be}\]

Find the mass number of Beryllium

Beryllium is given as \({ }^{9}_{4}\) Be, so its mass number (number of protons and neutrons) is A_Be = 9.

Substitute the formula for the nuclear radius

Now, we substitute the formula for the nuclear radius for each element: \[R_0 A_{Ge}^{1/3} = 2 R_0 A_{Be}^{1/3}\]

Divide by R_0

Divide both sides by \(R_0\): \[A_{Ge}^{1/3} = 2 A_{Be}^{1/3}\]

Substitute the mass number of Beryllium

We can now substitute the mass number of Beryllium (A_Be = 9) in the equation: \[A_{Ge}^{1/3} = 2 (9)^{1/3}\]

Calculate the cube of the equation

Take the cube of both sides of the equation: \[(A_{Ge}^{1/3})^3 = [2 (9)^{1/3}]^3\]

Calculate the mass number of Germanium

Simplify the equation to find the mass number of Germanium: \[A_{Ge} = 2^3 \times 9 = 8 \times 9 = 72\]

Identify the correct option

The number of nucleons (protons and neutrons) in Ge is 72. The correct option is: (A) 72

Key concepts

Nuclear Radius

The nuclear radius is a critical concept in nuclear physics that describes the size of an atomic nucleus. Each nucleus has a distinct radius, dependent on its mass number. The formula to determine the nuclear radius for a nucleus is given by:\[ R = R_0 A^{1/3} \]
Here,

• \( R_0 \) is a radius constant, approximately 1.2 fermi (fm), where 1 fm equals \( 10^{-15} \) meters.
• \( A \) represents the mass number, which is the total count of protons and neutrons.
• The term \( A^{1/3} \) reflects how the radius scales with the size of the nucleus. As the mass number increases, so does the nuclear radius, but at a diminishing rate due to the cube root.

One fascinating property is that nuclear radius changes relatively smoothly as different elements are considered, allowing for predictions based on changes in mass number. This formula assists scientists in estimating the size of atoms as their compositions change.

Mass Number

In nuclear physics, the mass number (denoted as \( A \)) is an essential concept. The mass number of an atomic nucleus represents the total number of nucleons, which include both protons and neutrons. It is pivotal when describing specific isotopes of elements. For instance, the mass number differentiates between two isotopes of the same element because they can have the same number of protons but different neutron numbers.

Several point highlights include:

• The mass number does not directly correspond to the atomic weight of an element because electrons have negligible mass comparatively.
• The mass number is integral to identifying an isotope’s identity in scientific research and applications.
• Understanding the mass number supports calculations involving nuclear reactions, as it helps in balancing nuclear equations and determining reaction outputs.

Understanding the mass number gives insight into the nucleus's mass and stability characteristics. The spirit of nuclear chemistry revolves significantly around the interactions between these nucleons that the mass number quantifies.

Ge Nucleus

The Ge nucleus refers to the atomic nucleus of the chemical element Germanium. It is denoted in isotopic form as \({}^{72}_{32}\text{Ge}\), indicating a mass number of 72, calculated through the steps of our exercise.

Important characteristics of the Ge nucleus include:

• Germanium, as a semi-metal, is located in group 14 of the periodic table and holds 32 protons, determined by its atomic number.
• The mass number of 72 specifies the total of protons and neutrons, making it essential in identifying this specific isotope of Germanium.
• Knowing the number of nucleons allows scientists to predict nuclear properties, such as stability, radioactive decay, and potential isotopic applications.

Typically, the Ge nucleus can undergo various nuclear reactions and is significant in semiconductor technology and nuclear physics research due to its unique properties.

Be Nucleus

The Be nucleus is integral in discussions of nuclear physics, particularly when exploring atomic structure aspects. Beryllium is represented with the symbol \( {}^{9}_{4}\text{Be} \), where

• The mass number 9 signifies total nucleons, comprising 4 protons and 5 neutrons.
• The atomic number is 4, indicating the number of protons.
• Beryllium is a light metal, and it exhibits properties unique to group 2 elements in the periodic table.

In the original exercise, this nucleus's properties were fundamental in determining the relative size of the Ge nucleus. Beryllium's small nucleus makes it a suitable reference in size comparisons of atomic structures. Understanding the Be nucleus helps grasp the interactions between subatomic particles in lighter elements, aiding nuclear physics and chemical property studies.