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