Question

Consider the following statements (A) and (B) and identify the correct answer given below:
Statement A: Positive values of packing fraction implies a large value of binding energy
Statement B: The difference between the mass of the nucleus and the mass number of the nucleus is called packing fraction.
A) A and B are correct
B) A and B are false
C) A is true, B is false
D) A is false, B is true

Answer 1

Hint: Firstly, we must know the definition of packing fraction and binding energy. We must carefully write these definitions down and compare them to the statements provided in the question. Check if any manipulation is required in the equations formed from the definition.

Complete step by step solution:
First, we will see what is the definition of pacing fraction:
The difference between the actual isotopic mass of the nucleus and the mass number of the nucleus is called the packing fraction.
Hence, Statement B is correct.
Now, from the definition of packing efficiency, we get the equation:
$P.F.=\dfrac{\Delta m}{A}=\dfrac{m-A}{A}$ ,
Where $P.F.$ stands for Packing Fraction, $\Delta m$ stands for mass defect, $A $ is the mass number (number of protons + electrons), and $m $ is the actual isotopic mass.
Now statement A says positive values of packing fraction implies a large value of binding energy.
Binding energy is given by: $B.E. = \Delta m\cdot {{c}^{2}}$

If sign of packing fraction is negative, $\Delta m < A$

$\Rightarrow $mass defect is converted into energy
And if the sign of packing efficiency is positive, $\Delta m>A$
$\Rightarrow $energy is being converted into the mass defect
So, the sign of packing efficiency doesn’t tell us anything about the binding energy, i.e. whether the binding energy will be small or large is not determined by the sign of packing efficiency.
Hence, statement A is wrong.

Therefore, the correct statement is (D), A is false and B is true.

Note: Whenever we are given questions like these (which of the statements are correct), we must not directly assume and jump to the conclusion that any of the statements are correct. We must always check the given statements with true statements. The knowledge of correct definitions is important to solve such questions, so the definitions must not be ignored.