Question

The disintegration energy $Q$ for the change $_{Z}^{A}X\xrightarrow{-\alpha }_{Z-2}^{A-4}D+Q$is given by $Q=[M(_{Z}^{A}X)-M(_{Z-2}^{A-4}D)-M(_{2}^{4}He)]\cdot {{C}^{2}}$, where $M$ represents the mass of respective nuclei reported in parentheses and $C$ is the speed of light. For $\alpha $- decay:
A. $Q$ = +ve
B. $Q$ = -ve
C. $Q$ = 0
D. not related to $Q$

Answer 1

Hint: Think about what happens when $\alpha $- decay occurs, whether the mass of the parent nucleus is reduced due to loss in mass or due to loss in energy. Considering the definitions of mass defect and binding energy will also help.

Complete answer:
The disintegration energy, sometimes also known as the Q-value, is the energy that is absorbed or released when a nuclear reaction takes place. The Q-value is positive if the reaction is exothermic and negative if the reaction is endothermic.
In this question, the option ‘D. not related to $Q$’ can be ruled out in the beginning since $\alpha $- decay is very much related to the disintegration energy since it is a byproduct.
Now, using the given formula, and simplifying it a bit, we have to find whether the mass of the parent nucleus is greater than, equal to, or less than the masses of the daughter nuclei. The Q-value will be positive, 0, or negative respectively. Since the square of any real number is positive, ${{C}^{2}}$will not affect the sign of the Q-value.
Now think about the definitions of mass defect and binding energy. Mass defect is the difference between the theoretical and observed values of nuclei. This difference is taken care of by the binding energy, this energy keeps the nucleons together. This energy is found to be greater in nuclei having bigger atomic masses.
Whenever a nucleus disintegrates by $\alpha $- decay some the binding energy that keeps the nucleus together also gets released. The daughter nuclei will have their own binding energies but they will be less than that of the parent nucleus. The consequence of this is that even after adding the masses of the products together, the sum will be less than the mass of the parent nucleus.
Hence, the answer is ‘A. $Q$ = +ve’
So, the correct answer is “Option A”.

Note: The law of conservation of mass may state that the masses of the reactants and products are always equal but when we reach the nuclear level, Einstein’s famous equation $E=m{{c}^{2}}$ comes into picture stating that energy has mass. Thus, all the energy that is released will also have mass and will result in a net loss of observable mass. Hence, the answer is ‘A $Q$ = +ve’ and not ‘C $Q$ = 0’