Question

Let $A$ be the mass number, $Z$ be the atomic number and $N$ be the neutron number of a nucleus. Then the statement which is always true, is

• A. $A^2 \ge NZ$
• B. $A \ge 2N$
• C. $A \ge 2Z$
• D. $AN \ge Z^2$

Hint:

Using simple counterexamples like Hydrogen-1 ($A=1, Z=1, N=0$) and Helium-3 ($A=3, Z=2, N=1$) lets you quickly eliminate incorrect options in nuclear physics relations.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The mass number $A$, atomic number $Z$, and neutron number $N$ describe the composition of a nucleus.
We need to find an algebraic relationship among these three variables that holds true for all possible nuclei.

Step 2: Key Formula or Approach:
For any nucleus, the mass number is the sum of protons (atomic number) and neutrons:
\[ A = Z + N \]
All three quantities $A$, $Z$, and $N$ are non-negative integers.

Step 3: Detailed Explanation:

• Let us analyze each option mathematically:

• Option (A): $A^2 \ge NZ$:
Substitute $A = Z + N$:
\[ (Z + N)^2 = Z^2 + 2NZ + N^2 \]
We want to compare this with $NZ$:
\[ Z^2 + 2NZ + N^2 \ge NZ \]
\[ Z^2 + NZ + N^2 \ge 0 \]
Since $Z \ge 0$ and $N \ge 0$, all terms $Z^2$, $NZ$, and $N^2$ are non-negative.
Thus, $Z^2 + NZ + N^2 \ge 0$ is always true for any nucleus. This means $A^2 \ge NZ$ is always true.

• Option (B): $A \ge 2N$:
This simplifies to $Z + N \ge 2N \implies Z \ge N$.
However, for heavy stable nuclei (such as Lead-208, where $Z = 82, N = 126$), we have $N \gt Z$. Thus, this is not always true.

• Option (C): $A \ge 2Z$:
This simplifies to $Z + N \ge 2Z \implies N \ge Z$.
However, for light nuclei (such as Helium-3, where $Z = 2, N = 1$), we have $Z \gt N$. Thus, this is not always true.

• Option (D): $AN \ge Z^2$:
For Hydrogen-1 ($^1\text{H}$), we have $A = 1, Z = 1, N = 0$.
Evaluating the expression:
\[ (1)(0) \ge 1^2 \implies 0 \ge 1 \]
This is false. Thus, this option is not always true.



Step 4: Final Answer:
The statement which is always true is $A^2 \ge NZ$.