Atomic masses of two heavy atoms are `A_1` and `A_2`. Ratio of their respective nuclear densities will be approximately

Options

A. (a) `A_1/A_2`
B. (b) `(A_1/A_2)^(1/3)`
C. (c) `(A_2/A_1)^(1/3)`
D. (d) `1`

Explanation

To solve the problem of finding the ratio of nuclear densities of two heavy atoms with atomic masses \( A_1 \) and \( A_2 \), we can follow these steps:

### Step 1: Understand the concept of density
Density (\( \rho \)) is defined as mass (\( m \)) divided by volume (\( V \)):
\[ \rho = \frac{m}{V} \]

### Step 2: Express the mass of the nucleus
For a nucleus, the mass can be approximated by the number of nucleons (which is equal to the mass number \( A \)) multiplied by the mass of a single nucleon (\( m_n \)):
\[ m = A \cdot m_n \]

### Step 3: Express the volume of the nucleus
The volume of a nucleus can be modeled as a sphere, given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the nucleus.

### Step 4: Relate the radius to the mass number
The radius of the nucleus can be expressed in terms of the mass number \( A \) as:
\[ r = r_0 A^{1/3} \]
where \( r_0 \) is a constant.

### Step 5: Substitute the radius into the volume formula
Substituting \( r \) into the volume formula gives:
\[ V = \frac{4}{3} \pi (r_0 A^{1/3})^3 = \frac{4}{3} \pi r_0^3 A \]

### Step 6: Substitute mass and volume into the density formula
Now, substituting the expressions for mass and volume into the density formula:
\[ \rho = \frac{A \cdot m_n}{\frac{4}{3} \pi r_0^3 A} \]

### Step 7: Simplify the density expression
Notice that the mass number \( A \) cancels out:
\[ \rho = \frac{m_n}{\frac{4}{3} \pi r_0^3} \]

### Step 8: Conclusion about nuclear density
Since the expression for density does not depend on \( A \), we conclude that the nuclear density is approximately the same for both heavy atoms:
\[ \frac{\rho_1}{\rho_2} \approx 1 \]

Thus, the ratio of their respective nuclear densities will be approximately equal.