Two nuclei have their mass numbers in the ratio of `1:3`. The ratio of their nuclear densities would be

Options

A. `(3)^(1//3):1`
B. `1:1`
C. `1:3`
D. `3:1`

Explanation

To solve the problem of finding the ratio of nuclear densities of two nuclei with mass numbers in the ratio of 1:3, we can follow these steps:

### Step 1: Understand the formula for nuclear density
Nuclear density (D) is defined as the mass (M) of the nucleus divided by its volume (V). The formula can be expressed as:
\[ D = \frac{M}{V} \]

### Step 2: Express mass in terms of mass number
The mass of a nucleus can be approximated as the product of the mass number (A) and the mass of a nucleon (which is approximately constant). Thus, we can write:
\[ M = m \cdot A \]
where \( m \) is the average mass of a nucleon.

### Step 3: Express volume in terms of radius
The volume (V) of a nucleus can be expressed in terms of its radius (R):
\[ V = \frac{4}{3} \pi R^3 \]

### Step 4: Relate radius to mass number
The radius of a nucleus can be related to its mass number by the empirical formula:
\[ R = R_0 A^{1/3} \]
where \( R_0 \) is a constant.

### Step 5: Substitute radius into the volume formula
Substituting the expression for radius 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:
\[ D = \frac{m \cdot A}{\frac{4}{3} \pi R_0^3 A} \]
The mass number \( A \) cancels out:
\[ D = \frac{m}{\frac{4}{3} \pi R_0^3} \]

### Step 7: Conclude that density is constant
From the above expression, we see that the density \( D \) is independent of the mass number \( A \). Therefore, the density remains constant for different nuclei.

### Step 8: Calculate the ratio of densities
Given that the two nuclei have mass numbers in the ratio of 1:3, their densities will be:
\[ \text{Density Ratio} = \frac{D_1}{D_2} = 1 \]

### Final Answer
The ratio of their nuclear densities is \( 1:1 \).

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