For uranium nucleus how does its mass vary with volume?

Options

A. (a) `m prop V`
B. (b) `m prop 1//V`
C. (c) `m prop sqrt(V)`
D. (d) `m prop V^2`

Explanation

To determine how the mass of a uranium nucleus varies with its volume, we can follow these steps:

### Step 1: Understand the relationship between the radius and mass number
The radius \( R \) of a nucleus is related to its mass number \( A \) by the formula:
\[ R = R_0 A^{1/3} \]
where \( R_0 \) is a constant (approximately \( 1.2 \times 10^{-15} \) m).

### Step 2: Calculate the volume of the nucleus
The volume \( V \) of a spherical nucleus can be calculated using the formula for the volume of a sphere:
\[ V = \frac{4}{3} \pi R^3 \]
Substituting the expression for \( R \) from Step 1:
\[ V = \frac{4}{3} \pi (R_0 A^{1/3})^3 = \frac{4}{3} \pi R_0^3 A \]
This shows that the volume \( V \) is directly proportional to the mass number \( A \).

### Step 3: Relate mass to mass number
The mass \( M \) of the nucleus can be expressed in terms of the mass number \( A \) and the mass of a single nucleon \( m_n \):
\[ M = A \cdot m_n \]
Here, \( m_n \) is approximately constant.

### Step 4: Establish the relationship between mass and volume
Since we have established that:
- \( V \propto A \) (from Step 2)
- \( M \propto A \) (from Step 3)

We can conclude that the mass \( M \) is directly proportional to the volume \( V \):
\[ M \propto V \]

### Conclusion
Thus, for a uranium nucleus, the mass varies directly with its volume.

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