Assuming that protons and neutrons have equal masses, calculate how many times nuclear matter is denser than water. Take mass of a nucleon `=1.67xx10^(-27)kg and R_(0)=1.2xx10^(-15)m`.

Explanation

To calculate how many times nuclear matter is denser than water, we will follow these steps:

### Step 1: Define the mass of a nucleon and the radius of the nucleus
Given:
- Mass of a nucleon (m) = \(1.67 \times 10^{-27} \, \text{kg}\)
- Radius of the nucleus \(R_0 = 1.2 \times 10^{-15} \, \text{m}\)

### Step 2: Define the volume of the nucleus
The volume \(V\) of a spherical nucleus can be calculated using the formula for the volume of a sphere:
\[ V = \frac{4}{3} \pi r^3 \]
Where \(r\) is the radius of the nucleus. The radius can be expressed in terms of the mass number \(A\):
\[ r = R_0 A^{1/3} \]
Thus, the volume becomes:
\[ V = \frac{4}{3} \pi (R_0 A^{1/3})^3 = \frac{4}{3} \pi R_0^3 A \]

### Step 3: Calculate the mass of the nucleus
The mass of the nucleus \(M\) is given by the product of the mass of a nucleon and the mass number \(A\):
\[ M = m \times A \]

### Step 4: Calculate the density of the nucleus
The density \(\rho\) of the nucleus can be calculated using the formula:
\[ \rho = \frac{M}{V} \]
Substituting the expressions for \(M\) and \(V\):
\[ \rho = \frac{m \times A}{\frac{4}{3} \pi R_0^3 A} = \frac{3m}{4\pi R_0^3} \]

### Step 5: Substitute the values
Now, substituting the values of \(m\) and \(R_0\):
\[ \rho = \frac{3 \times (1.67 \times 10^{-27} \, \text{kg})}{4 \pi (1.2 \times 10^{-15} \, \text{m})^3} \]

### Step 6: Calculate the numerical value
Calculating the denominator:
\[ (1.2 \times 10^{-15})^3 = 1.728 \times 10^{-45} \, \text{m}^3 \]
Thus,
\[ \rho = \frac{3 \times 1.67 \times 10^{-27}}{4 \times 3.14 \times 1.728 \times 10^{-45}} \]
Calculating this gives:
\[ \rho \approx 2.307 \times 10^{17} \, \text{kg/m}^3 \]

### Step 7: Density of water
The density of water \(\rho_w\) is given as:
\[ \rho_w = 10^3 \, \text{kg/m}^3 \]

### Step 8: Calculate how many times nuclear matter is denser than water
To find how many times nuclear matter is denser than water, we calculate:
\[ \frac{\rho}{\rho_w} = \frac{2.307 \times 10^{17}}{10^3} = 2.307 \times 10^{14} \]

### Final Answer
The nuclear matter is approximately \(2.307 \times 10^{14}\) times denser than water.

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