The nucleus radius of `._(8)O^(16)` is `3 xx 10^(-15)m`. Find the density of nuclear matter.

Answer

To find the density of nuclear matter for the nucleus of \( _{8}^{16}O \), we can follow these steps:

### Step 1: Calculate the Volume of the Nucleus

The volume \( V \) of a 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.

Given:
- Radius \( r = 3 \times 10^{-15} \) m

Substituting the value of \( r \):

\[ V = \frac{4}{3} \pi (3 \times 10^{-15})^3 \]

### Step 2: Calculate the Mass of the Nucleus

The mass of the nucleus can be calculated using the number of nucleons and the mass of a single nucleon.

The nucleus of \( _{8}^{16}O \) has 16 nucleons (8 protons and 8 neutrons). The mass of a nucleon is approximately \( 1.66 \times 10^{-27} \) kg.

Thus, the total mass \( M \) of the nucleus is:

\[ M = 16 \times (1.66 \times 10^{-27}) \text{ kg} \]

### Step 3: Calculate the Density of Nuclear Matter

The density \( \rho \) of nuclear matter is given by the formula:

\[ \rho = \frac{M}{V} \]

Substituting the values of \( M \) and \( V \) calculated in the previous steps:

\[ \rho = \frac{16 \times (1.66 \times 10^{-27})}{\frac{4}{3} \pi (3 \times 10^{-15})^3} \]

### Step 4: Solve for Density

Now, we can compute the values:

1. Calculate the volume \( V \):

\[ V = \frac{4}{3} \pi (3 \times 10^{-15})^3 \approx 1.13 \times 10^{-44} \text{ m}^3 \]

2. Calculate the mass \( M \):

\[ M = 16 \times (1.66 \times 10^{-27}) \approx 2.656 \times 10^{-26} \text{ kg} \]

3. Finally, calculate the density \( \rho \):

\[ \rho = \frac{2.656 \times 10^{-26}}{1.13 \times 10^{-44}} \approx 2.35 \times 10^{17} \text{ kg/m}^3 \]

### Final Answer

The density of nuclear matter for the nucleus \( _{8}^{16}O \) is approximately:

\[ \rho \approx 2.35 \times 10^{17} \text{ kg/m}^3 \]

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