At a given temperature internal energy of a monoatomic, diatomic and non - linear gas is `(U_(0)` each. Find their translational and rotiational kinetic energies separately.

Explanation

To solve the problem, we need to find the translational and rotational kinetic energies for monoatomic, diatomic, and non-linear (polyatomic) gases given that their total internal energy is \( U_0 \).

### Step-by-Step Solution:

1. **Understanding Internal Energy**:
The total internal energy \( U \) of a gas is related to its kinetic energy. For an ideal gas, the internal energy is the sum of translational and rotational kinetic energies.

2. **Monoatomic Gas**:
- A monoatomic gas has only translational degrees of freedom.
- The degrees of freedom for a monoatomic gas is 3 (all translational).
- Therefore, the rotational kinetic energy is 0.
- The translational kinetic energy can be calculated as:
\[ K_{trans} = U_0 \]
- Thus, for monoatomic gas:
- Translational Kinetic Energy: \( K_{trans} = U_0 \)
- Rotational Kinetic Energy: \( K_{rot} = 0 \)

3. **Diatomic Gas**:
- A diatomic gas has 5 degrees of freedom: 3 translational and 2 rotational.
- The translational kinetic energy is given by:
\[ K_{trans} = \frac{3}{5} U_0 \]
- The rotational kinetic energy is given by:
\[ K_{rot} = \frac{2}{5} U_0 \]
- Thus, for diatomic gas:
- Translational Kinetic Energy: \( K_{trans} = \frac{3}{5} U_0 \)
- Rotational Kinetic Energy: \( K_{rot} = \frac{2}{5} U_0 \)

4. **Non-linear (Polyatomic) Gas**:
- A non-linear gas has 6 degrees of freedom: 3 translational and 3 rotational.
- The translational kinetic energy is given by:
\[ K_{trans} = \frac{3}{6} U_0 = \frac{1}{2} U_0 \]
- The rotational kinetic energy is also:
\[ K_{rot} = \frac{3}{6} U_0 = \frac{1}{2} U_0 \]
- Thus, for non-linear gas:
- Translational Kinetic Energy: \( K_{trans} = \frac{1}{2} U_0 \)
- Rotational Kinetic Energy: \( K_{rot} = \frac{1}{2} U_0 \)

### Summary of Results:
- **Monoatomic Gas**:
- Translational Kinetic Energy: \( K_{trans} = U_0 \)
- Rotational Kinetic Energy: \( K_{rot} = 0 \)

- **Diatomic Gas**:
- Translational Kinetic Energy: \( K_{trans} = \frac{3}{5} U_0 \)
- Rotational Kinetic Energy: \( K_{rot} = \frac{2}{5} U_0 \)

- **Non-linear Gas**:
- Translational Kinetic Energy: \( K_{trans} = \frac{1}{2} U_0 \)
- Rotational Kinetic Energy: \( K_{rot} = \frac{1}{2} U_0 \)