According to equipartition law of energy each particle in a system of particles have thermal energy E equal to

Options

A. `E=k_(B)T`
B. `E=(1//2)k_(B)T`
C. `E=3k_(B)T`
D. `E=(3//2)k_(B)T`

Explanation

To solve the question regarding the equipartition law of energy, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Equipartition Law of Energy**:
The equipartition law states that in a system at thermal equilibrium, the total energy is distributed equally among all degrees of freedom of the particles in the system.

2. **Identify Degrees of Freedom**:
Each degree of freedom corresponds to a specific way in which a particle can store energy. For example, in three-dimensional space, a particle has three translational degrees of freedom (movement in x, y, and z directions).

3. **Energy Associated with Each Degree of Freedom**:
According to the equipartition theorem, the energy associated with each degree of freedom (denoted as \( E_F \)) is given by the formula:
\[ E_F = \frac{1}{2} k_B T \]
where \( k_B \) is the Boltzmann constant and \( T \) is the absolute temperature in Kelvin.

4. **Total Thermal Energy for Each Particle**:
If a particle has \( f \) degrees of freedom, the total thermal energy \( E \) for that particle can be expressed as:
\[ E = f \cdot E_F = f \cdot \left( \frac{1}{2} k_B T \right) \]

5. **Conclusion**:
Therefore, according to the equipartition law of energy, each particle in a system of particles has thermal energy \( E \) equal to:
\[ E = \frac{f}{2} k_B T \]

### Final Answer:
The thermal energy \( E \) for each particle in a system according to the equipartition law is given by:
\[ E = \frac{f}{2} k_B T \]