State and Prove Law of Equipartition of Energy.
The law of equipartition of energy states that, in thermal equilibrium, each degree of freedom in a system contributes an average energy of 1/2 kT, where k is the Boltzmann constant and T is the temperature. It provides insights into energy distribution and is essential in statistical mechanics and thermodynamics.
Statement of the Law : The law of equipartition of energy states that in thermal equilibrium, the average energy associated with each degree of freedom of a system is \(\frac{1}{2} kT\), where \(k\) is the Boltzmann constant and \(T\) is the temperature of the system.
Proof : Let us consider a system consisting of \(N\) particles, each possessing \(f\) degrees of freedom. The total energy of the system, denoted by \(E\), can be expressed as the sum of individual particle energies:
\[E = E_1 + E_2 + E_3 + \ldots + E_i + \ldots + E_N.\]
Here, \(E_i\) represents the energy associated with the \(i\)-th particle.
According to the law of equipartition of energy, each quadratic term in the total energy expression contributes an equal amount of energy, \(\frac{1}{2} kT\), to the average energy of each degree of freedom.
For instance, for a particle moving freely in three dimensions, the energy \(E_i\) can be written as:
\[E_i = \frac{1}{2} m(v_x^2 + v_y^2 + v_z^2),\]
where \(m\) is the mass of the particle, and \(v_x\), \(v_y\), and \(v_z\) are the velocities in the \(x\), \(y\), and \(z\) directions, respectively.
Applying the law of equipartition, each degree of freedom contributes an average energy of \(\frac{1}{2} kT\) to the total energy. As a result, the total contribution of all particles to the average energy is \(f\) times the average energy per degree of freedom:
\[E_{\text{average}} = f \times \frac{1}{2} kT.\]
Therefore, the average energy associated with each degree of freedom of the system is \(\frac{1}{2} kT\), as stated by the law of equipartition of energy.
The law of equipartition of energy provides valuable insights into the behavior of systems at thermal equilibrium and helps determine the distribution of energy among different modes of motion. It has significant implications in fields such as classical and quantum statistical mechanics, thermodynamics, and spectroscopy, enabling the calculation of thermodynamic properties and the understanding of the specific heat capacities of gases and solids.
Conclusion : The law of equipartition of energy states that in thermal equilibrium, each degree of freedom of a system contributes an average energy of \(\frac{1}{2} kT\) to the total energy, where \(k\) is the Boltzmann constant and \(T\) is the temperature. This principle, supported by rigorous mathematical reasoning, has far-reaching implications in understanding the distribution of energy and determining various thermodynamic properties. The law of equipartition serves as a foundational concept in statistical mechanics and finds applications in diverse areas of physics and chemistry.
Also Read : Write the Zeroth Law of Thermodynamics.
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