Give an account of the various theories of specific heat of solid. Discuss any of them in detail.

The Dulong-Petit's Law is one of the earliest theories formulated to explain the specific heat capacity of solids. This theory originates from classical mechanics and proposes that the specific heat capacity of a solid is constant and can be calculated using the formula: 3R, where R is the universal gas constant. This implies that each mole of any solid substance should have a specific heat capacity of approximately 24.94 J/mol·K. While this rule works well for many metals at room temperature, it fails to accurately predict the behavior of specific heat at lower temperatures. The reason for this discrepancy lies in the fact that the Dulong-Petit's Law does not account for the quantized nature of energy levels in solid materials, leading to significant deviations from observed values.

Einstein introduced a significant improvement over the Dulong-Petit's Law by incorporating the concept of quantized energy levels. According to Einstein, atoms in a solid vibrate at discrete, quantized energy levels. The specific heat capacity in Einstein’s theory is given by the formula: \[ C = 3R \frac{(\theta/E_a)^2 e^{\theta/E_a}}{(e^{\theta/E_a}-1)^2} \], where \( \theta \) is the Einstein temperature, a characteristic temperature dependent on the vibrational frequency of the atoms, and \( E_a \) represents the activation energy of the system. The Einstein model provided a better fit for experimental data, especially at low temperatures, compared to the classical model. However, it still had limitations as it considered only one vibrational frequency for all atoms, which is not entirely accurate in real-world scenarios.

The Debye theory is a further refinement that extends Einstein’s model by considering a spectrum of vibrational frequencies up to a maximum value known as the Debye frequency. The specific heat capacity in this theory is given by: \[ C = 9R \frac{T^3}{\theta_D^3} \times \frac{3}{e^{(\theta_D/T)}-1} \], where \( \theta_D \) is the Debye temperature. Debye’s theory accounts for low-temperature behavior using a cubic relation \( C \propto T^3 \) and approaches the constant value predicted by Dulong-Petit's Law at high temperatures. This model successfully predicts the specific heat for a wide range of temperatures, making it a more accurate and comprehensive theory for describing the heat capacity of solids.

The Modern Quantum Theory provides the most accurate predictions for specific heat capacity by integrating recent advancements in statistical mechanics and quantum field theory. This theory incorporates interactions at various energy levels and considers the complex interactions between particles within the solid. Through this approach, the Modern Quantum Theory accounts for all quantum states and transitions, capturing both low and high-temperature behavior accurately. It provides an all-encompassing model that includes contributions from lattice vibrations (phonons) and electron movements, offering precise results for different materials and conditions.

Specific heat capacity is a critical property of solids that indicates the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree Celsius. It varies with temperature and the material’s atomic structure. Understanding specific heat capacity is essential as it affects various practical applications, including material selection in engineering, thermal management in electronics, and understanding natural phenomena. The specific heat of solids is particularly influenced by the vibrational modes of the atoms within the solid. Different theories, such as Dulong-Petit's Law, Einstein theory, Debye theory, and the Modern Quantum Theory, have progressively advanced our understanding by incorporating classical mechanics, quantized energy levels, and comprehensive quantum mechanics.