Model Answer
0 min readIntroduction
Thermodynamics, a cornerstone of physics and geology, governs the energy transformations in physical systems. The first and second laws of thermodynamics are fundamental principles that dictate the behavior of energy and entropy. Understanding these laws is crucial for interpreting geological processes, such as phase transitions in minerals, geochemical reactions, and the evolution of planetary interiors. This answer will derive the expression dU = TdS - PdV, representing the change in internal energy (dU) in terms of temperature (T), entropy (S), pressure (P), and volume (V), assuming a reversible process.
Derivation of dU = TdS - PdV
The derivation begins with the fundamental definitions of the first and second laws of thermodynamics in their differential forms.
1. First Law of Thermodynamics
The first law states that the change in internal energy (dU) of a system is equal to the heat added to the system (dQ) minus the work done by the system (dW):
dU = dQ - dW
2. Second Law of Thermodynamics
The second law defines entropy (S) as a state function whose change (dS) is equal to the heat added to the system (dQ) divided by the absolute temperature (T) during a reversible process:
dS = dQ/T
Therefore, dQ = TdS (for a reversible process)
3. Work Done (dW)
For a reversible process involving changes in volume, the work done by the system is given by:
dW = P dV
4. Substituting into the First Law
Now, substitute dQ = TdS and dW = P dV into the first law equation:
dU = TdS - P dV
Thus, we have derived the expression dU = TdS - PdV, which relates the change in internal energy to changes in entropy, temperature, pressure, and volume for a reversible process.
Explanation of Terms
- U (Internal Energy): The total energy contained within a system, including the kinetic and potential energies of its molecules.
- S (Entropy): A measure of the disorder or randomness of a system.
- T (Temperature): A measure of the average kinetic energy of the molecules in a system.
- P (Pressure): The force exerted per unit area.
- V (Volume): The amount of space occupied by a system.
Reversible Process Assumption
It is crucial to note that this derivation is valid only for reversible processes. A reversible process is an idealized process that occurs infinitely slowly, allowing the system to remain in equilibrium at all times. In reality, all natural processes are irreversible to some extent. However, the concept of reversible processes is useful for theoretical calculations and provides a benchmark for understanding the efficiency of real processes.
Applications in Geology
This equation has significant applications in geology, including:
- Geochemical Thermodynamics: Predicting the stability of minerals under different conditions of temperature and pressure.
- Petrology: Understanding the phase transitions that occur during the formation of igneous and metamorphic rocks.
- Geophysics: Modeling the thermal evolution of the Earth's interior.
Conclusion
In conclusion, the expression dU = TdS - PdV is a fundamental relationship in thermodynamics, derived directly from the first and second laws. This equation provides a powerful tool for understanding and predicting the behavior of geological systems. The assumption of a reversible process is critical for the validity of this derivation, and its applications span a wide range of geological disciplines. Further understanding of thermodynamic potentials like enthalpy and Gibbs free energy builds upon this foundational relationship.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.