1. Laws of Thermodynamics, Statistical Mechanics, BS 7th semester

### **Laws of Thermodynamics for BS Physics (with Differential Calculations)** Thermodynamics is a fundamental branch of physics that deals with the transfer of energy and the relationships between heat, work, and internal energy in a system. The laws of thermodynamics provide a deep understanding of how energy conversion occurs and how systems approach equilibrium. In this enhanced article, we will explore the four laws of thermodynamics with a focus on differential form and their significance in advanced physics. --- ### **Zeroth Law of Thermodynamics: Thermal Equilibrium** The Zeroth Law of Thermodynamics forms the basis for defining temperature. It states: **If two thermodynamic systems are each in thermal equilibrium with a third system, they are also in thermal equilibrium with each other.** In mathematical terms, if system A is in equilibrium with system B, and system B is in equilibrium with system C, then: \[ T_A = T_B = T_C \] This allows for the establishment of a temperature scale. The concept of thermal equilibrium is crucial for measuring temperature and ensuring consistent thermodynamic states. --- ### **First Law of Thermodynamics: Conservation of Energy** The First Law is the mathematical expression of **energy conservation**. It asserts that the total energy of an isolated system remains constant, although energy can be transformed from one form to another. The internal energy \( U \) of a system changes with the heat \( Q \) added to the system and the work \( W \) done by the system. In its differential form, the First Law can be written as: \[ dU = \delta Q - \delta W \] Where: - \( dU \) = Infinitesimal change in internal energy. - \( \delta Q \) = Infinitesimal heat added to the system. - \( \delta W \) = Infinitesimal work done by the system. If the system performs **quasi-static** work (such as expansion), the work done is given by: \[ \delta W = P dV \] Thus, the First Law becomes: \[ dU = \delta Q - P dV \] Where \( P \) is the pressure and \( dV \) is the infinitesimal change in volume. This is a crucial relation when dealing with processes such as isothermal (constant temperature) and adiabatic (no heat exchange) transformations. #### **Example (Isothermal Expansion of an Ideal Gas):** For an ideal gas undergoing an isothermal expansion (constant \( T \)), the internal energy \( U \) remains constant, so the heat added to the system is entirely converted into work: \[ \delta Q = PdV = nRT \frac{dV}{V} \] Where \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the absolute temperature. Integrating this equation over the volume change gives: \[ Q = nRT \ln \frac{V_f}{V_i} \] Where \( V_f \) and \( V_i \) are the final and initial volumes, respectively. --- ### **Second Law of Thermodynamics: Entropy and Direction of Processes** The Second Law introduces the concept of **entropy** \( S \) and explains the irreversibility of natural processes. It can be stated as: **In any spontaneous process, the total entropy of an isolated system always increases.** This is expressed in differential form as: \[ dS \geq \frac{\delta Q}{T} \] For a **reversible process**, the equality holds (\( dS = \frac{\delta Q}{T} \)). For **irreversible processes**, the inequality applies. Entropy is a measure of the disorder or randomness in a system, and it tends to increase in any real process. #### **Entropy in a Reversible Process:** For an ideal gas in a reversible process, the differential change in entropy is given by: \[ dS = \frac{dQ_{\text{rev}}}{T} \] Integrating this for a process where heat is added or removed gives the total change in entropy: \[ \Delta S = \int_{T_i}^{T_f} \frac{dQ_{\text{rev}}}{T} \] Entropy tends to increase in natural processes, leading to the concept of **irreversibility** in real systems. For example, heat flows spontaneously from a hot body to a cold body, never the reverse. #### **Clausius Inequality:** For any cyclic process, the inequality: \[ \oint \frac{\delta Q}{T} \leq 0 \] This expresses that in a cyclic process, the total entropy change cannot be negative, which means the efficiency of any heat engine is limited. #### **Carnot Efficiency:** The maximum efficiency \( \eta_{\text{max}} \) of a heat engine operating between two thermal reservoirs is given by: \[ \eta_{\text{max}} = 1 - \frac{T_C}{T_H} \] Where \( T_C \) is the temperature of the cold reservoir and \( T_H \) is the temperature of the hot reservoir. No real engine can exceed this efficiency. --- ### **Third Law of Thermodynamics: Absolute Zero and Entropy** The Third Law of Thermodynamics is concerned with the behavior of entropy at very low temperatures. It states: **As the temperature of a system approaches absolute zero, the entropy of the system approaches a constant minimum.** Mathematically, as \( T \to 0 \): \[ S \to S_0 \] For a perfect crystalline substance, \( S_0 = 0 \), meaning the entropy of a perfect crystal at absolute zero is zero. This law has profound implications for the unattainability of absolute zero since removing the last bit of energy from a system becomes increasingly difficult. ### **Implication of the Third Law:** As the temperature approaches absolute zero, the thermal energy of particles in the system reduces, and they approach their minimum energy configuration. This implies that the system becomes perfectly ordered, and the entropy reaches a minimum value. In practice, reaching absolute zero is impossible due to the exponential increase in the work required to extract energy as the system approaches 0 K. Thus, the Third Law highlights the fundamental limit to how cold a system can become. --- ### **Conclusion** The four laws of thermodynamics provide the foundation for understanding energy transformations and the behavior of systems in both reversible and irreversible processes. By integrating the differential forms of these laws, we gain a precise mathematical framework for describing how heat, work, and entropy behave under various conditions. 1. **Zeroth Law** allows us to define temperature and thermal equilibrium. 2. **First Law** demonstrates the conservation of energy in differential form and the relation between internal energy, heat, and work. 3. **Second Law** introduces entropy and the inherent irreversibility of real-world processes, alongside Carnot's limit on engine efficiency. 4. **Third Law** reveals the behavior of systems as they approach absolute zero, and the impossibility of reaching that point. The detailed analysis of these laws, supported by differential calculations, underscores their critical role in thermodynamics and physical systems in general. These laws are not only essential in the study of physics but also have widespread applications in engineering, chemistry, and cosmology.