⚡️College Physics III – Thermodynamics, Electricity, and Magnetism Unit 3 – First Law of Thermodynamics

Key Concepts

• First Law of Thermodynamics states energy cannot be created or destroyed, only converted from one form to another
• Internal energy $(U)$ of a system is the sum of kinetic and potential energies of its particles
• Heat $(Q)$ is the transfer of energy between systems due to temperature differences
• Work $(W)$ is the transfer of energy between systems due to force acting over a distance
• Change in internal energy $(\Delta U)$ equals heat added to the system $(Q)$ minus work done by the system $(W)$: $\Delta U = Q - W$
  • Positive $Q$ means heat is added to the system, increasing its internal energy
  • Positive $W$ means work is done by the system, decreasing its internal energy
• Thermodynamic processes include isothermal (constant temperature), isobaric (constant pressure), isochoric (constant volume), and adiabatic (no heat exchange)
• Specific heat capacity $(c)$ is the amount of heat required to raise the temperature of a substance by one degree per unit mass

Historical Context

• The First Law of Thermodynamics emerged in the 19th century during the Industrial Revolution
• It built upon earlier work by scientists such as Robert Boyle, Joseph Black, and James Joule
• Rudolf Clausius introduced the concept of internal energy and formulated the First Law in 1850
• The law was crucial in understanding and optimizing heat engines, which powered the Industrial Revolution
  • Heat engines convert thermal energy into mechanical work (steam engines, internal combustion engines)
• The First Law laid the foundation for the development of thermodynamics as a scientific discipline
• It provided a unifying framework for understanding various thermal phenomena and energy transformations
• The law has since been applied to a wide range of systems, from microscopic particles to large-scale processes

Fundamental Principles

• Conservation of energy: Energy cannot be created or destroyed, only converted from one form to another
• Equivalence of heat and work: Heat and work are two forms of energy transfer that can be interconverted
• State functions: Internal energy $(U)$ is a state function, meaning its value depends only on the current state of the system, not on the path taken to reach that state
  • Changes in internal energy $(\Delta U)$ are independent of the process, only depending on initial and final states
• Reversibility: A reversible process is one that can be reversed without any net change to the system or its surroundings
  • Reversible processes are idealized and serve as a theoretical limit for real processes
• Irreversibility: Most real processes are irreversible, involving dissipation of energy and an increase in entropy
• Thermal equilibrium: When two systems are in thermal equilibrium, there is no net heat transfer between them
  • They have the same temperature and no longer exchange energy

Mathematical Formulation

• The First Law of Thermodynamics is expressed as: $\Delta U = Q - W$
  • $\Delta U$: Change in internal energy of the system
  • $Q$: Heat added to the system (positive if heat is added, negative if heat is removed)
  • $W$: Work done by the system (positive if work is done by the system, negative if work is done on the system)
• For infinitesimal changes: $dU = \delta Q - \delta W$
  • $dU$: Infinitesimal change in internal energy
  • $\delta Q$: Infinitesimal heat added (inexact differential)
  • $\delta W$: Infinitesimal work done (inexact differential)
• Work done by a system during volume change: $W = -\int_{V_1}^{V_2} P dV$
  • $P$: Pressure
  • $V$: Volume
  • Negative sign indicates work is done by the system when volume increases
• Heat capacity: $C = \frac{Q}{\Delta T}$
  • $C$: Heat capacity (amount of heat required to raise the temperature of a substance by one degree)
  • $Q$: Heat added
  • $\Delta T$: Change in temperature
• Specific heat capacity: $c = \frac{C}{m}$
  • $c$: Specific heat capacity (heat capacity per unit mass)
  • $m$: Mass of the substance

Applications in Real Systems

• Heat engines: The First Law governs the efficiency and performance of heat engines (internal combustion engines, steam turbines)
  • Efficiency $(\eta)$ is the ratio of work output to heat input: $\eta = \frac{W}{Q_H}$
  • Carnot efficiency is the theoretical maximum efficiency for a heat engine operating between two temperatures: $\eta_C = 1 - \frac{T_C}{T_H}$
• Refrigerators and heat pumps: The First Law also applies to refrigeration cycles and heat pumps
  • Coefficient of Performance (COP) measures the efficiency of these devices
  • For refrigerators: $COP_R = \frac{Q_C}{W}$, where $Q_C$ is the heat removed from the cold reservoir
  • For heat pumps: $COP_{HP} = \frac{Q_H}{W}$, where $Q_H$ is the heat delivered to the hot reservoir
• Phase transitions: The First Law is used to analyze phase transitions (melting, vaporization) and calculate the energy required for these processes
  • Latent heat is the energy absorbed or released during a phase transition without a change in temperature
  • Latent heat of fusion $(L_f)$ is the energy required to melt a substance
  • Latent heat of vaporization $(L_v)$ is the energy required to vaporize a substance
• Thermochemistry: The First Law is applied in thermochemistry to study heat changes during chemical reactions
  • Enthalpy $(H)$ is a state function related to internal energy: $H = U + PV$
  • Change in enthalpy $(\Delta H)$ equals the heat exchanged at constant pressure: $\Delta H = Q_P$
  • Hess's Law states that the total enthalpy change for a reaction is independent of the pathway

Experimental Demonstrations

• Joule's experiment: James Joule demonstrated the equivalence of heat and work using a paddle wheel apparatus
  • The paddle wheel was rotated by a falling weight, stirring water in an insulated container
  • The temperature increase of the water was proportional to the work done by the falling weight
  • This showed that mechanical work could be converted into heat, with a fixed relationship between the two
• Rubens' tube: A Rubens' tube is a long, gas-filled tube with small holes along its length, used to visualize standing sound waves
  • When sound waves are introduced, the gas molecules oscillate, causing periodic variations in pressure and temperature
  • These temperature variations are made visible by the pattern of flames along the tube, demonstrating the conversion of sound energy into heat
• Calorimetry: Calorimeters are devices used to measure heat transfer and determine the specific heat capacities of substances
  • A simple calorimeter consists of an insulated container filled with water, into which a heated object is placed
  • By measuring the temperature change of the water and knowing the mass of the object and water, the specific heat capacity can be calculated using the First Law
• Thermodynamic cycles: Various thermodynamic cycles (Carnot, Otto, Diesel) can be demonstrated using models or simulations
  • These cycles illustrate how heat engines and refrigerators operate, with energy being transferred between reservoirs and work being done
  • The efficiency of these cycles can be analyzed using the First Law and the concept of reversibility

Common Misconceptions

• Confusion between heat and temperature: Heat is a form of energy transfer, while temperature is a measure of the average kinetic energy of particles in a system
  • Objects with different temperatures can have the same amount of heat, depending on their mass and specific heat capacity
• Belief that energy is "used up" or "lost": The First Law states that energy is conserved, not destroyed
  • When energy seems to be "lost," it is actually converted into other forms (heat, sound, etc.) that may be less apparent or useful
• Misunderstanding of the signs of heat and work: The sign convention for heat $(Q)$ and work $(W)$ in the First Law can be confusing
  • Heat added to the system is positive, while heat removed is negative
  • Work done by the system is positive, while work done on the system is negative
• Assumption that all processes are reversible: Most real processes are irreversible due to factors like friction, heat loss, and non-equilibrium conditions
  • Reversible processes are ideal and serve as a theoretical limit for efficiency
• Confusion between state functions and path functions: Some quantities (internal energy, enthalpy) are state functions, while others (heat, work) are path functions
  • Changes in state functions depend only on the initial and final states, while path functions depend on the specific process

Connections to Other Laws

• Second Law of Thermodynamics: The Second Law introduces the concept of entropy and states that the total entropy of an isolated system always increases
  • It provides a direction for thermodynamic processes and limits the efficiency of heat engines
  • The Second Law is closely related to the irreversibility of real processes
• Third Law of Thermodynamics: The Third Law states that the entropy of a perfect crystal at absolute zero is zero
  • It provides a reference point for calculating entropy changes and predicting the behavior of materials at low temperatures
• Zeroth Law of Thermodynamics: The Zeroth Law defines thermal equilibrium and states that if two systems are in thermal equilibrium with a third system, they are also in thermal equilibrium with each other
  • It provides the basis for measuring temperature and establishing temperature scales
• Conservation of Mass: In closed systems, the First Law of Thermodynamics is often combined with the conservation of mass
  • This leads to the concept of mass-energy equivalence, as expressed in Einstein's famous equation $E = mc^2$
• Ideal Gas Law: The Ideal Gas Law $(PV = nRT)$ relates pressure, volume, temperature, and the amount of an ideal gas
  • It is derived from the kinetic theory of gases and is consistent with the First Law of Thermodynamics
  • The First Law can be applied to analyze processes involving ideal gases, such as isothermal, isobaric, and adiabatic processes