Detailed Notes: The First Law of Thermodynamics

This document provides a comprehensive overview of the First Law of Thermodynamics, its fundamental concepts, related state functions, and applications, as derived from the provided physical chemistry textbook.

1. Introduction to the First Law

The First Law of Thermodynamics is a statement of the conservation of energy. It introduces the concept of internal energy (U) as a fundamental state function of a system. It defines how energy is transferred between a system and its surroundings through heat (q) and work (w).

• Internal Energy (U): An extensive state function whose existence is postulated by the First Law. It represents the total energy stored within the system at the molecular level (translational, rotational, vibrational, electronic, and intermolecular potential energies).
• Total Energy (E): For a macroscopic body, E=K+V+U, where K is macroscopic kinetic energy and V is macroscopic potential energy. In most thermodynamic applications, systems are at rest with no external fields, so K=0 and V=0, leading to E=U.

2. Classical Mechanics (Review)

Thermodynamic concepts of work and energy originated in classical mechanics.

• Newton’s Second Law: F=ma (Force = mass × acceleration).
• Work (w): Energy transfer due to a macroscopic force acting through a distance.
  • Infinitesimal work: dw=Fx​dx.
  • Finite work: w=∫x1​x2​​F(x)dx.
  • Sign Convention: w>0 if work is done on the system by the surroundings; w<0 if work is done by the system on the surroundings.
  • SI Unit: Joule (J). 1 J=1 N⋅m=1 kg⋅m2/s2.
• Kinetic Energy (K): K=21​mv2. The work-energy theorem states w=ΔK.
• Potential Energy (V): Energy associated with the position of a body in a conservative force field (e.g., gravitational potential energy V=mgh).
• Conservation of Mechanical Energy: If only conservative forces act, K+V (total mechanical energy) remains constant.

3. P-V Work

This is the most common type of work in thermodynamics, associated with changes in the system’s volume.

• Reversible P-V Work: A process where the system is always infinitesimally close to equilibrium (i.e., infinitesimal imbalance of forces).
  • Infinitesimal work done on the system: dwrev​=−Psys​dV.
  • Finite work done on the system: wrev​=−∫V1​V2​​Psys​dV.
  • Path Dependence: The value of wrev​ depends on the specific path (process) taken between initial and final states, meaning it’s a line integral.
• Irreversible P-V Work: Occurs when there are finite unbalanced forces or turbulence (e.g., rapid expansion into a vacuum).
  • Cannot be calculated from Psys​ directly, as Psys​ is not uniform.
  • For a frictionless piston where piston starts and ends at rest: wirrev​=−∫V1​V2​​Pext​dV.

4. Heat

Heat (q) is defined as energy transfer between a system and its surroundings due to a temperature difference.

• Sign Convention: q>0 if heat flows into the system; q<0 if heat flows out of the system.
• Heat Capacity (Cpr​): Cpr​=dqpr​/dT. This relates heat flow to temperature change for a specific process (pr).
  • Specific Heat Capacity (cP​): Heat capacity per unit mass (cP​=CP​/m). dqP​=mcP​dT.
• Unit of Heat: Traditionally calorie (cal). Now, the SI unit is the Joule (J). 1 cal=4.184 J (thermochemical calorie).

5. The First Law of Thermodynamics (Formal Statement)

For a closed system at rest in the absence of external fields, the change in internal energy (ΔU) for any process is given by:
ΔU=q+w

• U is a State Function: The change in internal energy (ΔU) depends only on the initial and final states of the system (U2​−U1​) and is independent of the path taken.
• q and w are Path Functions: Heat and work are forms of energy transfer and their values depend on the specific path (process) followed.
• Cyclic Process: For any process where the system returns to its initial state, ΔU=0. However, q and w are generally non-zero for a cyclic process.

6. Enthalpy (H)

Enthalpy (H) is a derived state function useful for processes occurring at constant pressure.

• Definition: H=U+PV.
• H as a State Function: Since U, P, and V are all state functions, H is also a state function.
• Constant-Pressure Process (closed system, P-V work only):
  ΔH=qP​
  • This is a crucial relationship: the heat absorbed or released in a constant-pressure process equals the change in the system’s enthalpy.
• Constant-Volume Process (closed system, P-V work only):
  ΔU=qV​
  • In this case, since dV=0, w=0.
• Solids and Liquids: For condensed phases (solids and liquids) at low or moderate pressures, volume changes are small. Therefore, the PV term’s change is often negligible, leading to ΔH≈ΔU.

7. Heat Capacities

These quantities describe how the internal energy and enthalpy of a system change with temperature.

• Heat Capacity at Constant Pressure (CP​):
  CP​=(∂T∂H​)P​
• Heat Capacity at Constant Volume (CV​):
  CV​=(∂T∂U​)V​
• Both CP​ and CV​ are extensive state functions and are always positive for stable systems.
• Relation between CP​ and CV​:
  CP​−CV​=[(∂V∂U​)T​+P](∂T∂V​)P​

8. The Joule and Joule-Thomson Experiments

These experiments were designed to measure changes in temperature with volume or pressure at constant internal energy or enthalpy, respectively, providing insights into intermolecular forces.

• Joule Experiment (Free Expansion into Vacuum):
  • Process: Gas expands into an evacuated chamber via adiabatic walls.
  • Characteristics: q=0, w=0.
  • Result: ΔU=0. The internal energy remains constant.
  • Joule Coefficient (μJ​): μJ​=(∂V∂T​)U​.
  • Relationship: (∂V∂U​)T​=−CV​μJ​. For an ideal gas, μJ​=0, implying (∂U/∂V)T​=0.
• Joule-Thomson Experiment (Throttling through Porous Plug):
  • Process: Gas flows slowly through a porous plug between regions of constant high and low pressure, under adiabatic conditions.
  • Characteristics: q=0.
  • Result: ΔH=0. The enthalpy remains constant.
  • Joule-Thomson Coefficient (μJT​): μJT​=(∂P∂T​)H​.
  • Significance: μJT​>0 means cooling upon expansion (used in gas liquefaction); μJT​<0 means warming.

9. Perfect Gases and the First Law

A perfect gas is an idealized gas that satisfies two conditions:

1. Obeys the ideal-gas equation of state: PV=nRT.
2. Its internal energy depends only on temperature: (∂U/∂V)T​=0. (This also implies (∂H/∂P)T​=0.)

• Key Relations for Perfect Gases:
  • U=U(T) and H=H(T).
  • dU=CV​dT and dH=CP​dT.
  • CV​ and CP​ depend only on T.
  • Mayer’s Relation: CP​−CV​=nR (or CP,m​−CV,m​=R).
  • μJ​=0 and μJT​=0.
• Specific Processes in Perfect Gases:
  • Reversible Isothermal Process (ΔT=0):
    • ΔU=0, ΔH=0.
    • q=−w.
    • wrev​=−nRTln(V1​V2​​)=−nRTln(P2​P1​​).
  • Reversible Adiabatic Process (q=0):
    • dU=dw⟹CV​dT=−PdV.
    • Assuming CV​ is constant: CV,m​ln(T1​T2​​)=Rln(V2​V1​​) or T2​/T1​=(V1​/V2​)R/CV,m​.
    • Defining heat-capacity ratio (γ) as γ=CP​/CV​: P1​V1γ​=P2​V2γ​.
    • w=ΔU=CV​(T2​−T1​).
  • Adiabatic Expansion into Vacuum (Joule Experiment for Perfect Gas):
    • q=0, w=0.
    • ΔU=0, ΔH=0.
    • ΔT=0 (temperature remains constant).

10. Calculation Summary of First-Law Quantities

This section summarizes how to calculate q, w, ΔU, and ΔH for common processes in closed systems with only P-V work.

• Reversible Phase Change (Constant T, P):
  • q=ΔH (latent heat, must be measured).
  • w=−PΔV.
  • ΔU=q+w.
• Constant-Pressure Heating/Cooling (no phase change):
  • qP​=∫T1​T2​​CP​(T)dT=ΔH. (Holds whether reversible or irreversible, as H is a state function).
  • w=−PΔV.
  • ΔU=qP​+w.
• Constant-Volume Heating/Cooling (no phase change):
  • w=0.
  • qV​=∫T1​T2​​CV​(T)dT=ΔU. (Holds whether reversible or irreversible, as U is a state function).
  • ΔH=ΔU+Δ(PV).
• Any Perfect Gas Change of State:
  • ΔU=∫T1​T2​​CV​(T)dT.
  • ΔH=∫T1​T2​​CP​(T)dT.
  • q and w are path-dependent. If reversible, w=−∫PdV=−∫(nRT/V)dV, where T varies with V as per the path.
• Isothermal Reversible Process (Perfect Gas):
  • ΔU=0, ΔH=0.
  • q=−w.
  • w=−nRTln(V2​/V1​)=−nRTln(P1​/P2​).
• Adiabatic Reversible Process (Perfect Gas):
  • q=0.
  • w=ΔU.
  • ΔU=∫T1​T2​​CV​(T)dT.
  • For constant CV​: P1​V1γ​=P2​V2γ​ and T1​V1γ−1​=T2​V2γ−1​.
• Adiabatic Expansion into Vacuum (Perfect Gas):
  • q=0, w=0.
  • ΔU=0, ΔH=0.
  • ΔT=0.

11. State Functions and Line Integrals

• State Function (Exact Differential): A property whose change depends only on the initial and final states, not the path.
  • ∫12​db=b2​−b1​ (path-independent).
  • ∮db=0 (for any cyclic process).
  • Examples: U,H,T,P,V.
• Path Function (Inexact Differential): A property whose value depends on the path taken.
  • ∫12​db (path-dependent).
  • ∮db=0 (generally for a cyclic process).
  • Examples: q,w.

12. Molecular Nature of Internal Energy (Qualitative)

Internal energy (U) is the sum of various forms of molecular energy:

• Translational Kinetic Energy (Utr​): Energy of molecules moving through space. For 1 mole of gas, Utr,m​=23​RT. Primarily depends on temperature.
• Rotational Kinetic Energy (Urot​): Energy of molecules rotating. For 1 mole of linear gas, Urot,lin,m​=RT; for nonlinear, Urot,nonlin,m​=23​RT. Primarily depends on temperature (at moderate and high temperatures).
• Vibrational Energy (Uvib​): Energy of atoms oscillating within molecules. Contributes significantly at higher temperatures. Has a “zero-point energy” even at 0 K.
• Electronic Energy (Uel​): Energy associated with electron distribution. Very large energy gaps usually exist, so it remains constant for most processes unless chemical reactions or high-energy excitations occur.
• Intermolecular Potential Energy (Uintermol​): Energy due to forces between molecules.
  • Small (but non-zero) for gases at low pressures.
  • Significant for gases at high pressures, and very large for liquids and solids due to closer molecular proximity.
  • Generally negative (attractions lower energy).
• Rest-Mass Energy (Urest​): Energy associated with the mass of electrons and nuclei (mc2). A constant unless nuclear reactions occur.

Heat capacities (CP​, CV​) are related to how many of these energy modes can absorb energy as temperature increases. More modes accessible generally means higher heat capacities.