Thermodynamics: Energy, Entropy, and Thermodynamic Potentials

Energy Concepts: Useful vs. Usable Energy

• Energy is the broad umbrella concept; it can be categorized into what is useful (usable) vs. what is not useful in a given context.
• Examples of useful energy sources that commonly appear in engineering systems:
  • Potential mechanical energy
  • Energy stored in chemical bonds
  • Electric energy and electric fields
  • Magnetic fields
  • Energy in elastic materials (e.g., stretched springs)
  • Energy in pressurized gases
• What is a joule? A few practical exemplars to visualize scale:
  • A joule can represent the kinetic energy of a 2 kg mass moving at 1 m/s, since $E_k = \tfrac{1}{2} m v^2 = \tfrac{1}{2} \cdot 2 \cdot 1^2 = 1\ \text{J}.$
  • It can also represent the heat required to raise the temperature of 1 g of water by $0.24^{\circ}\mathrm{C}$, since for water $Q = c m \Delta T$ with $c \approx 4.186\ \mathrm{J/(g\,^{\circ}!C)}$, so $Q = 4.186 \times 1 \times 0.24 \approx 1.0\ \mathrm{J}.$
• Distinguishing energy, work, and heat:
  • Energy is the overarching concept.
  • Work is the portion of energy that is useful for performing tasks in a system.
  • Heat relates to energy transfer due to a temperature difference.
• In practical terms, identifying sources of usable energy is crucial when designing systems and analyzing energy flows.

Temperature, Heat, and Entropy: How they relate

• Temperature concept:
  • Temperature is proportional to the level of thermal motion; higher temperature corresponds to more thermal motion, more thermal energy, and more randomness.
• Relationship among heat, entropy, and temperature:
  • When heat is added to a system, entropy typically increases.
  • Entropy is a measure of chaos or randomness in a system.
  • For reversible processes, the fundamental relation is $\mathrm{d}Q_{\text{rev}} = T\,\mathrm{d}S.$ Hence, changes in entropy are tied to heat transfer and temperature.
• Entropy and temperature are connected to other thermal metrics (e.g., enthalpy) through their definitions and through the laws of thermodynamics.

The Four Thermodynamic Potentials and Their Differentials

• Internal energy (U):
  • Related to entropy and volume.
  • Differential form: $\mathrm{d}U = T\,\mathrm{d}S - P\,\mathrm{d}V.$
• Enthalpy (H):
  • Definition: $H = U + P V.$
  • Differential form: $\mathrm{d}H = T\,\mathrm{d}S + V\,\mathrm{d}P.$
• Helmholtz free energy (F):
  • Definition: $F = U - T S.$
  • Differential form: $\mathrm{d}F = -S\,\mathrm{d}T - P\,\mathrm{d}V.$
• Gibbs free energy (G):
  • Definition: $G = H - T S = (U + P V) - T S.$
  • Differential form: $\mathrm{d}G = -S\,\mathrm{d}T + V\,\mathrm{d}P.$
• Common variables used in these relations:
  • $T$ = temperature,
  • $S$ = entropy,
  • $P$ = pressure,
  • $V$ = volume.
• Additional note: The equations above reflect standard differential relationships for a simple compressible system and form the basis for many thermodynamic derivations.

First and Second Laws of Thermodynamics (Sign Conventions and Concepts)

• First Law (energy conservation):
  • In differential form, the fundamental balance is $\mathrm{d}U = \delta Q - \delta W.$
  • Here, grecchi:
  • \delta Q>0 means heat is added to the system.
  • \delta W>0 means work is done by the system on the surroundings (i.e., energy leaves the system as work).
  • If the system does work on the surroundings, the term subtracts from internal energy; if heat is added, internal energy increases.
  • For PV-work, $\delta W = P\,\mathrm{d}V,$ so the relation becomes $\mathrm{d}U = \delta Q - P\,\mathrm{d}V.$
• Energy balance and open/closed systems:
  • The total energy of a system is the sum of internal energy and any other forms (e.g., kinetic, potential) depending on the context.
  • The energy balance concept is tied to open vs. closed systems and is central to formulating mass and energy balances in engineering applications.
• Second Law (entropy and spontaneity):
  • For an isolated system, entropy cannot decrease; it must increase or stay the same: entropy increases in spontaneous processes and remains constant for reversible ones.
  • Entropy is a measure of disorder or randomness and is related to heat transfer via entropy changes.
• Entropy and heat relation (reiteration):
  • Changes in entropy are proportional to heat transfer at the system's temperature: $\mathrm{d}S = \frac{\delta Q_{\text{rev}}}{T}.$
• Enthalpy and pressure work terminology in context:
  • Enthalpy includes the pressure-volume work component and thus records the system’s heat content plus the energy needed to create the system under pressure.

How the Key Equations are Derived (Overview of Derivation Sketch)

• From the base relation for internal energy and the product rule:
  • Start with $H = U + P V.$ Taking a total differential and using the product rule yields a path to the enthalpy differential:
  • Using the first law, substitute for (\mathrm{d}U = \delta Q - P\,\mathrm{d}V) and re-arrange to express in terms of entropy and pressure-volume changes.
  • If the process is reversible, substitute (\delta Q = T\,\mathrm{d}S) to obtain the canonical form for the enthalpy differential:
  • $\mathrm{d}H = T\,\mathrm{d}S + V\,\mathrm{d}P.$
• Deriving Helmholtz and Gibbs forms follows similarly by combining the definitions with the product rule and the first law, then substituting for heat via reversible relations when needed:
  • Helmholtz: starting from (F = U - T S) and differentiating, then substituting (\mathrm{d}U = T\,\mathrm{d}S - P\,\mathrm{d}V) and applying the product rule on any T and V dependence, leading to
  • $\mathrm{d}F = -S\,\mathrm{d}T - P\,\mathrm{d}V.$
  • Gibbs: starting from (G = H - T S) and differentiating, substituting the expressions for (\mathrm{d}H) and (\mathrm{d}S) gives
  • $\mathrm{d}G = -S\,\mathrm{d}T + V\,\mathrm{d}P.$
• The derivations emphasize:
  • The interconnections among energy, entropy, temperature, pressure, and volume.
  • How the four potentials provide convenient descriptions of a system under different constraints (e.g., constant T and V for F, constant T and P for G).

Quick Reference: Summary of Key Equations

• Internal energy differential:
  • $\mathrm{d}U = T\,\mathrm{d}S - P\,\mathrm{d}V.$
• Enthalpy differential:
  • $\mathrm{d}H = T\,\mathrm{d}S + V\,\mathrm{d}P.$
• Helmholtz free energy differential:
  • $\mathrm{d}F = -S\,\mathrm{d}T - P\,\mathrm{d}V.$
• Gibbs free energy differential:
  • $\mathrm{d}G = -S\,\mathrm{d}T + V\,\mathrm{d}P.$
• First law (basic form):
  • $\mathrm{d}U = \delta Q - \delta W.$
  • For PV-work: $\mathrm{d}U = \delta Q - P\,\mathrm{d}V.$
• Relation between heat and entropy (reversible):
  • $\delta Q_{\text{rev}} = T\,\mathrm{d}S.$
• Key definitions (recap):
  • $H = U + P V,$
  • $F = U - T S,$
  • $G = H - T S.$
• Common variables: $T$ (temperature), $S$ (entropy), $P$ (pressure), $V$ (volume).

Practical Notes and Takeaways

• When designing systems, identify sources of usable energy and account for how energy transfers occur via heat and work.
• Keep straight the sign convention: energy balance uses "+Q" for heat added to the system and "+W" for work done by the system on surroundings; thus $\mathrm{d}U = \delta Q - \delta W.$
• The four thermodynamic potentials provide convenient descriptions under different constraints and are foundational tools for engineering analysis and optimization.
• The relationships among U, H, F, and G connect microscopic states (S, T) and macroscopic conditions (P, V) in a compact mathematical framework that underpins many applied problems.