Conservation of energy Study Guide

📖 Core Concepts Energy Conservation Law – In an isolated system the total energy never changes; it only transforms or moves between forms. Closed vs. Isolated System – Closed: energy can cross the boundary as heat or work. Isolated: no exchange of energy or matter at all. Mass–Energy Equivalence – Special relativity unifies mass and energy: \(E = m c^{2}\). Rest mass can be turned into other energy forms. First Law of Thermodynamics – For a closed system: \(\displaystyle \delta Q - \delta W = \Delta U\). Heat and work are energy transfers; internal energy \(U\) is a state property. Noether’s Theorem – Every continuous symmetry gives a conserved quantity; time‑translation symmetry ⇒ energy conservation. Mechanical‑Equivalent of Heat – Heat and mechanical work are interchangeable; the caloric theory (heat cannot be created/destroyed) is false. --- 📌 Must Remember Law Statement: Total energy of an isolated system = constant. First‑Law Formula: \(\delta Q - \delta W = \Delta U\). Work for Simple Compression: \(\displaystyle \delta W = P\,dV\). Heat for Reversible Process: \(\displaystyle \delta Q = T\,dS\). Relativistic Energy: \(\displaystyle E = m c^{2}\). Noether ⇒ Energy: Time‑translation invariance ↔ energy conservation. Perpetual‑Motion‑First‑Kind: Impossible; would violate the conservation law. Experimental Precision: Verified to \(10^{-15}\) in nuclear experiments. --- 🔄 Key Processes Applying the First Law (Closed System) Identify heat added (\(\delta Q\)) and work done by the system (\(\delta W\)). Compute \(\Delta U = \delta Q - \delta W\). Work in a Quasi‑Static Compression Integrate \(W = \int P\,dV\) between initial and final volumes. Heat Transfer in a Reversible Process Use \(Q = \int T\,dS\) with known temperature–entropy path. Mass‑to‑Energy Conversion (e.g., annihilation) Convert rest mass to energy via \(E = m c^{2}\). Add any kinetic or potential energy of the products for total energy balance. Checking Symmetry for Conservation Determine if the system’s Lagrangian is invariant under time translation. If yes → energy is conserved; if not → energy may flow to/from external agents. --- 🔍 Key Comparisons Energy vs. Mass Conservation Classical: Treated separately. Relativistic: Unified; total mass‑energy is conserved. Closed System vs. Isolated System Closed: Energy can cross boundaries (heat/work). Isolated: No energy exchange; total energy strictly constant. Caloric Theory vs. Energy Theory Caloric: Heat is a conserved fluid, cannot be created. Energy: Heat can be generated from work (Joule’s experiments). Time‑Independent vs. Time‑Dependent Hamiltonian Time‑independent: Energy expectation value stays constant. Time‑dependent: Energy can change; Noether’s theorem does not guarantee conservation. --- ⚠️ Common Misunderstandings “Energy can disappear” – Only appears to disappear when converted to a form not accounted for (e.g., thermal loss). “Mass is always conserved” – In nuclear reactions, rest mass converts to other energy forms; only mass‑energy is conserved. “All heat is “lost” energy – Heat is a legitimate energy transfer; the first law accounts for it just like work. “Perpetual motion machines are possible if friction is removed” – Even with zero friction, the first law forbids net energy output without input. --- 🧠 Mental Models / Intuition “Energy as a Ledger” – Treat energy like money: you can transfer, spend, or convert it, but the total balance in a closed account never changes. “Four‑Vector Conservation” – In relativity, think of energy and momentum as two components of a single conserved “energy‑momentum” package. “Symmetry → Conservation” – If you can shift something in time without altering the physics, the system must keep its energy “budget” unchanged. --- 🚩 Exceptions & Edge Cases Explicitly Time‑Dependent Potentials – Energy is not conserved unless you include the external source/sink. Expanding Universe (FLRW metric) – Global vacuum energy appears to change; strict global conservation may not hold in cosmology. Open Systems with Mass Flow – First law gains a term \(\sumi \dot{m}i hi\) for enthalpy carried in/out. --- 📍 When to Use Which First Law (Thermodynamics) – Use for any problem involving heat, work, and internal energy changes in closed or open systems. \(E = mc^{2}\) – Apply when mass is converted to radiation or kinetic energy (nuclear, particle‑annihilation, astrophysical). \( \delta W = P\,dV\) – Use for quasi‑static mechanical work on gases or pistons. \( \delta Q = T\,dS\) – Use for reversible heat transfer calculations. Noether’s Theorem – Use to justify conservation laws when identifying symmetries in Lagrangian/Hamiltonian formulations. --- 👀 Patterns to Recognize “Heat ↔ Work” – Whenever a mechanical device (e.g., falling weight) produces a temperature rise, look for a Joule‑type energy balance. Quadratic Velocity Dependence – Kinetic energy always appears as \(\frac12 m v^{2}\); check any “vis viva” statements for this pattern. Mass Defect → Energy Release – In nuclear reactions, the missing mass multiplied by \(c^{2}\) gives the released energy. Time‑Invariant Lagrangian – Spot a constant‑in‑time Lagrangian → automatically know energy is conserved. --- 🗂️ Exam Traps Confusing \( \delta Q\) and \(Q\) – \(\delta Q\) denotes an infinitesimal heat transfer; using \(Q\) for a finite amount can lead to sign errors. Sign Convention for Work – Many textbooks define work done on the system as positive; the first law above uses work by the system as positive. Watch the convention given in the problem. Neglecting Mass Flow Enthalpy – In open‑system problems, forgetting the \(\sum \dot{m} h\) term yields an apparent violation of the first law. Assuming Energy Conservation in Expanding Space – Cosmology questions may deliberately test the limits of global conservation; look for wording about “expanding universe” or “vacuum energy”. Mixing Caloric and Energy Theories – Answers that treat heat as a conserved substance (caloric) are outdated and incorrect. ---