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An adiabatic process has no heat transfer across the chosen boundary throughout the process: $\delta Q=0$, rather than merely zero net heat after incoming and outgoing transfers cancel. It is a condition on energy transfer, whereas an isothermal process holds temperature constant. A sealed cylinder can approach an adiabatic process through insulation or through motion short compared with the time needed for appreciable heat transfer. Its temperature can nevertheless change because work crosses the boundary.^[ref]

Take the trapped gas as a closed system, with fixed amount of substance and composition and no net change in bulk kinetic energy or potential energy between the endpoints. Define $Q_{\mathrm{in}}>0$ as heat entering and $W_{\mathrm{out}}>0$ as work delivered to the surroundings. The first law of thermodynamics givesHere $U$ is internal energy. An adiabatic compression normally supplies work, so $W_{\mathrm{out}}<0$; an adiabatic expansion against a resisting load delivers work, so $W_{\mathrm{out}}>0$. An adiabatic process need not preserve internal energy and need not make the closed system an isolated system. ^[ref]

For a reversible thermodynamic process, $dS=\delta Q_{\mathrm{rev}}/T$ at positive absolute temperature $T$. A reversible adiabatic process therefore has $dS=0$: it is an isentropic process. Irreversible adiabatic processes can instead generate entropy; free expansion is the existing counterexample, with its calculation in entropy change in free expansion. Conversely, constant entropy alone does not certify an adiabatic process or reversibility: outward heat can export the entropy generated internally. For a closed system the macroscopic entropy balance iswhere $T_b$ is the absolute boundary temperature where each heat transfer occurs and $S_{\mathrm{gen}}$ is internal entropy production. With no heat transfer, this reduces to $\Delta S=S_{\mathrm{gen}}$; reversibility makes $S_{\mathrm{gen}}=0$. ^[ref]

A motivating experiment measures the gas temperature directly during compression and expansion, instead of inferring it from how warm a cylinder feels. PASCO's authored TD-8565 manual describes a sealed piston cylinder, a fine nickel-wire thermometer, an absolute pressure sensor and a position readout for volume. A computer records all three channels together. The wire's electrical resistance changes with temperature; its small size permits a rapid response. ^[ref]

Use ordinary trapped air and the apparatus's designed travel stops; mount the base securely, close the valve during a stroke, and record the actual volume limits. John Hart's university demonstration uses a large transparent cylinder with a low-heat capacity thermocouple, a pressure sensor, a travel gauge and a safety stop. It explicitly keeps the release valve closed during the demonstration. The basic observable is a rise in temperature on compression and a fall on loaded expansion, alongside the corresponding pressure and volume changes. Hart discusses wall thermal conduction, sensor response and leakage as reasons real records differ from the reversible model. ^[ref]

For a proposed rapid-versus-slow comparison, let the air and walls return to the same initial thermal equilibrium before each run, then compare the same starting and finishing volume values with different stroke durations. Read temperature, absolute pressure and position continuously, including the hold after motion stops. A rapid compression gives a temporary temperature rise; holding the smaller volume lets heat leave and the temperature approach the surroundings again. Slow compression in good thermal contact allows that heat transfer during the stroke and approaches an isothermal compression. Loaded expansion reverses the signs: the rapid stroke cools, while heat entering during slow motion reduces the cooling. PASCO explicitly supports rapid nearly adiabatic processes and slow isothermal processes; the matched-run controls here are a proposed extension, not a reported set of measurements. ^[ref]

Insulation suppresses thermal conduction; it does not prove that heat transfer vanishes. As proposed controls, repeat the matched strokes with and without a removable insulating jacket, record room and wall temperatures, measure fixed-position drift, and check valve closure and seal leakage. Calibrate the thermometer and position readout independently, including any connected dead volume; account for the heat capacity of the probe rather than treating it as a passive observer. Hart explicitly identifies the small air sample relative to the probe and walls, sensor inertia and seal leakage as limitations. A fast display cannot eliminate those physical effects. ^[ref]

There are two distinct timing requirements. Let ${\tau }_{\mathrm{eq}}$ describe internal equilibration, ${\tau }_{\mathrm{probe}}$ the thermometer response, $t_{\mathrm{move}}$ the stroke duration and ${\tau }_{\mathrm{heat}}$ the effective heat transfer time constant. A useful sufficient window for a resolved, nearly reversible adiabatic process iswith negligible friction. The left inequality permits a quasistatic process and a usable readout; the right limits heat leakage. A violent stroke may satisfy only the right inequality. PASCO reports a roughly $30$–$50\mathrm{ms}$ lag of its temperature channel behind the other channels, so apparent simultaneous samples require care. If no such window exists, analyze equilibrium endpoints with the first law of thermodynamics, and regard the path as approximate. ^[ref]

The diagram's model is deliberately a hold after the stroke, when boundary work is zero. For constant whole-sample heat capacity at constant volume $C_V$ and effective thermal conductance $K>0$, it assumesThis is Newton's law of cooling with one uniform gas temperature and a fixed surrounding temperature; actual wall and probe dynamics can require more than one time constant. At $t/{\tau }_{\mathrm{heat}}=0.1$, the fraction of the initial temperature anomaly erased is $1-e^{-0.1}\simeq 0.0952$; at one time constant it is about $0.6321$. These are analytical fractions, not experimental tolerances or evidence that the preceding stroke was reversible. Measuring the hold helps estimate heat leakage independently of the motion.

The usefulness of an adiabatic process is that it separates mechanical work from heat transfer. Its reversible limit connects temperature changes, mechanical stiffness and entropy in one model: Rüchardt experiment and Clément-Desormes experiment already use that connection to measure heat capacity ratio. The contrast with free expansion explains the deeper point: suppressing heat does not determine how much work is exchanged or how much entropy is produced.

Anharmonicity is departure from the motion of a harmonic oscillator, often caused by nonquadratic terms in a potential energy. It can make the oscillation frequency depend on amplitude and generate higher harmonics. A quartic oscillator has one such anharmonic potential energy.

For unit mass, incoming speed $v>0$, impact parameter $b>0$, and potential energy $\mp \kappa /r$, put $q=\kappa /v^2>0$. At closest approach, conservation of energy and conservation of angular momentum give the displayed radii and tangential speeds $u_{\mp }=bv/p_{\mp }$. Consequently $p_{-}{p}_{+}=b^2$ and $u_{-}{u}_{+}=v^2$. These relations compare equal force magnitudes. The attractive radial case $b=0$ is a singular collision, so the speed-product statement does not extend to that endpoint.

The portion of a fluid system's potential energy that can be released by admissible rearrangement relative to a stable reference state. In a uniformly stratified incompressible fluid, the quadratic density is $A={\rho }_0{N}^2{\xi }_z^2/2={\rho }_0{b}^2/(2N^2)$, where ${\xi }_z$ is vertical fluid displacement and $b$ the buoyancy perturbation. It is positive for stable stratification and measures the restoring energy of internal gravity waves.

Baroclinic instability converts potential energy of a stratified sheared basic flow into growing disturbances. In the Eady model, coupled Boundary Rossby waves provide a useful mechanism: their induced velocities displace boundary buoyancy anomalies, and suitable wavelengths permit phase locking. A single neutral boundary wave does not by itself imply exponential instability.

Joseph Bertrand's theorem states that, among smooth attractive central potentials admitting stable circular orbits, requiring every bound orbit to be closed singles out the inverse-distance potential energy and the isotropic quadratic potential energy, up to additive constants. Rational radial-to-angular frequency ratios near a circular orbit provide only a necessary local closure test; finite-amplitude closure is the additional content. In $U=B{r}^n$, the radial-to-angular frequency ratio near a stable circle is $\sqrt{n+2}$, and stability requires $nB>0$ and $n>-2$. The all-bound-orbit cases are $n=-1$ and $n=2$.

A bond dipole represents the separation of positive and negative electric charge associated with a chemical bond. In a neutral two-point charge model put $+q$ at ${\mathbf{r}}_A$ and $-q$ at ${\mathbf{r}}_B$, with $q>0$. Its physical electric dipole moment isIf $B$ is the negative end, the physical electric dipole moment points from $B$ to $A$. The customary crossed-tail chemical arrow points towards the negative, more electronegative end, from $A$ to $B$. It therefore has the opposite direction. State the convention before doing vector addition; use physical electric dipole moments in the interaction potential energy $U=-\mathbf{p}\cdot \mathbf{E}$, where $\mathbf{E}$ is the electric field.^[ref]

For a fixed neutral molecule, its complete electric dipole moment follows from all atomic nuclei and the electron density, not just point charges assigned to sites. If $n(\mathbf{r})$ is the expected number of electrons per unit volume, the corresponding charge density is ${\rho }_e(\mathbf{r})=-en(\mathbf{r})$; the nuclear terms contribute separately. Partitioning that distribution into bond dipoles is a model. Lone pairs and electric polarization within a chemical atom can contribute as well. A measured molecular electric dipole moment cannot uniquely recover every individual bond dipole or partial atomic charge.

A bond-dissociation energy specifies the energy needed for a stated dissociation into separated fragments. An energy from the lowest vibrational state, an energy from the bottom of a potential energy well, and a finite-temperature molar dissociation enthalpy require different conventions; do not silently mix them in an electronegativity formula.

Which fragments, in which quantum states, and at which temperature? For a neutral gaseous molecule with a specified two-electron covalent bond, the usual thermochemical convention is homolysis,Each fragment receives one of the two bonding electrons; the dots denote unpaired electrons. The final fragments are isolated from each other, with their own relaxed molecular geometry and specified electronic quantum states. The energy cost includes this relaxation, rather than merely stretching the initial chemical bond while freezing everything else. For a neutral parent, heterolysis instead produces ${\mathrm{A}}^{+}(g)+{\mathrm{B}}^{-}(g)$, or the opposite assignment of electric charges; those are different final states and different energies. A value for gaseous neutral fragments cannot be transferred to ions, a chemical solution, or a solid without accounting for those changed conditions.^[ref]

The distinction is useful beyond terminology. Pauling electronegativity compares consistently defined bond-dissociation energies; photodissociation can reveal a specified ground-state threshold; a thermal chemical reaction balance requires compatible enthalpies. A threshold is also different from an activation energy: it compares initial and final quantum states, whereas a barrier can obstruct the path between them.

The bond force constant is the curvature of the effective potential energy at the equilibrium bond length. For a diatomic bond stretch, it plays the role of a spring constant. Its definition is local: two wells with the same curvature can have different bond-dissociation energies.

For a diatomic molecule, let $V(r)$ be the electronic potential energy at bond length $r$, with the atomic nuclei treated as fixed while finding the electronic energy. Choose zero at infinitely separated chemical atoms in the specified electronic quantum states. If the minimum occurs at $r_e$, the bond potential-well depth isThe bottom of this curve describes an equilibrium bond length, not an attainable motionless vibrational quantum state. Changing the zero of potential energy shifts both endpoints equally and leaves $D_e$ unchanged. A quantum harmonic oscillator approximates the curve near its minimum but rises without limit, so that local approximation alone cannot supply a dissociation asymptote. A Morse potential supplies both a minimum and a finite separated-fragment limit.^[ref]

The Born-Oppenheimer approximation separates electronic and nuclear motion. At leading order, fixed nuclear positions and electric charges specify an electronic potential energy curve, on which the atomic nuclei move with their own masses.

An external pressure changes the balance even if the system remains confined. For a material boundary acted on by spatially uniform inward pressure $P_s$, the surface force contributes $-P_s{\int }_{\partial \mathcal{V}}\mathbf{r}\cdot \mathbf{n}dS=-3P_s\mathcal{V}$, by the divergence theorem. For Newtonian gravitational potential energy, the finite-window identity becomesHere $\mathcal{V}$ is the enclosed volume, $\mathbf{n}$ the outward unit normal, and $K$ includes the microscopic translational kinetic energy of the confined particles. Do not also add the same random motions as a separate internal pressure term. A nonuniform external stress needs its surface integral; a fixed observational boundary crossed by matter also needs flux terms.^[ref]

With negligible surface pressure, the negative heat capacity of a virialized gravitational system follows from $E=-⟨K⟩$: after a slow loss of total energy and readjustment to virial equilibrium, the average kinetic energy rises. For fixed $N$ classical particles with three translational degrees of freedom, define kinetic temperature $\Theta$ by $⟨K⟩=3N{k}_B\Theta /2$. Along this isolated equilibrium sequence, $dE/d\Theta =-3N{k}_B/2$. The system contracts and its binding potential energy becomes more negative, supplying both the outgoing energy and the increased kinetic energy. This is an endpoint comparison, not a claim that every transient contraction obeys the average relation. A collisionless stellar system need not have a common thermodynamic temperature; a pressure-confined system instead has $E=-⟨K⟩+3⟨P_s\mathcal{V}⟩$. A stable canonical ensemble in contact with a large thermal reservoir does not support this negative heat capacity branch.^[ref]

An orbit that remains within a bounded spatial region for all times for which the motion is defined. In an attractive inverse-distance central potential with zero potential energy at infinity, a noncolliding orbit is bound when its total mechanical energy is negative. That sign test depends on the potential and its energy origin; it is not a universal definition of bounded motion. A closed orbit repeats its position and velocity, whereas a general bound orbit can precess without closing.

Calorimetry measures heat transfers. A calorimeter, a calibrated heat capacity and an accounting of unwanted heat transfer connect an observed temperature change to an energy balance, as in heat-engine calorimetry and heat-pump calorimetry.

The central idea is to measure a response that is easier to observe than the transfer itself: a known heat capacity turns a temperature change into an energy estimate. Mixing calorimetry can determine an unknown specific heat capacity, while a chemical reaction can be studied by measuring the response of its surroundings. The useful observable is a change in temperature, together with a stated thermodynamic system boundary; a hot object does not contain a separately conserved store of heat.

For a closed system with unchanged bulk kinetic energy and potential energy, let $Q_{\mathrm{in}}$ be net inward heat and $W_{\mathrm{on}}$ net inward work. The first law of thermodynamics givesIf the boundary includes a hot sample, receiving water and the participating vessel, their mutual heat transfer is internal. With no external input, the total storage change is minus the signed outward heat leakage, not the positive amount transferred from sample to water. A calorimeter calibration with an immersed resistor instead supplies electrical work across this boundary; Joule heating converts it into internal energy. Choosing only the receiving water as the boundary changes which transfers count as external.

The mechanical constraint decides which storage quantity the readout represents. For a closed system at fixed volume, with no other work, $Q_V=\Delta U$. At constant external pressure, with only expansion work $p\Delta V$, $Q_p=\Delta H$, where $H=U+pV$ is enthalpy. Thus the corresponding warming balances use heat capacity at constant volume or heat capacity at constant pressure. Electrical or shaft work must be included separately when present. These endpoint balances do not require the actual exchange to be a reversible thermodynamic process.

A covered cup open to atmospheric pressure approximately realizes the second constraint when escaping material is negligible. A sealed rigid reaction vessel realizes the first; its surrounding water, vessel and immersed parts form the calibrated receiver. The reaction's outward heat equals the receiver's gain only after losses and ignition input have been accounted for. The distinction concerns the reaction boundary, even if its receiver is open to atmospheric pressure.^[ref]

A historical quantitative result appears in James Prescott Joule's On the Mechanical Equivalent of Heat (1850): Table IX gives $772.692$ foot-pounds of mechanical work per warming of one pound of water by one degree Fahrenheit, with weighing reduced to vacuum. His final rounded statement specifies water between $55$ and $60$ degrees Fahrenheit. He acknowledged residual vibration and sound as possible upward bias. This is an original reported conversion result, not the definition of the modern joule; the apparatus and resting controls are described in Joule's paddle-wheel experiment.^[ref]

For an electron packet, a sufficiently positive gate electric potential attracts electrons beneath the insulating layer of a MOS capacitor. The corresponding potential energy is $U=-e\varphi$, where $e>0$ is the magnitude of elementary charge: a larger electric potential gives a deeper potential well for electrons. A drawing of potential energy must therefore reverse the vertical sense of a drawing of electric potential.

A simplified three-phase transfer raises the adjacent gate's electric potential before lowering the old holding gate. This overlaps the accessible regions, then leaves the packet under the new gate. Repeat with the next gate to move it forward. Isolated wells with an intervening barrier would store the packet without transferring it.^[ref]

A chemical bond associates chemical atoms or groups of chemical atoms into a sufficiently stable structure to treat it as a distinct entity. The concept covers connections within molecules and in extended solids; see the IUPAC Gold Book, chemical bond. It is useful because composition alone does not specify how the chemical atoms are connected, and those connections help determine a chemical substance's properties and possible chemical reactions.

In a covalent bond, electrons are shared between chemical atoms. Two hydrogen chemical atoms can form a molecular hydrogen molecule, with its electrons interacting with both atomic nuclei. In the usual description of this diatomic molecule, the potential energy as a function of nuclear separation has a minimum: moving the nuclei too close or too far raises the energy. See 7.2 Covalent Bonding.

Separating bonded chemical atoms requires energy; forming the corresponding chemical bonds releases energy. Whether an overall chemical reaction releases energy depends on both the chemical bonds broken and those formed. The lines in the stoichiometry diagram indicate connections schematically, rather than literal rods or a sequence of microscopic events.

A bounded trajectory in the damped Binet equation with nonzero angular momentum asymptotically becomes circular. Energy loss is nonnegative in magnitude and vanishes only for zero radial velocity. The centrifugal effective potential energy excludes collision; bounded radius forces indefinitely increasing forward angular distance, so the reciprocal-radius oscillation decays. Escaping trajectories are excluded by the boundedness hypothesis.

For a particle of fixed energy $E$, a classically allowed region has positive classical kinetic energy $E-V(x)$. For a constant potential energy, the stationary wavefunction is a combination of $e^{ikx}$ and $e^{-ikx}$, where $k=\sqrt{2m(E-V)}/\hslash$ is the wavenumber. In a varying potential energy this local value underlies the WKB approximation away from turning points.

A classically forbidden region would require negative classical kinetic energy, but a quantum wavefunction need not vanish there. For constant potential energy, it is $A{e}^{qx}+B{e}^{-qx}$ with $q=\sqrt{2m(V-E)}/\hslash$. Boundary matching determines both coefficients. Beside a hard wall at $D$, a nonzero forbidden-region solution is proportional to $\sinh [q(D-x)]$, and therefore has no interior nodes. Across a finite barrier, both exponentials are generally needed; a pure decaying exponential is appropriate only when the other boundary condition permits it.

Classical mechanics describes the motion of bodies by assigning positions and velocities and specifying how their interactions change those quantities. A point mass is the simplest model: its size and internal structure are ignored, leaving a mass and a moving position. A rigid body retains orientation and rotational kinetic energy. Choosing a model means deciding which features affect the question being asked; a sliding block may be treated as a point mass, whereas predicting its rolling without slipping requires its moment of inertia.

Three complementary questions organize the subject. Newton's second law asks how the net force determines the instantaneous acceleration. The work-energy theorem asks how that force, acting through a displacement, changes kinetic energy. Conservation of mechanical energy asks which combinations of position and velocity are accessible when the interactions admit a time-independent potential energy and no other force supplies net power. These descriptions are compatible, but answer different questions: an energy calculation may find a maximum speed without first solving for position at every time.

The harmonic oscillator below makes this unity visible. A restoring force produces a repeating position trace; the corresponding phase plane has a closed ellipse; kinetic energy and potential energy exchange while their sum stays fixed. Lagrangian mechanics and Hamiltonian mechanics provide further ways to organize the same dynamics, especially when coordinates or constraints make individual forces awkward to track.

The scope of classical mechanics also matters. Newtonian mechanics approximates motion at speeds small compared with the speed of light; special relativity changes that description when this approximation fails. Quantum mechanics is needed for phenomena for which a classical assignment of simultaneous position and momentum is inadequate. These limitations do not prevent classical mechanics from being a useful predictive model within its regime. ^[ref]