Raoult’s Law and Henry’s Law for Dilute Solutions

In a system at thermodynamic equilibrium, the electrochemical potential, \(\bar{\mu}_i\), for any given charged species `i` must be uniform throughout all phases. If a gradient exists (\(\nabla \bar{\mu}_i \neq 0\)), ions will spontaneously move from regions of higher potential to lower potential, creating a current or flux, until this gradient is eliminated and equilibrium is re-established.

Sublimation is a laboratory technique for purifying compounds, particularly volatile solids. The impure solid is gently heated, often under reduced pressure, causing it to sublime directly into a gas. This gas then deposits as a pure solid on a cooled surface, known as a cold finger, leaving the non-volatile impurities behind in the original container.

The Kutta-Joukowski theorem quantifies the lift force generated by an airfoil. It states that the lift per unit span (\(L'\)) is directly proportional to the fluid density (\(\rho\)), the free-stream velocity (\(V\)), and the circulation (\(\Gamma\)) around the body: \(L' = \rho V \Gamma\). This links the abstract concept of circulation to the physical force of lift.

The boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant. Introduced by Ludwig Prandtl, this concept simplifies fluid dynamics problems by dividing the flow into two regions: the thin boundary layer where viscosity dominates and the outer region where inviscid flow theory can be applied.

The Reynolds-Averaged Navier-Stokes (RANS) equations are time-averaged equations of motion for turbulent fluid flow. This approach, called Reynolds decomposition, separates flow variables into a mean and a fluctuating component. The averaging process introduces an additional term, the Reynolds stress tensor, which represents the effect of turbulence and must be modeled to achieve closure, making simulations computationally tractable.

The absolute electrochemical potential of a single electrode cannot be measured. Instead, potentials are measured relative to a universally accepted reference. The Standard Hydrogen Electrode (SHE) is the primary reference, assigned a potential of exactly 0 volts under standard conditions (1 M \(H^+\) concentration, 1 bar \(H_2\) gas, 298 K). All other standard electrode potentials are reported relative to this zero point.

Thermite possesses a very high activation energy, making it stable at room temperature and difficult to ignite. Ignition requires reaching temperatures of approximately 1,300 °C (2,400 °F). This is typically achieved not with a direct flame but with an intermediate, high-temperature initiator like a burning magnesium ribbon or a specially designed pyrotechnic fuse, which provides the necessary localized energy to start the reaction.

Energy is not continuous but comes in discrete packets called quanta. The energy \(E\) of a single quantum of electromagnetic radiation (a photon) is directly proportional to its frequency \(\nu\). This relationship is defined by the Planck-Einstein relation, \(E = h\nu\), where \(h\) is the Planck constant. This concept fundamentally challenged classical physics.

A statistical ensemble is a conceptual tool consisting of a large number of virtual copies of a system, each representing a possible microstate. By averaging properties over all systems in the ensemble, one can calculate macroscopic observables. The main types are the microcanonical (isolated system with fixed N, V, E), canonical (closed system, fixed N, V, T), and grand canonical (open system, fixed µ, V, T).

The equivalence principle is a cornerstone of general relativity. It posits that the effects of a uniform gravitational field are indistinguishable from the effects of uniform acceleration. This means that an observer in a windowless, free-falling elevator experiences weightlessness, just like an observer in deep space, far from any gravitational source, forming the conceptual basis for gravity as a geometric phenomenon.