Lennard-Jones Potential

Complex redox reactions can be balanced using the half-reaction method. The overall reaction is split into two separate half-reactions: one for oxidation and one for reduction. Each half-reaction is balanced for atoms and charge independently, often by adding H+, OH-, and H2O in aqueous solutions. Finally, the electron counts are equalized and the half-reactions are combined.

A Bingham plastic is a viscoplastic material that behaves as a rigid body at low stresses but flows as a viscous fluid at high stress. It is characterized by a yield stress, \(\tau_0\), which is the minimum shear stress required to initiate flow. Below this threshold, it does not deform. Common examples include toothpaste, mayonnaise, and clay slurries.

This is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is a linear partial differential equation for the wavefunction, \(\Psi(x, t)\). The time-dependent version is \(i\hbar\frac{\partial}{\partial t}\Psi = \hat{H}\Psi\), where \(\hat{H}\) is the Hamiltonian operator, representing the total energy of the system.

Thixotropy and rheopexy describe time-dependent changes in viscosity. A thixotropic fluid's viscosity decreases over time under constant shear stress (e.g., yogurt becomes runnier when stirred). A rheopectic (or anti-thixotropic) fluid's viscosity increases over time under constant shear (e.g., some lubricants). These effects are reversible; the fluid returns to its original state after a period of rest.

A quantum mechanical phenomenon where a wavefunction can propagate through a potential energy barrier. Classically, a particle lacking sufficient energy to surmount a barrier would be reflected. However, due to the wave-like nature of particles, there is a non-zero probability that the particle can appear on the other side of the barrier, effectively 'tunneling' through it.

General relativity predicts that the path of light is bent by gravity. When light from a distant source passes a massive object like a galaxy or a star, its path is deflected. This phenomenon, known as gravitational lensing, can magnify, distort, or create multiple images of the background source, acting like a cosmic telescope for observing the distant universe.

Redox (reduction-oxidation) reactions involve a transfer of electrons between chemical species. One species undergoes oxidation (loses electrons), while another undergoes reduction (gains electrons). These two processes always occur simultaneously. The species that loses electrons is the reducing agent, and the one that gains electrons is the oxidizing agent. This fundamental concept underpins electrochemistry and many biological processes.

All quantum entities, such as photons and electrons, exhibit both particle and wave properties. Depending on the experimental setup, they can behave like a localized particle or a distributed wave. The de Broglie hypothesis states that any particle with momentum \(p\) has an associated wavelength \(\lambda = h/p\), where \(h\) is Planck's constant.

Shear-thinning, or pseudoplasticity, is a non-Newtonian behavior where a fluid's viscosity decreases with an increasing rate of shear stress. Many common substances, like ketchup or paint, exhibit this property. At rest, they are thick, but they flow more easily when shaken, stirred, or spread. This is often modeled by the Ostwald–de Waele power-law relationship.

Quantum statistics modifies classical statistical mechanics to account for the indistinguishability of identical particles. It splits into two types: Fermi-Dirac statistics for fermions (half-integer spin particles like electrons), which obey the Pauli exclusion principle, and Bose-Einstein statistics for bosons (integer spin particles like photons), which can occupy the same quantum state. This distinction is crucial at low temperatures and high densities.

It is impossible to simultaneously know with perfect accuracy certain pairs of complementary physical properties of a particle. The most common example is the position \(x\) and momentum \(p\). The principle states that the product of their uncertainties, \(\Delta x\) and \(\Delta p\), must be greater than or equal to a specific value: \(\Delta x \Delta p \ge \frac{\hbar}{2}\).

A non-Newtonian fluid is one whose viscosity changes under an applied shear stress. Unlike a Newtonian fluid where viscosity is constant, its flow properties are not described by a linear relationship between shear stress (\(\tau\)) and shear rate (\(\dot{\gamma}\)). This dependency can manifest as shear-thinning (viscosity decreases with stress) or shear-thickening (viscosity increases with stress).