Mohr’s Circle for 3D Stress

Electromotive force (EMF) and electric potential difference (voltage) are distinct concepts, though both are measured in volts. EMF is the work per unit charge done by a non-conservative force (e.g., chemical reaction, changing magnetic field) to move charge within a source. Potential difference is the work per unit charge done by the conservative electrostatic field between two points.

Sublimation, the direct phase transition from solid to gas, occurs at temperatures and pressures below a substance's triple point. The triple point is the unique thermodynamic state where the solid, liquid, and gas phases can coexist in equilibrium. On a phase diagram, the sublimation curve separates the solid and gas phases, originating at the origin and ending at the triple point.

The Mach number (M) is a dimensionless quantity representing the ratio of flow velocity past a boundary to the local speed of sound: \(M = v/a\), where v is the flow velocity and a is the speed of sound. It is the primary indicator of compressibility effects. As Mach number approaches and exceeds 1, the density of the air changes significantly, altering aerodynamic forces.

The Nernst equation relates the reduction potential of a half-cell (or the total voltage of an electrochemical cell) to the standard electrode potential, temperature, and the activities (often approximated by concentrations) of the chemical species undergoing redox. The equation is \(E = E^{\circ} - \frac{RT}{nF} \ln Q\), where Q is the reaction quotient.

The critical temperature is the temperature above which a distinct liquid phase cannot be formed, regardless of the applied pressure. Each gas has a unique critical temperature. To liquefy a gas, it must first be cooled below this temperature. This concept, established by Thomas Andrews, is fundamental to understanding the conditions required for any liquefaction process.

Rayleigh scattering is the elastic scattering of light by particles much smaller than the light's wavelength. This phenomenon is responsible for the blue color of the daytime sky. Shorter, blue wavelengths of sunlight are scattered more effectively by the nitrogen and oxygen molecules in the atmosphere than longer, red wavelengths, causing the sky to appear blue from an observer's perspective.

This foundational formula connects the macroscopic thermodynamic quantity of entropy (S) with the number of possible microscopic arrangements, or microstates (W), corresponding to the system's macroscopic state. The equation, \(S = k_B \ln W\), reveals that entropy is a measure of statistical disorder or randomness. The constant \(k_B\) is the Boltzmann constant, linking energy at the particle level with temperature.

Mohr's circle is a two-dimensional graphical representation of the Cauchy stress tensor. It visualizes the transformation of normal stress (\(\sigma_n\)) and shear stress (\(\tau_n\)) on an arbitrarily oriented plane at a point. The abscissa of each point on the circle is the normal stress, and the ordinate is the shear stress, allowing for easy determination of principal stresses.

The Reynolds number (\(\text{Re}\)) is a dimensionless quantity in fluid mechanics used to predict flow patterns by representing the ratio of inertial forces to viscous forces. Low Reynolds numbers characterize smooth, orderly laminar flow, while high Reynolds numbers indicate chaotic, eddy-filled turbulent flow. It is crucial for determining the dynamic behavior of a fluid and for scaling experiments.

Shear-thickening, or dilatancy, is a non-Newtonian behavior where viscosity increases with the rate of shear stress. A classic example is a suspension of cornstarch in water (oobleck), which feels liquid when stirred slowly but becomes almost solid when struck or stirred rapidly. This phenomenon is caused by the jamming of suspended particles under high shear.

In electrochemical cells like batteries and fuel cells, EMF is generated by chemical reactions. The separation of charge is driven by oxidation and reduction reactions occurring at two different electrodes. The maximum EMF of a cell is related to the change in Gibbs free energy (\(\Delta G\)) of the reaction by \(\mathcal{E} = -\frac{\Delta G}{nF}\), where \(n\) is moles of electrons and \(F\) is the Faraday constant.