The Nernst Equation

For a general three-dimensional state of stress, the analysis is represented by three Mohr's circles. These circles are drawn in the \(\sigma_n - \tau_n\) plane using the three principal stresses (\(\sigma_1, \sigma_2, \sigma_3\)) as diameters. The largest circle, defined by \(\sigma_1\) and \(\sigma_3\), encloses the other two and determines the absolute maximum shear stress, \(\tau_{abs max} = (\sigma_1 - \sigma_3)/2\).

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.

Motional EMF is generated when a conductor moves through a magnetic field. The magnetic component of the Lorentz force, \(\mathbf{F} = q(\mathbf{v} \times \mathbf{B})\), acts on the charge carriers within the conductor, causing them to move and create a charge separation. This separation establishes an electric field and a potential difference. The resulting EMF is given by the line integral \(\mathcal{E} = \oint (\mathbf{v} \times \mathbf{B}) \cdot d\mathbf{l}\).

The Hampson-Linde cycle liquefies gases like air using the Joule-Thomson effect combined with regenerative cooling. High-pressure gas is cooled in a counter-flow heat exchanger by the cold, low-pressure gas returning from the expansion valve. This progressively lowers the temperature at the valve until it drops below the critical point and liquefaction begins, forming the basis for the modern gas liquefaction industry.

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.

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.

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 Zeeman effect is the splitting of an atomic spectral line into multiple components when an external static magnetic field is applied. This splitting occurs because the magnetic field interacts with the magnetic dipole moment associated with the atom's orbital and spin angular momentum, shifting the energy levels of its electrons, a phenomenon crucial for probing atomic structure.

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.