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With specific values for C a and K a this quadratic equation can be solved for x. Assuming [4] that pH = −log 10 [H +] the pH can be calculated as pH = −log 10 x. If the degree of dissociation is quite small, C a ≫ x and the expression simplifies to = and pH = 1 / 2 (pK a − log C a).
Following an initial solution, iteration of the procedure may generate additional terms of an asymptotic expansion providing a more accurate solution. [1] [2] An early example of the dominant balance method is the Newton polygon method. Newton developed this method to find an explicit approximation for an algebraic function.
A Markov process is called a reversible Markov process or reversible Markov chain if there exists a positive stationary distribution π that satisfies the detailed balance equations [13] =, where P ij is the Markov transition probability from state i to state j, i.e. P ij = P(X t = j | X t − 1 = i), and π i and π j are the equilibrium probabilities of being in states i and j, respectively ...
x is the reactor tube axial position, m; dx the differential thickness of fluid plug; the index i refers to the species i; F i (x) is the molar flow rate of species i at the position x, mol/s; D is the tube diameter, m; A t is the tube transverse cross sectional area, m 2; ν is the stoichiometric coefficient, dimensionless
where p* is the vapor pressure of the pure component. At first sight, Raoult's law appears to be a special case of Henry's law, where H v px = p*. This is true for pairs of closely related substances, such as benzene and toluene, which obey Raoult's law over the entire composition range: such mixtures are called ideal mixtures.
The global balance equations can then be partitioned to give a set of local balance equations (also known as partial balance equations, [2] independent balance equations [7] or individual balance equations [8]). [1] These balance equations were first considered by Peter Whittle. [8] [9] The resulting equations are somewhere between detailed
Per standard procedure in taking the nonrelativistic limit, we let c → ∞, so that the factor (+ ()) (+ ()) (()) Therefore, in the nonrelativistic limit the Tolman–Oppenheimer–Volkoff equation reduces to Newton's hydrostatic equilibrium: = () = () = () (we have made the trivial notation change h = r and have used f(Ρ,ρ) = 0 to express ...
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity , but it can be generalized to apply to any extensive quantity .