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One use of conjugate acids and bases lies in buffering systems, which include a buffer solution. In a buffer, a weak acid and its conjugate base (in the form of a salt), or a weak base and its conjugate acid, are used in order to limit the pH change during a titration process. Buffers have both organic and non-organic chemical applications.
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}
Conjugate gradient, assuming exact arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2). In mathematics , the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations , namely those whose matrix is positive-semidefinite .
In solution chemistry, it is common to use H + as an abbreviation for the solvated hydrogen ion, regardless of the solvent. In aqueous solution H + denotes a solvated hydronium ion rather than a proton. [9] [10] The designation of an acid or base as "conjugate" depends on the context. The conjugate acid BH + of a base B dissociates according to
E1cB rev is when the first step is reversible but the formation of product is slower than reforming the starting material, this again results from a slow second step (k −1 [conjugate acid] ≫ k 2). E1cB irr is when the first step is slow, but once the anion is formed the product quickly follows (k 2 ≫ k −1 [conjugate acid]).
The conjugate transpose of a matrix with real entries reduces to the transpose of , as the conjugate of a real number is the number itself. The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2 × 2 {\displaystyle 2\times 2} real matrices, obeying matrix addition and multiplication: a + i b ≡ ...
Since the rate of a reaction is only determined by its slowest step, the rate at which the leaving group "leaves" determines the speed of the reaction. This means that the better the leaving group, the faster the reaction rate. A general rule for what makes a good leaving group is the weaker the conjugate base, the better the leaving group.
In chemistry, protonation (or hydronation) is the adding of a proton (or hydron, or hydrogen cation), usually denoted by H +, to an atom, molecule, or ion, forming a conjugate acid. [1] (The complementary process, when a proton is removed from a Brønsted–Lowry acid, is deprotonation.) Some examples include The protonation of water by ...