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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.}
Two elements , are conjugate if there exists an element such that =, in which case is called a conjugate of and is called a conjugate of . In the case of the general linear group GL ( n ) {\displaystyle \operatorname {GL} (n)} of invertible matrices , the conjugacy relation is called matrix similarity .
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
The Hermitian conjugate of the Hermitian conjugate of anything (linear operators, bras, kets, numbers) is itself—i.e., (†) † =. Given any combination of complex numbers, bras, kets, inner products, outer products, and/or linear operators, written in bra–ket notation, its Hermitian conjugate can be computed by reversing the order of the ...
The pH after the equivalence point depends on the concentration of the conjugate base of the weak acid and the strong base of the titrant. However, the base of the titrant is stronger than the conjugate base of the acid. Therefore, the pH in this region is controlled by the strong base. As such the pH can be found using the following: [1]
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 ≡ ...
In fact, Galois showed more than this. He also proved that if ζ is a reduced quadratic surd and η is its conjugate, then the continued fractions for ζ and for (−1/η) are both purely periodic, and the repeating block in one of those continued fractions is the mirror image of the repeating block in the other. In symbols we have