What is Conjugate: Definition and 259 Discussions

A conjugate acid, within the Brønsted–Lowry acid–base theory, is a chemical compound formed when an acid donates a proton (H+) to a base—in other words, it is a base with a hydrogen ion added to it, as in the reverse reaction it loses a hydrogen ion. On the other hand, a conjugate base is what is left over after an acid has donated a proton during a chemical reaction. Hence, a conjugate base is a species formed by the removal of a proton from an acid, as in the reverse reaction it is able to gain a hydrogen ion. Because some acids are capable of releasing multiple protons, the conjugate base of an acid may itself be acidic.
In summary, this can be represented as the following chemical reaction:

Acid + Base ⇌ Conjugate Base + Conjugate Acid

Johannes Nicolaus Brønsted and Martin Lowry introduced the Brønsted–Lowry theory,
which proposed that any compound that can transfer a proton to any other compound is an acid, and the compound that accepts the proton is a base. A proton is a nuclear particle with a unit positive electrical charge; it is represented by the symbol H+ because it constitutes the nucleus of a hydrogen atom, that is, a hydrogen cation.
A cation can be a conjugate acid, and an anion can be a conjugate base, depending on which substance is involved and which acid–base theory is the viewpoint. The simplest anion which can be a conjugate base is the solvated electron whose conjugate acid is the atomic hydrogen.

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  1. D

    Integrating the Complex conjugate of z with respect to z

    Im doing a bit of contour integration, and a question came up with a term in it am unsure of how to do: in its simplest form it would be \int\bar{z}dz where z is a complex number and \bar{z} is it's conjugate. Hmm i can't get the formatting to work out properly.. :S
  2. J

    Definition of Complex Conjugate

    Hey all, I was just curious: Why is the conjugate of a complex number (a + bi) defined as (a - bi)? If we instead change the sign of the real part (-a + bi), we still get a real number when we multiply the two. Is there a particular significance to the current definition...
  3. R

    Can conjugate images be used in place of dark field filters?

    I was working on some dark field filters and I was wondering if you could substitute the black spot in the centre of a dark field filter for a completely black filter with a light passing hole in the middle. Sort of an inverted dark field filter. I seem to remember reading something like this...
  4. D

    How to Multiply by a Conjugate with Multiple Terms?

    multiplying by a conjugate?? Homework Statement I am dealing with a limit, however, I am not sure how to multiply by a conjugate when there are two variables and three terms on top. For example lim [3h + sqrt( 2x +2h) - sqrt( 2x)] / h h->0 i don't need help solving it, i just...
  5. L

    Proving $\hat{O}\hat{O}^\dagger$ is Hermitian

    pretty simple question. have to prove \hat{O} \hat{O}\dagger is a Hermitian operator. i found that \left( \int \int \int \psi^{\star}(\vec{r}) \hat{O} \hat{O}^{\dagger} \phi(\vec{r}) d \tau \right)^{\star} = \int \int \int \phi^{\star}(\vec{r}) \hat{O}^{\dagger} \hat{O} \phi(\vec{r}) d...
  6. G

    Conjugate e^(iz): Solving for z = nπ

    Homework Statement I want to show that \exp \left( i\bar{z} \right) = \overline{\exp \left( iz \right)} if and only if z = n\pi for any integer n. Homework Equations The Attempt at a Solution Utilizing Euler's formula, I got \cos \bar{z} = \cos z and...
  7. W

    Conjugate Limit: $\frac{\sqrt{x}-1}{x-1}$

    $\lim_{x\to 1}\frac{\sqrt{x}-1}{x-1}$ Homework Statement Calculate the limit of \lim_{x\to 1}\frac{\sqrt{x}-1}{x-1}. Homework Equations As above. The Attempt at a Solution Have tried to multiplicate with the conjugate.
  8. M

    PH of buffer with addition of conjugate base

    Homework Statement Consider 1L of a solution that is 0.100 M in acetic acid and 0.100 M in sodium acetate. What will the pH be, when 0.0900 mol of sodium acetate are added? Ignore changes in volume. acetic acid pka = 1.75 E-5 Homework Equations HA --> H+ + A- pH=pka +log A-/HA...
  9. U

    If a complex converges, then it's conjugate converges.

    Homework Statement Prove that z_n -> z_0 if and only if ~(z_n) -> ~(z_0) as n goes to infiinity. ~(z_n) is the conjugate of z_n. Homework Equations The Attempt at a Solution |~(z_n) - ~(z_0) | = | ~(z_n) + ~(-z_0)| <= |~(z_n)| + |~(-z_0) | = |z_n| + |z_0| <= and I...
  10. D

    2nd order ode w/complex conjugate roots

    x''+x'+2x=0 x(0)=2 x'(0)=0 I've taken the characteristic equation and reduced the roots to 1/2 +- Sqrt(7/4)i of the form a +- bi (i = sqrt(-1) Then i put the homogeneous solution into the form of e^{}at*(B1cos(bt)+B2sin(bt)) for B1 i used the first i.c. and found that B1=2...
  11. F

    Similarity (conjugate) math problem

    my question is what is meant by saying that A and B are similar where A and B are nxn matrices with entries in field F, also show that is A~B then A^2~b^2 first bit i have the definition for A~B is they are nxn matrices in the same field and there exists a non-singular square matrix P such...
  12. S

    Conjugate Variables: Uncertainty Relation Explained

    Why there is an uncertainty relation between conjugate variables? what exactly are conjugate variables?
  13. M

    Conjugate of every variable is denoted

    Hi I have received PM so I would like to post this question for scientific debate here conjugate of every variable is denoted by _. 1.define f_:R3->R3 by f_(x_)=y_, where y1=x1+(x2)2+(x3-1)2, y2=(x1)2+x2+((x3)2-3.x3), y3=(x1)3+(x2)2+x3.prove that f_ is locally invertible about x_=(0,0,1)...
  14. Amith2006

    Complex conjugate of wave function

    A wave function(psi) is a mathematical quantity which gives complete information about the state of a system at a particular instant of time. But what information does the complex conjugate of a wave function(psi*) give? Does it represent the same state as psi? Or does it just have a...
  15. V

    Complex conjugate & lp space

    I have two questions Homework Statement Show that the function defined by complex conjugation is not differentiable. Homework Equations If z = x + iy then the complex conjugate of z is x - iy The Attempt at a Solution f'(z) = lim z -> z1 \frac{f(z) - f(z1)}{z - z1} So I...
  16. Y

    Hermitian Conjugate: 5-6i

    What is the Hermitian conjugate of a complex #, say, 5+6i??
  17. E

    Help Solve a Math Problem: Find the Conjugate of 2-5i

    Hi today I will be constructing a ratio thing for math. Our teacher asked us to survey as many people as we can. I am going to ask a simple math question and if you can answer it with 100% confidence. This is just a little survey I am not trying to trick you in doing my homework, I already know...
  18. M

    Show that complex conjugate is also a root of polynomial with real coefficients

    Homework Statement Suppose that f(x) is a polynomial of degree n with real coefficients; that is, f(x)=a_n x^n+ a_(n-1) x^(n-1)+ …+a_1 x+ a_0, a_n,… ,a_0∈ R(real) Suppose that c ∈ C(complex) is a root of f(x). Prove that c conjugate is also a root of f(x) Homework Equations...
  19. H

    Is Complex Conjugate Integration Valid on Contour |z|=1?

    i was thinking this over, and i'd like to know if the following statement is valid: \oint\overline{z}dz = \oint1/zdz over the contour \left|z\right|=1 any thoughts?
  20. D

    Complex conjugate variables as independent variables in polynomial equations

    Hi, Is there any trick to treat complex conjugate variables in polynomial equations as independent variables by adding some other constraint equation ? Say, we have polynomial equation $f(x,x^{*},y,...) = 0$. where x^{*} is the complex conjugate of variable $x$. I might think of taking $x = r...
  21. camilus

    How can I solve a complex conjugate problem for my friend in college algebra?

    Homework Statement sun, I don't know why I am stuggling with a simple freakin problem. Its not even for me its for my friend who's in college algebra, but for some reaon I can't get the correct answer. {9 - 11i \over 6i} The Attempt at a Solution I multiplied by the conjugate twice and...
  22. C

    Complex conjugate an independent variable?

    It's very commong to use z and z* as two independent variables, differentiating with respect to one while keeping the other constant. Can you please give me some intuitive insight into this method, and why it works so well? Because every time I see this my first thought is that z and z* are NOT...
  23. E

    Complex Conjugate of Fourier Transform

    Hello All, As I understand it, the wavefunction Psi(x) can be written as a sum of all the particle's momentum basis states (which is the Fourier transform of Psi(x)). I was woundering if the wavefunction's complex conjugate Psi*(x) can be written out in terms of momentum basis states, similar...
  24. B

    Variational methods conjugate points

    (a will be alpha and b will be beta) Let y=y(x,a,b) be a general solution of Euler's equation, depending on two parameters a and b. Prove that if the ratio (subdifferential y/subdifferential a)/(subdifferential y/subdifferential b) is the same at the points, the points are conjugate. I...
  25. B

    Variational methods - conjugate of function

    Let F:H->R bar be a function and F*:H->R bar its conjugate. Fix aEH and show that the conjugate of the new function G(u)=F(u-a) is G*(u*)=F*(u*)+<a,u> Verify the case where F:R^2->R, F(x)=1/2(x)^2 and a = (2,-1) I don't really know how to show this. please help
  26. A

    Conjugate and Inverse Quaternions

    Does an inverse exist for every quaternion under multiplication??
  27. B

    Conjugate points of extermals of functional

    Show that the extermals of any functional of the form integ (a->b) F(x,y') dx have no conjugate points. Not sure how to start this question, any help would be appreciated
  28. Pythagorean

    Is the Complex Conjugate of Sin Equal to Sin of the Conjugate?

    Homework Statement does (\sin{z})^* = \sin{z^*}? (where z is a complex number) Homework Equations \sin{z} = \frac{1}{2} (e^{iz} - e^{-iz}) The Attempt at a Solution (\sin{z})^* = \frac{1}{2} (e^{iz} - e^{-iz})^* = \frac{1}{2} (e^{-iz^*} - e^{iz^*}) = -\frac{1}{2} (e^{iz^*} - e^{-iz^*}) =...
  29. M

    What is the missing piece in proving A=0 when A(A*)=0?

    Let A be a nxn matrix. Prove that if (A*)A=0 then A=0. What if A(A*) = 0? A* is the conjugate transpose of A. When I write out the expansion formula, I cannot conclude that every entry of A is zero. What am I missing?
  30. R

    Hermitian Conjugate of Matrix Explained

    Simple question, and pretty sure I already know the answer - I just wanted confirmation, Considering the Hermitian Conjugate of a matrix, I understand that A^{+} = A where A^{+} = (A^{T})^{*} Explicitly, (A_{nm})^{*} = A_{mn} Would this mean that for a matrix of A, where A is a...
  31. D

    Hermitian Conjugate of an Operator

    Homework Statement a = x + \frac{d}{dx} Construct the Hermitian conjugate of a. Is a Hermitian? 2. The attempt at a solution <\phi|(x+\frac{d}{dx})\Psi> \int\phi^{*}(x\Psi)dx + <-\frac{d}{dx}\phi|\Psi> I figured out the second term already but need help with first term... am...
  32. I

    Are the Helicoid and Catenoid Conjugate Minimal Surfaces?

    Homework Statement Show that the helicoid and the catenoid are conjugate minimal surfaces Homework Equations the helicoid is given by the parameterization X(u,v) = (asinh(v)*cosu, asinh(v)*sinu, au) = (x1, x2, x3) the catenoid is given by the parameterization Y(u,v) = (acoshv*cosu, a...
  33. K

    Proving the Complex Conjugate of a Matrix Determinant

    Homework Statement show that det(A)=(detA)*= det(A$) where * denotes complex conjugate and $ means transpose Homework Equations The Attempt at a Solution Please help me to start the problem.I am not getting a way.
  34. A

    Hermitian Conjugate of Spinors: Is it Equal?

    I just have a simple question about hermition conjugates of spinors. Is the hermitian conjugate of: \epsilon \sigma^\mu \psi^\dagger equal to: -\psi \sigma^\mu \epsilon^\dagger where both psi and epsilon are 2-component spinors of grassmann numbers?
  35. G

    Does one zero equal its algebraic conjugate?

    I'm studying abstract algebra and never got the hang of ideas like these. Currently I am trying to grasp: Show that the splitting field of a quadratic polynomial P(x) = Ax^x + Bx + C, with zeros \alpha and \alpha^{'} is V_\alpha = Q*1 + Q * \alpha. Q is of course the set of rationals. I...
  36. C

    Proving Isomorphism of Complex Number Conjugate: General Algebraic Systems

    I am working through this algebra book and some of the problems. The chapter this comes out of is General Algebraic Systems and the section is Isomorphisms. I am new to proofs and maths higher than calculus I so I am not sure if I am following the text or not. There aren't any solutions and this...
  37. T

    Need help with Proof and conjugate roots theorem

    Need Assistance w/ existence of conjugate roots in polynomial Prove: given that f(x) = Anx^n + An-1X^n-1+ ...A1X + A0 , and An does not = 0 => if f(x) has a root of the form (A+Bi), then it must have a root of the form (A-iB). (complex roots) So far I've come up with the conjugate roots...
  38. K

    Conjugate of either sin or cos

    okay, so this particular equation involves me writing conjugate of either sin or cos, but hows that possible considering they both are real in the given problem? maybe i should convert sin and cos into their exponential form first? but then wt would be the conjugate of this...
  39. E

    Please help -Dirac delta potential-, Hermitian Conjugate

    Please help! -Dirac delta potential-, Hermitian Conjugate Im trying to solve problem 2.26 from Griffiths (1st. ed, Intro to Q.M.). Its about the allowed energy to double dirac potential. I came up with a final equation that is trancedental. (After I separate the even and odd solution of psi.)...
  40. G

    Is the conjugate of an integral the integral of the conjugate in L2 space?

    In the complex conjugate of an integral equal to the integral of the complex conjugate? If so, is there an easy way to show this? Thanks.
  41. R

    Is My Proof Correct? Understanding the Complex Modulus Inequality

    I would like to prove \mid z + \overline{z} \mid \leq 2 \mid z \mid The first way I could think of: \begin{multline} RHS^2 - LHS^2\\ =4\mid z \mid^2 - \mid z + \overline{z} \mid ^ 2\\ =4z\overline{z} - (z + \overline{z})(\overline{z}+z)\\ =4z\overline{z} -...
  42. S

    .Which Acid Has the Strongest Conjugate Base?

    Which acid of the following set has the strongest conjugate base? a. CH4 b. NH3 c. H2O d. HF Is this correct? Any help is appreciated
  43. S

    Acid Strength and Conjugate Bases

    1. Which Bronsted-Lowry acid has the strongest conjugate base? a. HBr b. HClO4 c. HF d. HI 2. What is the strongest acid among the following? a. HF b. HCl c. HBr d. HI Correct? Thanks.
  44. N

    How does one conjugate psi(x,0)

    I am perplexed. My professor told me that I can conjugate: \psi \left( x,0 \right) However, I am confused. There is no time dependence, there is no i. It is like saying z = a + 0i z* = a - 0i It is rather useless to even say that and it seems also useless to conjugate something with...
  45. T

    Conjugate Elements of a Symmetric Group

    Is the the following definition correct? Two elements a and b of a group G are said to be CONJUGATE if there exists g in G such that a=gbg^{-1}. For instance, show that all elements in the symmetric group S5 of order 6 conjugate.
  46. E

    Hermitian conjugate of differential operator

    Hi everyone! How can I find the Hermitian conjugate of the differential operator D, with D psi = 1/i dpart/dpart(x) psi? I know you can do this with partial integration starting from <phi|D|psi>* = <phi|D+|psi> but how exactly does it work? I'm sorry for using such an ugly...
  47. D

    What Determines the Strength of a Conjugate Acid-Base Pair?

    Uggh finally did all the chem problems and studied for the upcoming final but there are 2 questions left unanswered. MAybe someone can help me out with this: If given a lists of acids, how would one know which 1 is the weakest conjugate base? For example, HF, HNO_2, H_2CO_3, H_3BO_3, HCl...
  48. Q

    How to take a complex conjugate of a 5*5 matrices?

    I'm doing a take home final and wanted reassurance that I'm doing the problem right. the question involves taking <Sz>of |psi>. I know it's <psi|Sz|psi>. I've never done it for a spin 2 particle which is a 5*5 matrix.Do i just take the complex conjugate of the values without switching their...
  49. B

    How to Identify Lewis Acids and Bases and Their Conjugate Base Formulas?

    Whats the formula for this conjugate bases? how do I get them? a) HSO3tothe(-) b) NH3 c) HCL and how is the lewis acid and lewis base identified? thanks
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