What is Critical point: Definition and 56 Discussions

In thermodynamics, a critical point (or critical state) is the end point of a phase equilibrium curve. The most prominent example is the liquid–vapor critical point, the end point of the pressure–temperature curve that designates conditions under which a liquid and its vapor can coexist. At higher temperatures, the gas cannot be liquefied by pressure alone. At the critical point, defined by a critical temperature Tc and a critical pressure pc, phase boundaries vanish. Other examples include the liquid–liquid critical points in mixtures.

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  1. yecko

    Vapor-compression refrigeration system - boiling & critical point

    Is it the lower the boiling point the better? As room pressure and temperature already can change the phase without working at vacuum? And the higher critical point means it is hard to reach super critical phase? But there is transcritical CO2 cycle, which the critical point is so low, and...
  2. M

    B Critical point of a piecewise function

    If the function is not differentiable at point. Can we consider this point is critical point to the function? f(x) = (x-3)^2 when x>0 = (x+3)^2 when x<0 he asked for critical points in the closed interval -2, 2
  3. karush

    MHB 1.8.1 AP Calculus find k for critical point

    $\displaystyle g'=2xe^{kx}+e^{kx}kx^2$ we are given $ x=\dfrac{2}{3}$ then $\displaystyle g'=\dfrac{4}{3}e^\left(\dfrac{2k}{3}\right)+e^\left({\dfrac{2k}{3}}\right)\dfrac{4k}{9}$ ok something is ? aren't dx supposed to set this to 0 to find the critical point did a desmos look like k=-3 but ...
  4. jfizzix

    A Does the Material Cu2OSeO3 Have a Solid-Liquid Critical Point?

    In many phase diagrams of a single substance, there is a triple point, where the solid, liquid and gas phases coexist in equilibrium, and there is a liquid-gas critical point beyond which, the transition between liquid and gas becomes continuous, and the substance is known as a super-critical...
  5. maistral

    Cubic equation of state near critical point (~0.95 < Tr < 1)

    Hi. I have read somewhere that cubic equations of state seem to have convergence issues when vapor pressure calculations are done near the critical point. Sadly, I have forgotten where I have have read it :( I would like to ask some knowledgeable people regarding this, or can people point me I...
  6. R

    MHB Finding Critical Point: x for y=3e^(-2x)−5e^(-4x)

    y = 3e^(−2x) −5e^(−4x) y'= −6e^(−2x)+20e^(−4x) How do I find the critical point at x? The answer is (1/2)ln(10/3) but I don't know how to get that answer Thank you
  7. toforfiltum

    Newton's method to approximate critical point

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  8. lep11

    Prove that the critical point of f satisfies the equation

    Homework Statement a.) Let f(x)=(sin x)/x, x≠0. Define f(0) such that f is continuous at x=0. b.) Prove that if x0 is critical point of function f (f(0) defined as in part a), then |f(x0)|=1/(1+x02)-½ (Hint: use the basic properties of sine and cosine with given information.)The Attempt at a...
  9. N

    Finding critical points for phase planes

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  10. N

    Find the approximate linear ODE system

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  11. T

    T-dependence of heat of vaporization near critical point

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  12. S

    Gases compression at critical point. LNG (Methane) storage.

    There are claims or at least assumptions that: "upon approaching "critical point" pure gases become super compressible. You could compress them many times without pressure increase and store like this if you maintain exact temperature and pressure needed. It opens possibility to superdense gas...
  13. A

    Quantum fluctuations at critical point

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  14. Cosmophile

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  15. C

    Is this question missing a third time interval?

    Homework Statement Given critical points (3,-4) and (6,0); interval of increase (3, infinity); interval of decrease (-infinity, 3), find the local maxima/minima and sketch the graph. Homework Equations No relevant equations are given, I believe it's a simple sketch the graph. The Attempt at...
  16. V

    Solving Differential Eqs with Critical Point: Initial Conditions & Solutions

    Hello, I have two different discrepancies to this system: a) How and when is possible to have more solution of differential eq. or their system for same initial problem? For example this is happening in following system. It is written about this system: "Different value of constant \dot{M}=4\pi...
  17. C

    Gibbs phase rule at critical point

    What is the degrees of freedom at the critical point of a 1 component system such as water. in my test I put zero because the critical point is a POINT that is fixed per substance. is there any books that can back me up? I heard that my teacher said there was 1 degree of freedom which I am...
  18. N

    Thermodynamic PV-diagram: Why does a critical point =inflection point?

    http://en.wikipedia.org/wiki/Critical_point_%28thermodynamics%29 Look here please. Why is the first and 2nd order derivatives of pressure with regards to volume equal zero at the critical point for an isotherm phase curve? Why/how is it possible to expand a fluid at its critical point...
  19. C

    Find the D value of critical point and the type of critical point

    Homework Statement The function f(x,y) = [e^(-y^2)]cos(4x) has a critical point (0,0) Homework Equations Find the D value at the critical point. What type of critical point is it? (max, min, saddle or none) The Attempt at a Solution I know that to find the D value I must compute...
  20. fluidistic

    Is the triple point a kind of critical point? (thermodynamics)

    In a problem in Callen's book I was asked to say what was the latent heat of fusion at the triple point for ammonia. I answered "without performing any algebra, 0J". Because I remember a video I saw on youtube about the triple point () and now I read on wikipedia which seem to confirm that...
  21. R

    Finding absolute extrema with only 1 critical point

    Homework Statement Edit: Not absolute, just extrema I've already found the critical point to be (-1/2, -1/4, 1/2) with a value of -1/2. My only problem is finding whether this is a max or min. What technique do I use to find out? I don't believe I can use the 2nd derivative test because all...
  22. O

    What is the state of matter beyond the critical point?

    Starting from a gas state and reaching the liquid state by getting around the critical point in a p-T phase diagram, what happens (qualitatively) to the molecules (or atoms)? Does a fraction of them start to form bonds? This concept of "indistinguishable" liquid and gas is new to me.
  23. I

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  24. B

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  25. M

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  26. P

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    I admit I am a bit out of practice when it comes to DiffEq. I think I am either forgetting a simple step or getting my methods mixed up. Homework Statement The problem concerns a pendulum defined by d2θ/dt2 + (c/mL)(dθ/dt) + (g/L)sinθ = 0 where m=1, L=1, c=0.5, and of course g=9.8 After...
  27. L

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  28. L

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  29. GreenGoblin

    MHB Critical point of function of two variables

    Hello, Please help me solve this problem and help me find if I made a mistake? If you will, Thank you $f(x,y) = 2cos(2x) + sin(x^{2}-y^{2})$ Find all the first and second order derivatives, hence show the origin is a critical point and find which type of critical point First time attempting...
  30. D

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  31. A

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  32. N

    Analyzing a Continuous Decreasing Function: Critical Point at (4,2)

    Homework Statement Multiple Choice If f is a continuous, decreasing function on [0, 10] with a critical point at (4, 2), which of the following statements must be false? E (A) f (10) is an absolute minimum of f on [0, 10]. (B) f (4) is neither a relative maximum nor a relative minimum...
  33. T

    Finding the critical point and its nature. With solid attempt

    Finding the critical point and its nature. With solid attempt! Homework Statement Find all critical points of the function f(x, y) = xy2 - 2xy - 2x2 - 3x +7 and determine their nature. Homework Equations none The Attempt at a Solution I know that to find the critical points you must...
  34. S

    Thermodynamics: Critical Point

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  35. S

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    Homework Statement The critical point is defined as : At the critical point there is no change of state when pressure is increased or if heat is added. At the critical point the water and steam can't be distinguished, and there is no point referring to water or steam. Homework...
  36. N

    Why specific heat at near critical point equals differentiate twice Gibbs energy?

    Please teach me this: Why specific heat near critical point equals differentiate twice Gibbs free energy with respect to temperature, but not differentiate once with respect to temperature as usually doing. Thank you very much in advance.
  37. P

    Phase transitions between solid and liquid. Critical point CP

    Look at a simple single - component system. And PT diagram for this system. Suppose that supstance we looking at has three phases solid, liquid and gas. In that case we have critical point between liquid and gas phase, but not between solid and liquid phase. Why? Why solid - liquid coexistence...
  38. G

    Quality needed to pass through critical point (thermodynamics)

    Homework Statement A rigid tank contains 1 kg of saturated liquid and saturated vapor mixture at 400 kPa. What quality (x1) do you need if you want the P-v diagram to pass through the critical point? Use steam tables to answer this question. Homework Equations x = quality =...
  39. L

    Type and stability of critical point at (0,0)

    Consider the following system (where B is a real number, B is not equal to 0), x1' = -2x1 + (B+2)x2 x2' = Bx2 Depending on the value of B, the critical point at (0,0) can be of different type and/or stability. Describe the possible type/stability of the critical point for the different...
  40. L

    Index of a critical point in a vecor field

    Homework Statement This is from Henle's Combinatorial Introduction to Topology, section 9 question 2d). Here's the phase portrait: http://img69.imageshack.us/img69/70/phasepor.jpg Find the index of the point in the center of that picture.The Attempt at a Solution I used the fact that index =...
  41. M

    Non-Linear First Order ODE: Critical Point Linearization

    Homework Statement dx/dt = x - y + (x^2) - xy dy/dt = -y + (x^2) - Determine the critical points for the equation, - Determine the linearized system for each critical point and discuss whether it can be used to approximate the behaviour of the non-linear system. What is the type and...
  42. G

    Simple critical point question - how to specify?

    Homework Statement Classify the critical points subject to the constraints: 0<=x, y<=Pi Homework Equations fx=sin(y)sin(2x+y) fy=sin(x)sin(2y+x) The Attempt at a Solution Clearly one set of critical points will be (n*Pi,m*Pi) where m <=1 and n and m are all positive and...
  43. K

    Determining the type of critical point when the second derivative test fails

    Homework Statement For example with f(x,y) = x2y + xy2 Homework Equations The Attempt at a Solution Well I know there is a critical point at (0,0). So I calculated the second derivatives but they are all 0 here so that doesn't help. I also tried using the Taylor expansion to...
  44. G

    Critical point of two variable function

    Homework Statement Each of the functions f have a critical point at (0,0), however at this point the second derivatives are all zero. Determine te type of critical point at (0,0) in this case 1) f(x,y)=x2y+xy2 2)f(x,y) = x4+2*x3y+x2y2+y4 The Attempt at a Solution For...
  45. T

    Check answers to critical point, extrema questions

    Homework Statement If my answers to the questions in the attachments could be checked that would be great. All work is done, and the questions in the pictures are fairly quick and simple. I want to be sure I got them correct. Thank you! Homework Equations The Attempt at a Solution
  46. JJBladester

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  47. JJBladester

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  48. D

    Stability of Degenerate Critical Points: A Phase Plane Analysis

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  49. G

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  50. T

    Find critical point of interval problem

    Can you just say that x(4-x) <4 for x \in (0,2) ? You don't need to prove this? Or to prove this, just find critical point which is x = 2 ?
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