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. 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. 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. 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. 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. 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. 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. Newton's method to approximate critical point

Homework Statement Let ##F(x,y)=4sin(xy)+x^3+y^3## Use Newton's method to approximate the critical point that lies near ##(x,y)=(-1,-1)## Homework EquationsThe Attempt at a Solution I have a problem here because the derivative is not a square matrix. Hence, I can't find the inverse needed for...
8. 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. Finding critical points for phase planes

Homework Statement X'=x-y^2 Y'=x^2 -xy -2xHomework Equations Find the critical point of this system. The Attempt at a Solution I know one point is (0,0). When I tried it I got sqrt(x)= 2 and sqrt(x)= 1 but the point (1,1) doesn't fit later parts of the question. I'm also aware that since a...
10. Find the approximate linear ODE system

dx/dt = x-y^2 dy/dt= x^2 -xy -2x For each critical point, find the approximate linear OD system that is valid in a small neighborhood of it. I found the critical points which are (0,0),(4,2),(4,-2) but have no idea how to do the above question! please help!
11. T-dependence of heat of vaporization near critical point

The Wikipedia page on the heat of vaporization has a nice graph showing the heat of vaporization going to zero at the critical points of various substances. Is there a known form for the heat of vaporization as a function of temperature near the critical point? I imagine it is probably a power...
12. 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. Quantum fluctuations at critical point

According to wikipedia: "As for a classical second order transition, a quantum second order transition has a quantum critical point (QCP) where the quantum fluctuations driving the transition diverge and become scale invariant in space and time." I am confused about what this means. Why do the...
14. Critical Point when f is undefined

Homework Statement I am being asked to find the absolute extrema of the function f(x) = 3x^{2/3}- 2x, [-1,1] 2. The attempt at a solution I have taken the derivative and simplified it to: \frac {1- \sqrt [3] x}{\sqrt [3]x} = 0 Here is my dilemma: 0 lies in the interval, and I know that...
15. 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. 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. 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. 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. 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. 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. 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. 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. MHB A proof about maximum point, critical point and differentiation

Let $E\subset\mathbb{R}^n$ and $f: E\rightarrow\mathbb{R}$ be a continuous function. Prove that if $a$ is a local maximum point for $f$, then either $f$ is differentiable at $x = a$ and $Df(a) = 0$ or $f$ is not differentiable at $x = a$. Deduce that if $f$ is differentiable on $E^o$, then a...
24. Polynomials of odd degree can have no critical point.

Are these assertions true? I am referring to polynomials with real coefficients. 1. There exists of polynomial of any even degree such that it has no real roots. 2. Polynomials of odd degree have atleast one real root which implies that polynomial of even degree has atleast one critical...
25. Adiabatic gas Compression of Piston after Critical Point

Hello there, Sorry, if there's a thread about this already. Let's say if i have Piston which is fully insulated because as mentioned, it is an adiabatic compression. The pressure will increase as the volume of decrease. But what really happened when the system is compressed over the...
26. Determining stability about a critical point using eigenfunctions

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. Critical point exponents inequalities - The Rushbrooke inequality

The Rushbrooke inequality: H=0, T\rightarrow T_c^- C_H \geq \frac{T\{(\frac{\partial M}{\partial T})_H\}^2}{\chi_T} \epsilon=\frac{T-T_c}{T_c} C_H \sim (-\epsilon)^{-\alpha'} \chi_T \sim (-\epsilon)^{-\gamma'} M \sim (-\epsilon)^{\beta} (\frac{\partial M}{\partial T})_H \sim...
28. Critical point exponents inequalities - The Coopersmith inequolity

The Coopersmith inequolity: T=T_c, H\rightarrow 0^+ I'm confused by few things. What means H\rightarrow 0^+? And what difference will be if H\rightarrow 0^-? And what means T=T_c if we can't measure T_c in experiments? Then there is relation M \sim H^{\frac{1}{\delta}} That means if I...
29. 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. Finding the Critical Point of a Multivariable Function

Homework Statement find the critical point of, x^(2)+y^(2)+2x^(-1)y^(-1) Homework Equations none The Attempt at a Solution Ok so first we differentiate the function such that fx = 0 and fy=0 doing this yields, fx = 2x-2x^(-2)y^(-1) fy = 2y-2x^(-1)y^(-2) both set to...
31. Finding Critical Points: Where to Look and Why

Homework Statement Find any critical point of the following function. f(t)= 3 — lt-3l, [-1,5] The answer says it has a critical point where t=3. Homework Equations The Attempt at a Solution f(t)=3 — lt-3l f(t)=3-t+3, t≥3 f'(t)= -1 f(t)=3-(-t+3), t<3 f'(t) = 1...
32. 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. 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. Thermodynamics: Critical Point

1. Find Critical Point and Prove it 2. dP/dV =0 3. n/a Hey guys, so our professor was talking about critical points in class. He mentioned on our exam we would have to find a critical point and prove it. Anyhow none of our homework problems ask this, and very little info in...
35. Visualizing the critical point of a substance (thermodynamics)

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Differential Equation stability at critical point

Homework Statement This is the autonomous differential equation: x" - 2x' + 37x = 0 Solve the above DEQ and state whether the critical point (0,0) is stable, unstable, or semi-stable. Homework Equations Solution to the above DEQ is x = c1excos6x + c2exsin6x The Attempt at a...
47. Show that the origin is a critical point (linear differential equation)

Homework Statement Show that the origin (0,0) is a critical point. Write the linear differential equation in operator format and solve. Homework Equations x'' + 10x' + 25x = 0 The Attempt at a Solution I am not sure how to show that the origin is a critical point (without using...
48. Stability of Degenerate Critical Points: A Phase Plane Analysis

How can you determine the stability of a critical point which is degenerate?
49. How Do You Estimate the Critical Temperature and Pressure of a Fluid?

Homework Statement I am to estimate the critical temperature and critical pressure (where vapor and liquid merge) of a fluid. I am given that rho (the density), L (latent heat of vaporization), t_b (boiling temperature at 1atm) and P = 1atm (pressure) to be known.Homework Equations dG = -S dt...
50. 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 ?