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I got to read things like

In case of a first order phase transition, the volume and temperature change in a discontinuous manner. However for phase transitions of higher order the change in volume and temperature can be continuous.

I could not exactly understand the statement and thats why I thought what exactly it means by the order of a phase transition.

May I know what exactly it means by the order of phase transition?

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Did you read the Wikipedia page? – Qmechanic Jan 24 '13 at 15:55

In the graphic included in the related Wikipedia article, depicted is a type of space where the axes are thermodynamic variables. You can similarly just think of any two dimensional plot with a line drawn on it. The line can be understood to represent a function. Normally if you want to find the slope of a function, you take the first derivative of that function. If the first derivative of the function has a discontinuity (e.g. a point were the function is no longer well defined), and the variable in question is a thermodynamic variable, then by definition the phase transition is a first order phase transition (because the discontinuity occurs in the in the first derivative of the function).

Essentially, the scheme is using an understanding of the properties of derivatives of math functions in order classify the types of phase transitions. Without a proper understanding of the related mathematical concept, its understandable that this classification isn't immediately obvious. I hope this helps.

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Broadly speaking Hal is right. Let me try to make it a bit more transparent if you don't mind. The 1st derivative of the Gibbs free-energy of the system w.r.t T, at constant P, is the entropy of the system, while the derivative w.r.t P, at constant T, is the volume. If these quantities are discontinuous at the CP (critical point,) the phase transition is said to be first order, because of the singularity of the 1st derivative. On the other hand, the 2nd derivative w.r.t T, at constant P, relates to the specific heat of the system. If the specific heat is singular at the CP then the phase transition is said to be second order. Therefore, the terms 1st and 2nd order phase transition are general, and distinguish one type from the other, as the physics behind these two can relate to very different properties of condensed matter or other fields of physics. For example, at an elastic 1st order phase transition the volume and entropy of the crystal undergo abrupt change at CP. If the transition is 2nd order, the specific heat of the crystal grows singularly, and at the same time the speed of the vibrational mode leading the transition tends to zero (mode softening!)

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I can see the promise of insight in your answer, but could you help me in understanding few more things. Like I want to know what exactly is a 'phase'?What criteria does a given arrangement of matter, temperature and pressure need to satisfy in order to be called as a different phase? Is it so that the order of derivative of a thermodynamic function which becomes discontinuous is the order of phase transition? Can it happen that during a phase transition one variable witnesses discontinuity in the nth order derivative and at the same time another witnesses in mth order? N and m are int. – danny gotze Jan 30 '13 at 14:23
Phase of a system (eg matter) is the set of measurable physical properties it has, for given values of some thermodynamic variables (eg T and P.) In the context of an elastic phase transition, at some T the structure of the lattice can be cubic say, this is the high symmetry phase. If you now pass an RF acoustic wave through the crystal, and change T so that you can hit the CP, you can induce a phase transition in which the lattice will be tetragonal. This is the lower symmetry phase. The 1st and 2nd derivatives of the GFE relate to measurable physical quantities, hence their importance. – JKL Jan 30 '13 at 15:15
May I know what exactly you mean by an elastic phase transition ? could you illustrate your definition of phase by describing the solid phase and liquid phase? Like why a water at 75 degree centigrade and water at 65 degree centigrade is the same phase? – danny gotze Jan 30 '13 at 15:21
The RF acoustic wave excites a phonon in the crystal lattice, that corresponds to a non identity irreducible representation of the cubic lattice symmetry. This induces a corresponding irreducible strain tensor component, which violates the cubic symmetry, and causes a structural phase transition to the tetragonal phase, at the CP. That particular strain tensor component is the order parameter that drives the transition. I hope this helps(?) – JKL Jan 30 '13 at 15:27
An elastic phase transition is just one of the many types of phase transitions that can occur in nature. In these, the solid remains solid, but its crystal structure changes. Like in the superconducting phase transition, the solid wire remains solid, but its electrical resistivity drops abruptly to zero. The state changes you mentioned is yet another type of phase transition. The universe is believed to have undergone a number of phase transitions, in which it changed the way particles interacted with one another etc. – JKL Jan 30 '13 at 15:38

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