1st AND 2nd ORDER PHASE TRANSITION
- 1. Topic 1st Order and 2nd Order Phase Transition Subject Advance Solid State Physics Class BS Physics (Morning) 7th Semester
- 2. 1st Order and 2nd Order Phase transtion Phase Transition: Phase transition occur when a substance changes from a solid, liquid, or gas state to a different state. Every element and substance can make transition from one phase to another at a specific combination of temperature and pressure. There are two types of phase transition. (i)1st Order Phase Transition (ii) 2nd order Phase Transition
- 3. Landau Theory of Phase Transition This theory tells about how phase occur and what happens to the system when phase transition occurs. When phase transition occurs in a system then for this system at constant temperature and volume F (Helmholtz function) is minimum and the system is in equilibrium. In fact, F tend to approach a minimum value for all processes in universe.
- 4. Landau Theory of Phase Transition F is the thermodynamic potential at constant volume of a system and is given by F = U-TS Here U is internal energy T is temperature and S is entropy. Suppose F is minimum with respect to single order parameter x, which is different for different cases. e.g. x = magnetization of ferromagnetic substance, superconducting electrons in case of superconductor.
- 5. Landau Theory of Phase Transition The parameter x further depends on temperature. Here is an important thing that for each system to describe it as best as possible the F will be the function of T and x also, the system should be in equilibrium at constant T and V. According to Landau the free energy function or Helmholtz function is given as FL = U(x,τ) - τ S(x,τ)…………..(1)
- 6. Landau Theory of Phase Transition F (τ) is minimum at a particular value of x and T at values other than the particular value F (τ) has larger value. FL (x, τ0) <= FL(x, τ) at x ≠ x0……..(2) For most of ferromagnetic substances FL (x, τ) = f0(τ) + 1/2f2(τ)x2 + 1/4f4(τ)x4 +……… (3) f0(τ) , f2(τ) and f4(τ) are the coefficients of expansion and they are always function of temperature. For phase change only, f2(τ) changes and is given by f2(τ) = α (τ- τ0)…………………. (5)
- 7. Landau Theory of Phase Transition By putting the value of f2(τ) from equation 5 to 4 we get FL (x, τ) = f0(τ) + 1/2 α (τ- τ0) x2 + 1/4f4(τ)x4……………. (6) At equilibrium change in F with respect to x is zero so above equation becomes α (τ- τ0) x + f4 x3 = 0 from here we can obtain three values of x which are x= 0 and x2 = α/f4(τ0 – τ) for x = 0 f(τ) = f0(τ) , This is valid when temperature is about τ0
- 8. Landau Theory of Phase Transition For x = 0 f(τ) = f0(τ) , This is valid when temperature is about τ0 . x2 = α/f4(τ0 – τ) this value exist when τ is less than τ0. When x is imaginary we can’t decide what type transition occurs their. Putting value of x2 in equation 6 we get FL (x, τ) = f0(τ) - 1/2 α2 (τ- τ0)2 …………………….(7)
- 9. First Order Phase Transition A transition is said to be first order phase transition if a finite amount of heat which is called latent heat is supplied for transition. During such a transition, a system releases or absorbs a fixed (typically large) amount of energy per unit volume. This transition is also called discontinuous phase transition. Latent Heat: The heat required to convert a specific amount of solid into liquid and liquid into solid without change in temperature. e.g. for 1g of ice it is 80 calrie.
- 10. First Order Phase Transition When we want to convert solid into liquid e.g. ice into water then we supply latent heat Solid → Liquid → Gas N1 N2 N3 L= finite latent heat Here N1, N2 and N3 are number of particles in solid, liquid and gas phase respectively. Then Gibbs energy per particle is given by 𝐺 𝑁 = g ……………………. (1) Here G is Gibbs energy g is Gibbs energy per particle
- 11. First Order Phase Transition If we have N1 number of particles in solid state and N2 number of particles in liquid state then total number of particles are N which are remains constant and given by N = N1 + N2 …………………. (2) The Gibbs energy for these particles is given by G= G1 + G2 ……………………. (3) By using equation 1 and solving for 3 we ge G= N1g1 +N2 g2…………………. (4)
- 12. First Order Phase Transition From equation 2 we get ∂N = ∂N1 + ∂N2……………………. (5) As the total number of particles remains constant so, we get ∂N1 = - ∂N2 …………………………………… (6) By using equation 6 in 4 we get ∂N2 (g1 – g2) = 0 ……………………... (7) As ∂N2 ≠ 0 so g1 – g2 = 0 hence g1 = g2 Similarly for liquid to gas transition g3 = g2 At a certain temperature and pressure g1 = g2 = g3 at this point three states co-exist and it is called triple point.
- 13. First Order Phase Transition From above calculations we can say that Gibbs energy per particle for each state is constant in first order phase transition. Here G = H – TS Where H = enthalpy of the system.
- 14. First Order Phase Transition Enthalpy: The sum of internal energy of a system plus product of pressure and volume is called enthalpy. The graph on previous page shows that Gibbs function decreases when temperature increase. As the temperature increase enthalpy also increases but product of T and S increases sharply as compared to H. So Gibbs function decrease.
- 15. First Order Phase Transition If in a process pressure is constant then the process is said to be iso – baric. The Gibbs energy is given by G= Vdp – SdT ………………. (8) -( 𝑑𝐺 𝑑𝑇 )= 0 – SdT/d S= -( 𝑑𝐺 𝑑𝑇 )P……………… (9)
- 16. First Order Phase Transition This graph shows that entropy shows discontinuity with temperature. In first curve the temperature increases entropy also increases but a time comes when T becomes constant and entropy increases at this time latent heat is used to free the particles from their substance or it is used against the interatomic forces. After this entropy start to increase again.
- 17. 2nd Order Phase Transition The transition is said to 2nd order when no latent heat is supplied to the substance for transition. This transition is also called continuous phase transition. Example: (i) Conversion of ferromagnetic substance into paramagnetic and para to diamagnetic substances. If we increase the temperature of a ferromagnetic substance, then at a certain temperature it becomes paramagnetic upon further increase in temperature the para becomes diamagnetic.
- 18. 2nd Order Phase Transition (ii)Conversion of ordinary solid into Superconductor and vice versa. When we decrease the temperature of a substance to a certain level it becomes superconductor. No latent heat is supplied in this process. And when we increase the temperature to small extent the superconductor start to loss its superconducting electrons.
- 19. Graphical Behavior in 2nd Order Transition The graphical behavior of Gibbs function and entropy for 2nd order phase transition is shown below the discontinuity of curve is not observed in this case because no latent heat is supplied.
- 20. References • https://chem.libretexts.org/Fundamentals_of_Phase_Transitions • https://www.youtube.com/watch?v=TrCCQru2ru0 • https://www.youtube.com/watch?v=R9Xj3bII0jg • https://www.youtube.com/watch?v=L9WLBgUwvDA