Asian American Pacific Islander Month DDSD 2024.pptx
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.
