Stat chapt 5 Lec 1

Statistical Physics (PH-524) Section V

Phase Transitions and Critical Phenomenon Lecture 5.1

• Statistical mechanics by R K Pathria & Paul D. Beale, 3rd _{Edition, Elsevier}

• Concepts in Thermal Physics by S. J Blundell • Thermal Physics by Daniel V. Schroeder

Phase transformations

A discontinuous change in the properties of a substance as environment is changed infinitesimal.

Ice, water and steam are different phases of H₂O.

 Phase diagram of H₂O

• Stable Phases: Ice, water and steam

• Metastable phases can also Exists, e.g., water can be

“Supercooled”

• On phase diagram, lines represent the conditions under which two different phases can coexist in equilibrium.

• Pressure at which gas can coexist with solid or liquid phase is called vapour pressure.

• At T = 0.01o_{C and P = .006 bar, three phases coexist.} This point is called Triple point.

• Triple point of CO₂ lies above atm press. It is 5.2 bar.

• Note the difference in solid-liquid phase boundary for CO₂ and H₂O.

• Most substances are like CO₂. If we increase pressure,

melting temperature also increase (Positive slope). However, for ice it is not so. Applying pressure, lower

• Liquid-gas phase boundary always has a positive slope.

• If liquid and gas are in equilibrium, and if temp is raised, press also need to increase to keep liquid from vaporizing.

• However, as pressure is increased gas become more

dense. A point known as Critical point is reached where is not possible to distinguish between liquid and water. (we call it just fluid)

Critical point:

For H₂O: 374o_{C and 221 bar; For CO}

• Boiling point of 4He at atm press is 4.2K and critical point (5.2K,2.3 bar)

• Only element which remain liquid at absolute temperature.

• 4He has two phases: He I (Normal phase) & He II

Few important points

• For a thermally isolated system, the stable equilibrium situation is characterized by

Entropy S = Maximum

• For a system which is in contact with a heat reservoir such

that temperature is held constant

equilibrium corresponds to the situation when

Helmholtz free energy F = U-TS is minimum

• For a system which is in contact with a reservoir such that temperature and pressure is held constant

equilibrium corresponds to the situation when

• Stability conditions for a homogeneous substance

Consider a one-component system in single phase

Consider a small part A of this system having some fixed no of particles. Rest part can be considered as reservoir at temp T₀ and pressure P₀

Stability equilibrium applied to A

• If we allow temperature variations

• If we allow variations in volume

Density fluctuations

Condition for equilibrium of phases

We consider single component system with two phases say phase 1 and 2.

For example ice and water as two phases of H₂O

Suppose system is in contact with reservoir at temp T and press P. At equilibrium

We consider N1 and N2 as no of particles in phase 1 and 2, such that

---(1) Equilibrium implied that

=0

---(2) And therefore

Conservation of matter implied that

---(4) And therefore

---(5)

Thus from eq.(3)

=0 ---(6)

The Clausius–Clapeyron equation

Here we want to find an equation which describe the phase boundary between two phases

We start from the equilibrium condition

We move along the phase boundary such that and

 From eq.(7) note that, left side represent the slope of

phase equilibrium at point (T,p) and is related to change in the entropy and volume of substance on “crossing the line” at this point.

 _{As entropy change occur on moving from one phase to}

other heat is also absorbed.

 We define the “Latent heat of transformation” as the heat

Latent heat per particle

Eq.(9) helps to understand the nature of slope of line

separating solid-liquid phases in substances e.g. H2O and CO2.

As solid melts into liquid, entropy increase. Latent heat is absorbed and heat gets absorbed.

• In most cases, solid expands upon melting i.e.,

If i.e., substances contract on melting, for example in case of water.

 _{Exercise: Derive the equation for the phase boundary}

between liquid and gas, considering vapours as ideal gas.

 _{Exercise: Derive the equation for the phase boundary}

Classification of Phase transitions

• Order of the phase transition is determined from the

order of lowest differential of Gibbs free energy G which shows discontinuity at Tc.

• First order phase transition: If the first derivative

given by eq.(1) are discontinuous and hence, S and V, then phase transition will be of 1st _order.

Heat capacity, which is a 2nd _{derivative of G, will show a}

If the 2nd _{differential of G (i.e. heat capacity,}

compressibility) is discontinuous, then transition is called 2nd _{order phase transition.}

Examples: Superconducting transitions

Transitions other than 1st _{order are also known as}

Liquid-gas phase transition is a 1st _{order phase transition} except at critical point where it is continuous or 2nd _order.

In relation to phase transitions one usually use the parameter known as “order parameter” ( ).

For example, for liquid gas phase transition we can define density difference as order parameter

Figure shows behaviour of entropy for liquid gas phase transition.

In above is known as

evaporation heat or latent heat.

• 1st _{order transitions involve latent}

For 1st _{order phase}

In general