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*Phases of Matter:*

Matter or substance can exist in three familiar phases that is; solid phase, liquid phase, gaseous phase. Matter of some substances can also exist in two less familiar phases that is; super fluid phase, and plasma phase.

Solid Phase: Molecules are arranged in the closely packed form known as crystal. These molecules can only vibrate about their lattice point.

Liquid Phase: Molecules are close together and they take shape of container. Molecules of liquid, inside its volume, can move from place to place, rotate and vibrate.

Gaseous Phase: Molecules are extensively separated and free to move around freely.

Super fluid: The supercritical (or critical) fluid is the liquid/gas under extreme pressure.

Plasma: Plasma is the gas which is made up of free-floating ions and free electrons.

**Phase Diagram:**

Three regions for three phases are shown in diagram given below. Solid curves or lines symbolize boundary between two phases like fusion curve is boundary between solid phase and liquid phase. These lines are known as equilibrium lines. Implication of this is that under particular conditions of temperature and pressure, a substance can exist in equilibrium in more than one phase at the same time.

Triple Point: This is point where three equilibrium lines meet as indicated in phase diagram given below. At triple point, solid, liquid, and vapor phases of pure substance coexist in equilibrium. All substances have triple point except Helium.

Triple Point Temperature: this is temperature at which solid, liquid, and vapor phases coexist in equilibrium.

Triple Point Pressure: This is pressure at which solid, liquid, and vapor phases coexist in equilibrium.

Critical Point: This defines conditions of temperature and pressure beyond which it is no longer possible to differentiate liquid from the gas. Point is indicated in phase diagram and region beyond critical point is called as fluid region.

**Co-exist Phases:**

This is when more than one phase of the substance (like liquid-solid) exist side-by-side in equilibrium at same time. For instance, solid water and liquid water can coexist at 0^{0}C along procedure of fusion or melting. Gibbs energy (G) for two coexisting phases α and β of the pure substance are equal.

*Phase Transitions:*

Phase transition takes place when matter changes from one of the phases of matter to another. Process always involves withdrawal or addition of heat energy from or to matter. Using figure given above as illustration, phase transition takes place whenever any one of the curves in phase diagram is crossed. Phase transition for the pure substance takes place at constant temperature and pressure. Implication of this statement is that, for the pure substance dT = dP =0 during the phase change. Though, extensive thermodynamic coordinates or properties (like Volume) change suddenly because of the phase transition. Internal energy (U), enthalpy (H), and entropy (S) may also change during the phase transition.

**Latent Heat, L, during Phase Transition:**

Latent heat L is an amount of heat energy per mole which should be added or removed when the substance changes from one phase to another. If phase transition occurs reversibly, heat transfer (i.e. latent heat) per mole for transition from initial phase α to the final phase β is provided by

L = T(S^{β} - S^{α})

*Kinds of Phase Transition:*

The three kinds of phase transitions are: first order, second order and lambda phase transitions.

**First Order Phase Transition:**

Phase transitions which are recognizable with i.e. sublimation, vaporization and fusion are known as first order phase transition. They are known as first order as first order derivatives of Gibbs function are finite.

Thus, for first order phase transition:

- There are changes in entropy and volume, and
- The first-order derivatives of Gibbs function change discontinuously.

The specific heat capacity at constant pressure is infinite; this is due to temperature is constant during phase change (C_{P} = T∂S/∂T|_{P})

**Second Order Phase Transition:**

This is a phase transition in which second derivates of Gibbs are finite.

For order phase transition,

- T, P, G, S, and V (also H, U, and F) remain unchanged
- CP , β, and κ (i.e. from second derivatives of G) are finite

The only example of second order phase transition is transition for superconductor from superconducting to normal state in zero magnetic fields.

**Lambda phase transition:**

For lambda phase transition:

- T, P, and G remain constant,
- S and V (also U, H, and F) remain constant, and
- CP , β, and κ are infinite

The most interesting example of lambda transition is transition from ordinary liquid helium to super fluid helium at a temperature and corresponding pressure known as a lambda point.

*Gibbs Function during Phase Transition:*

The Gibbs function G doesn't change during phase transition. For coexisting phases,

dG|_{T,P} = 0

I.e. change in Gibbs at constant temperature and pressure is zero.

Two phases (e.g. liquid-gas) can coexist in equilibrium. For coexisting phases α and β of the pure substance

G^{α} = G^{β} = dG^{α} = dG^{β}

Gibbs function G is provided by equation:

dG = -SdT + VdP

After replacing values we get:

-S^{α}dT + V^{α}dP = -S^{β}dT + V^{β}dP

Rearranging to get:

dP/dT = (S^{β} - S^{α})/(V^{β} - v^{α})

Further solving we get:

dP/dT = L(T(v^{β} - V^{α}))

This equation is called as Clapeyron's equation.

If solid phase is labeled 1, liquid 2, and gas or vapor phase 3, equation can be written as follows:

For solid - vapor phase transition, we have

(dP/dT)_{13 }= L_{13}/(T(V^{3} - V^{1}))

Where L_{13} is latent heat of sublimation.

And for solid - liquid phase transition, we get:

(dP/dT)_{12 }= L_{12}/(V^{2} - V^{1})

Where L_{12} is latent heat of fusion

*Usefulness of Clapeyron's Equation:*

Equation can be integrated to get the expression for pressure as the function of temperature. If following assumptions holds i.e. if variation in latent heat can be negligible, and if one of the phases is vapor, and if vapor is assumed to be the ideal gas, and if specific volume of liquid or solid is neglected in comparison with that of vapor, the integration can be readily performed.

(dP/dT)_{23 }= L_{23}/T(RT/P)

∫dP/P = L_{23}/R∫dT/T^{2}

Then ln P = -L_{23}/RT + ln constant

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