Chapter 4 Physical transformations of pure substance

For the pure substances, a phase diagram is a map of the pressure and

temperatures at which each phase is the most stable.

Chemical potential (µ): a property that is at the centre of discussion of

phase transitions (physical reactions) and chemical reactions.

4.1 The stability of phases:

A phase of a substance is a form of matter that is uniform throughout

in chemical composition and physical state. (solid, liquid, gas allotropes.)

A phase transition, the spontaneous conversion of one phase into

another phase, occurs at a characteristic temperature for a given pressure.

The transition temperature, Ttrs, is the temperature at which the two phase

are in equilibrium and Gibbs energy is minimized at the prevailing pressure.

A transition that is predicted from th

ermodynamics to be spontaneous may occ

ur too slowly to be significant in practice.

Exp. C(Diamond) C (graphite)

is too slow to be detected under ambient

condition.

Thermodynamically unstable phase that

persist because the transition is kinetically

hindered are called metastable phase.

4.2 Phase boundaries

The lines separating the regions, which are called phase boundaries,

shows the values of p and T at which two phases coexist in equilibrium.

Vapor pressure: the pressure of a vapor in equilibrium with the liquid.

Sublimation vapor pressure: the vapor pressure of the solid phase.

The vapor pressure of a substance increases with the increase of

temperature because at higher temperature the Boltzmann distribution

populates more heavily the sates of higher energy .

(a). Critical points and boiling points

When liquid is heated in an open vessel, at the temperature at which its

vapor pressure would be equal to the external pressure, vaporization can

occur throughout the bulk of the liquid. The condition of free vaporization

throughout the liquid is called boiling.

Boiling Temperature: the temperature at which the vapor pressure of a

liquid is equal to the external pressure.

At 1.0 atm: normal boiling point, Tb; At 1.0 bar, standard boiling point.

Boiling does not occur when a liquid in a closed vessel. Instead,

the vapor pressure, and the density of the vapor rise continuously as

increase of temperature.

The temperature at which the surface d

isappears is the critical temperature, Tc.

The vapor pressure at the critical temp

erature is called the critical pressure, pc.

At and above the Tc, a single uniform

phase called a supercritical fluid fills the c

ontainer and interface no longer exist.

At T > Tc, no liquid phase.

The Tc is the upper limit for liquid.

(b). Melting point and triple points

Melting point: the temperature at which, under a specified pressure, the liquid and

solid phases coexist in equilibrium. (Freezing point; liquid solid)

At p = 1 atm, normal freezing (melting) point, Tf

At p = 1 bar, standard freezing (melting) point.

Triple point (T3): a point three phase boundaries

meet. (three phases coexist in

equilibrium)

T3 occurs at a single definite p and T

characteristic of the substance.

thermodynamic temperature scale

The T3 marks the lowest pressure at which a

liquid phase of a substance can exist.

4.3 Three typical phase diagrams

(a)Water

The solid-liquid boundary has a

negative slope, which means a

decrease of volume on melting.

(dp/dT) < 0

At high pressures, different

structure forms of ice come into

stability as the hydrogen bonds

between H2O modified by

pressure.

Hydrogen-bonding structure in ice and water:

Ice (d = 0.9 cm3/g) Water (d = 1.0 cm3/g)

The volume decreases on melting. (ΔVm < 0)

(b) Carbon dioxide:

The positive slope of the solid-liqu

id boundary. (dp/dT >0)

The volume increases on melting.

The triple point lies above 1 atm. T

o obtain the liquid, it is necessary to

exert a pressure > 5.11 atm.

When a 67 atm CO2 squirts(噴出 )

to 1 atm, only the snow-like solid CO2

was obtained

(c). Helium

He behaves unusually at low temperatur

e.

The solid and gas phase are never in

equilibrium however low the temperature.

Solid helium can be obtained by applyin

g pressure (Helium is too light).

4He (I = 0), has a liquid-liquid phase tran

sition at its -line.

Helium-I phase behaves like a normal li

quid, Helium-II is a superfluid (no viscosit

y).

Phase Stability and Phase Transitions

4.4 The thermodynamic criterion of equilibrium

Chemical potential µ = Gm, µ is a measure of potential that a substance

for bring about physical or chemical change in a system.

Phase I

Phase II

The 2rd Law:

At equilibrium, the chemical potential of a

substance is the same throughout a sample,

regardless of how many phases are

present.

In equilibrium,

µ(phase I) = µ(phase II)

4.5 The dependence of stability on the conditions

dG = VdP – SdT ; dµ = VmdP – SmdT

(a). Temperature dependence of phase stability

(µ/T)p = – Sm

As the temperature raises, the che

mical potential of a pure substance decr

eases: Sm > 0 for all substance.

Sm(g) > Sm(l) > Sm(s)

The easiest way of bring about a pha

se transition is by changing the tempera

ture.

dG = dH – TdS – SdT = dU + pdV + Vdp – TdS – SdT

= dq – pdV + pdV + VdP – TdS – SdT = TdS + Vdp – TdS – SdT

= VdP – SdT

(b). The response of melting to applied pressure

For most substances except H2O, it is as thought the pressure is preven

ting the formation of the less dense liquid phase.

(µ/p)T = Vm

An increase in pressure increases the chemical potential of any substanc

e because Vm > 0.

In general, Vm(g) > Vm (l) > Vm (s)

The pressure dependence of the chemical potential (dG = VdP )

(a). Vm(l) > Vm(s) (b). Vm(l) < Vm(s)

p Tf p Tf

(c). The effect of applied pressure on vapor pressure

When pressure is applied to a condensed phase, its vapor pressure rises: in effec

t, molecules are squeezed out (壓出 ) the phase and escape as a gas.

Pressure can be exerted on the conden

se phases mechanically or by subjecting i

t to the applied gas of an inert gas (partial

pressure).

p = p*exp(VmΔp/RT)

Δp: the applied pressure

If VmΔp/RT << 1,

p p*(1 + VmΔP/RT)

(p – p*)/ p* = VmΔp/RT

Justification 4.1:

In equilibrium µ(l) = µ(g) dµ(l) = dµ(g)

dµ(l) = Vm(l)dP (applied pressure) ; dµ(g) = Vm(g)dp (p: vapor pressure)

The gas phase behaves as an ideal gas, dµ(g) = (RT/p)dp

RT/pdp = Vm(l) dP RT (1/p)dp (p* p) = Vm(l) dP (p* ΔP + p*)

If Vm(l) = constant, and vapor pressure change (p – p*) << ΔP

RT ln (p/p*) = Vm(l) ΔP

Illustration 4.1:

For water, the molar volume = 18.1 cm3mol-1, when subjected to an increas

e in pressure of 10 bar, the vapor pressure has an increase of 0.73 %.

4.6 The location of phase boundary

Phase boundaries – the pressure and temperature at which two phases are in

equilibrium. µ(p, T) = µ(p, T)

(a) The slope of phase boundaries

Phase boundaries dp/dT

On the phase boundaries, the two phase

continue to be in equilibrium.

dµ = dµ

For each pashe, dµ = – SmdT + Vmdp

– S,m dT + VmdP = – S,m dT + V,m dp

(V,m – V,m)dp = (S,m – S,m)dT

dp/dT = ΔtrsS/ΔtrsV

(b). The solid-liquid boundary:

dp/dT = ΔfusH/TΔfusV (ΔfusS = ΔfusH/T)

ΔfusV: the change in molar volume on melting

ΔfusV is usually positive and always small.

dp = (ΔfusH/ΔfusV) 1/T dT

p p* + ΔfusH/ΔfusV ln (T/T*)

When T is close to T*

(ln T/T* = ln(1+ (T – T*)/T*)

(T – T*)/T*

p p* + ΔfusH/T*ΔfusV (T – T*)

(c). The liquid-vapor boundary:

dp/dT = ΔvapH/TΔvapV (ΔvapH > 0 ; ΔvapV > 0 and large)

dp/dT > 0 for vaporization, and hence the boiling temperature is more respo

nsive to pressure than the freezing temperature.

Cooking in a covered pot,

Pressure boiling temperature cooking rate

Because the molar volume of a gas is larger than

that of a liquid.

dp/dT = ΔvapH/T(RT/p)

dlnp/dT = ΔvapH/RT2 (Clausius-Clapeyron eq.)

p = p*ex x = ΔvapH/R (1/T – 1/T*)

(d) The solid-vapor boundaries

dp/dT = ΔsubH/TΔsubV

dp/dT = ΔsubH/T(RT/p)

dlnp/dT = ΔsubH/RT2

p = p*ex x = ΔsubH/R (1/T – 1/T*)

Due to the ΔsubH > ΔvapH

The dp/dT slope for sublimation curve is st

eeper than that for vaporization curve at simi

lar temperature.

4.7 The Ehrenfest Classification of Phase Transitions:

Other transitions:

Solid-solid, conducting-superconducting and fluid-superfluid.

At the transition from a phase to another phase :

(µ/p)T – (µ/P)T = V,m – V,m = trsV

(µ/T)p – (µ/T)p = –S,m + S, m = trsS = trsH/Ttrs

The first derivatives of the chemical potentials with respect to pressure

and temperature are discontinuous at transition.

A transition for which the first derivative of the chemical potential with re

spect to temperature is discontinuous is classified as a first-order phase tr

ansition.

4.7 The Ehrenfest Classification of Phase Transitions:

1st

2rd

A second-order phase transition in the Ehrenfest sense is one in which the

first derivation of with respect to temperature is continuous but its second d

erivative is discontinuous.

V, H, and S do not change at the transition. The Cp is discontinuous at tran

sition but does not become infinite there.

A conducting-superconducting transition in metals at low temperature is an

example of second-order transition.

The term -transition is applied to a phase transition that is not first order y

et the heat capacity becomes infinites at the transition temperature.

This type of transition includes

order-disorder transitions in alloys,

the onset ferromagnetism, and the

fluid-superfluid transition of liquid

helium.

Second-order phase transition and -transitions:

Suppose the two shorter dimensions increases more than the longer

dimension when the temperature is raised.

The tetragonal cubic phase transition has occurs, but as it has not

involved a discontinuity in the interaction energy between atoms or the

volume they occupy, the transition is not first-order.

The order-disorder transition in -brass

(CuZn) is an example of a -transition.

At T =0 the order is perfect, but islands

of disorder appear as the temperature is r

aised.

The islands form because the transition

is cooperative in the sense that, once tw

o atoms have exchanged locations, it is e

asier for their neighbors to exchange their

location. The island grow in extent, and m

erge throughout the crystal at the transitio

n temperature (724 K).

5.1 Inorganic Solid-State Clathrate Compounds

5.1.1 Clathrate Hydrates:

5.1.1.1 Formation:

Many relatively nonpolar compounds that do not hydrogen bonding

with water are capable of forming clathrate hydrates.

In contrast, species such as alcohols, acids and ammonia, which are capable of

forming hydrogen bonds, do not generally form hydrates.

Solid clathrate hydrate are formed under very speci

fic temperature (often well above the melting point of

pure ice) and pressure. Indeed, some known gas hyd

rate are stable to 31.5 oC.

This high degree of thermal stability is currently a significant problem in the

natural gas industry. Transport of natural gas (CH4) in pipeline is plagued by

blockages caused by the formation of clathrate hydrates.

Solutions: 1. Drying and warming the gas.

2. Addition of large quantities of kinetic (PVP; polyvinylpyrrolidinone) and th

ermodynamic (methanol, glycol, salt solutions) hydrate formation inhibitors.

5.1.1.2 Structure:

Ice does not possess any cavities capable of including guest molecules.

Hydrates: a template reaction occurs in which polyhedral cavities are form acc

ording to guest size.

The cavities are composed

extensively of fused five- and six-

membered hydrogen-bonded rings.

512 (5: five-member ring, 12: the

number of ring in the cage)

Almost all clathrate hydrates form two basic structure termed type I and

type II.

Type I clathrate hydrates Type II clathrate hydrates

Guest molecule

In general, small guests occupy the sma

ll 512 type cavities. The small difference be

tween the size of this cavity in structures I

and II can have a significant effect on whic

h structural type is adopted.

He, H2 and Ne are small enough to diff

use through the cage and do not form hydr

ates.

In the real hydrates, some cages remain

empty. Thus the real hydrates have more

water than the “ideal composition”.