[PDF] [PDF] Lecture 14 Equilibrium between phases (Ch5) - Rutgers Physics

[PDF] Water Phase Diagram

Under the singular conditions of temperature and pressure where liquid water, gaseous water and hexagonal ice stably coexist, there is a 'triple point' where both 

[PDF] Phase Diagrams and the Triple Point - Employees Csbsju

In other words, we find that at any given temperature, the pressure of water vapor in phase equilibrium with liquid water will have some definite, characteristic 

[PDF] Phase diagram of water Note: for H O melting point decreases with

Water has a high boiling point and high freezing point If water were “normal” it would be a gas at room temperature CH 4 , 

[PDF] Triple Point Water

Discussion: On a phase diagram, the point at which all three curves intersect is known as the triple point At this temperature and pressure, all three phases are in equilibrium The triple point of water is 273 16 K and 4 58 torr

[PDF] Lecture 14 Equilibrium between phases (Ch5) - Rutgers Physics

at the triple point – in this case, all three phases coexist at (T tr , P tr ) The shape of coexistence curves on the P-T phase diagram is determined by the condition: ( ) ( ) at the boiling point, the values of G for liquid water and water vapor

[PDF] Phase Diagrams

Pressure-Temperature Phase Diagram: Solid-gas coexistence curve ends at the triple point and the liquid-gas curve Example – Phase diagram of water

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Here are some example phase diagrams for carbon dioxide, argon and water Figure 4 Phase At the triple point, three phases are in equilibrium together:

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Lecture 14. Phases of Pure Substances (Ch.5)T

P The generic phase diagram of a substance in the P-Tcoordinates is shown above. Every point of this diagram is an equilibrium state; different states of the system in equilibrium are called phases. The lines dividing different phases are called the coexistence curves . Along these curves, the phases coexist in equilibrium, and the system is macroscopically inhomogeneous. All three coexistence curves can meet at the triple point- in this case, all three phases coexist at (T tr ,P tr

Up to now we have dealt almost

exclusively with systems consisting of a single phase. In this lecture, we will learn how more complicated, multi- phase systems can be treated by the methods of thermodynamics. The guiding principle will be the minimization of the Gibbs free energy in equilibrium for all systems, including the multi- phase ones.

The Coexistence Curves

Any system in contact with the thermal bath is governed by the minimum free energy principle.The shape of coexistence curves on the P-Tphase diagram is determined by the condition:

TPGTPG,,

21
- otherwise, the system would be able to decrease its Gibbs free energy by transforming the phase with a higher into the phase with lower . Two phases are in a state of diffusive equilibrium: there are as many molecules migrating from

1to 2 as

the molecules migrating from 2to 1.

TPTP,,

21
NG and, since

Also for equilibrium

between the phases: 21
TT - as for any two sub-systems in equilibrium 21
PP - the phase boundary does not move Along the coexistence curves, two different phases 1 and 2 coexist in equilibrium (e.g., ice and water coexist at T = 0 0

C and P = 1bar). The system

undergoes phase separation each time we cross the equilibrium curve (the system is spatially inhomogeneousalong the equilibrium curves). Though Gis continuous across the transition, Hdemonstrates a step-like behavior:

TSHTSPVUNG

STH (different phases have different values of the entropy)

Example: the Gas-Liquid Transformation

VTVTVTPT

NSTNUTSUFNF

NG Gas: U/Nterm is small and positive (kin. energy of a single molecule), T(S/N)term is large and positive is negative, and rapidlydecreases with increasing T. Liquid: U/Nterm is negative (attraction between molecules), T(S/N)term is smaller than that in gas and positive is also negative, and slowlyincreases with decreasing T. T gas liquid 0 phase transformation STG NP STTGG

TNPTNP0,,,,

0 S (water) = 70 J/K S (vapor) = 189 J/K Tableon page 404 (a very useful source of information) provides the values of Hand Gfor different phases of many substances. The data are provided per mole, at T=298 K and P=1 bar. For example, let's check that at the boiling point, the values of G for liquid water and water vapor are equal to each other: mo l

JmolKJKmolJG

KTliq /10242)/(7075/10237 33
373
molJmolKJKmolJG KTvap /10242)/(18975/106.228 33
373

Phases of Carbon

G

P (MPa)

12 graphite diamond

2.9 kJ

The phase equilibrium on the P,T-plane is determined by

TPTPTPGTPG,,,,

2121
or At normal conditions, graphite is more stable than diamond: G(graphite) = 0, G(diamond) = 2.9 kJ (diamonds are not forever...). What happens at higher pressures? VPG NT - since the molar volume of graphite is greater than the molar volume of diamond, G(graphite) will grow with pressure faster than G(diamond) [we neglected V = V(P)]

D. becomes more stable than G. only at

P > 1.5 MPa

STG NP VPPGG

PNTNT0,,,

0

With increasing T, the pressure range of graphite

stability becomes wider: STTGG

TNPTNP0,,,,

0 S (graphite) = 5.74 J/K, S (diamond) = 2.38 J/K, G T (K)

800 1300

graphite diamond

2.9 kJ

300

T = 300K

P = 1 bar

The First-Order Transitions

PS T=T 1 = c o ns t gas gas+liquid liqu i d liqu i d solid so lid

Because molecules aggregate differently

in different phases, we have to provide (or remove) energy when crossing the coexistence cu rves. The energy difference is called the latent heat ; c r ossing the coexistence curve, the system releases (absorbs) a latent heat L

The entropy of the

system changes abruptly: T 1 T L T Q S

The transitions which displays

a jump in entropy and a latent heat associated with this jump ar e cal l ed the firs t-orde r pha s e transit i ons

The "evaporation"

L is generall y greater than the " m elting" L (the disorder introduced by evaporation is greater than that introduced by melting). TT C S S gas S l i quid crit ical point mixedphase temper ature beyond critical point, gas is indistinguishable from liquid Q:

Can the critical point exist along the

melting coexistence curve?

The First-Order Transitions (cont.)

G T solid liquid gas

P,N = const

Pr. 5.9

On the graph

G T at P,N = const, the sl ope d G /d T is always negative: G T S T S = L T N P TG S T C P N P P TS T C Note that in the first-order transitions, the G(T) curves have a real meaning even beyond the intersection point, this results in metastability and hysteresis

There is usually an energy barrier

that prevents a transition occurring fr om the higher to the lower phase (e.g., gas, being cooled below T tr does not immediately condense, since surface energy makes the for m ation of very small droplets energetically unfavorable)

L. water can exi

s t at T far lower than the freez ing temperature: water in organic cells can avoid freezing down to -20 0

C in insects and down

to -47 0

C in plants.

Problem

(a) What is the heat of vaporization at this temperature? (b) The enthalpy of vapor under these conditions is

2680 J/g. Calculate the enthalpy of water under these conditions.

(c) Compute the Gibbs free energies of water and steam under these conditions.The entropy of water at atmospheric pressure

and 100 0

C is 1.3 J/g·K, and the entropy of water

vapor at the same Tand Pis 7.4 J/g K. (a) The heat of vaporization: L= TS = 373K6.1 J/g·K=2275 J/g (b) The differential of enthalpy dH= TdS+VdP. Hence, H water = H vapor -TdS=H vapor -L= (2680-2275)J/g = 405 J/g (c) Since G= H-TS, G water = H water -TS water = 405J/g - 373K 1.3J/g·K = -80J/g G vapor = H vapor -TS vapor = 2680J/g - 373K 7.4J/g·K = -80J/g

The Second Order Transitions

phase on one side and the gas phase on the other gradually decreases and finally disappears at (T C , P C ). The

T-driven

phase transition that occurs exactlyat the critical point is called a second-order phase transition. Unlike the 1 st order transitions, the 2 nd -order transition does not require any latent heat (L=0). In the higher-order transitions order-disorder transitionsor critical phenomena) the entropy is continuous across the transition. The specific heat C P

T(S/T)

P diverges at the transition (a cusp-like singularity).

Whereas in the 1

st -order transitions the G(T)curves have a real meaning even beyond the intersection point, nothing of the sort can occur for a 2 nd -order transition - the Gibbs free energy is a continuous function around the critical temperature. G T S T S=0

Second-order transition

TC P

The vaporization coexistence

curve ends at a point called the critical point(T C , P C ). As one moves along the coexistence curve toward the critical point, the distinction between the liquid NPP TSTC

The Clausius-Clapeyron Relation

TPGTPG

21

Along the phase boundary:

P T P

Tphase

boundary

TPGTPG

21

TPVTPVTPSTPS

dTdP 2121
- the slope is determined by the entropies and volumes of the two phases. The larger the difference in entropy between the phases - the steeper the coexistence curve, the larger the difference in molar volumes - the shallower the curve. (compare the slopes of melting and vaporization curves)

Since S

1 -S 2 = L/T(Lis the latent heat), we arrive at the Clausius-Clapeyron Relation: TVTTL dTdP

TLTVTVT

dPdT liqgas -since V gas > V liq , and L> 0for the "liquidgas" transformation, the boiling temperature increases withquotesdbs_dbs14.pdfusesText_20