Thermo & Stat Mech - Spring 2006 Cla 1 Thermodynamics and Statistical Mechanics Change of Phase
Dec 22, 2015
Thermo & Stat Mech - Spring 2006 Class 9
2
Thermodynamic Potentials
We know that for an isolated system, S ≥ 0. Therefore, any processes in an isolated system can only increase entropy, and the system will be in equilibrium when it reaches maximum entropy.
But what of a system that is not isolated?
Thermo & Stat Mech - Spring 2006 Class 9
3
Helmholtz Function
Suppose a system is in contact with a reservoir at temperature T. The system undergoes a process, and Q is transferred from reservoir to system. S0 is the entropy change of the reservoir, and S is the entropy change of the system.
S0 + S ≥ 0
Thermo & Stat Mech - Spring 2006 Class 9
4
Work
T
QS 0 so, S0 + S ≥ 0 becomes
fiffii
ifif
FFTSUTSUW
TSTSUUW
STUWQUW
QSTST
Q
)()(
)()(
or ,0
Thermo & Stat Mech - Spring 2006 Class 9
5
Helmholtz Function
W ≤ – F (Constant T)
If work is zero for the process,
F ≤ 0, or Ff ≤ Fi
System tends to go to lowest F.
At stable equilibrium when dF = 0
Thermo & Stat Mech - Spring 2006 Class 9
6
Gibbs Function
If contact with the reservoir keeps both temperature and pressure constant, the system goes to the lowest value of the Gibbs function. As before, TS ≥ Q, but in addition, W = PV. Then, Q = U + PV, or
U + PV – Q = 0
U + PV – TS ≤ 0
Thermo & Stat Mech - Spring 2006 Class 9
7
Gibbs Function
U + PV – TS ≤ 0
(Uf + PVf – TSf ) – (Ui + PVi – TSi ) ≤ 0
G ≤ 0
Gf ≤ Gi
Constant T and P.
Stable equilibrium when dG = 0
Thermo & Stat Mech - Spring 2006 Class 9
8
Gibbs Function
If non-mechanical work is done by the system, at constant T and P, then as with F,
Wnm ≤ G
Thermo & Stat Mech - Spring 2006 Class 9
9
Phase Transition
A phase transition, as from a liquid to a vapor, usually takes place at constant temperature and pressure. Therefore the system will go to the state of lowest Gibbs function. Let us see how the specific Gibbs function changes with temperature.
gis the vapor and g is the liquid.
Thermo & Stat Mech - Spring 2006 Class 9
11
Gibbs Function
dG = – SdT + VdP G(T, P)
VTP
TP
P
GV
T
GS
dPP
GdT
T
GdG
and
Thermo & Stat Mech - Spring 2006 Class 9
15
Transition
heatLatent )(
)()(
23
ssTvv
ss
dT
dP
dPvvdTss
dPvdTsdPvdTs
gdgd
gg
Thermo & Stat Mech - Spring 2006 Class 9
16
Clausius-Clapeyron Equation
liquid-Solid )(
vapor-Solid )(
vapor-Liquid )(
12
12
13
13
23
23
vvTdT
dP
vvTdT
dP
vvTdT
dP
Thermo & Stat Mech - Spring 2006 Class 9
17
Enthalpy and Latent Heat
du = đq – Pdv
At transition, u2 – u1 = l12 – P(v2 – v1)
l12 = (u2 + Pv2) – (u1 + P v1)
l12 = h2 – h1
Thermo & Stat Mech - Spring 2006 Class 9
18
Enthalpy and Latent Heat
ion)(subllimat vapor tosolid
ion)(vaporizat vapor toliquid
(fusion) liquid tosolid
13
23
12
hh
hh
hh
Thermo & Stat Mech - Spring 2006 Class 9
21
Problem
Consider a sealed steel container completely filled with water at 0ºC and pressure of one atmosphere. Lower the temperature to – 1ºC. What happens? Water starts to freeze, but tries to expand. That raises pressure, so freezing point is lowered. How much?
Thermo & Stat Mech - Spring 2006 Class 9
22
Freezing Problem
K 1
K 273
/kgm 1005.9
J/kg 1034.3
)(
so )(
35
512
1212
12
T
T
vv
vvT
TP
vvTdT
dP
Thermo & Stat Mech - Spring 2006 Class 9
23
Freezing Problem
atm 135
atm 134Pa/atm 101.01
Pa 1035.15
7
P
P
How much freezes?Call the fraction that freezes x.
Thermo & Stat Mech - Spring 2006 Class 9
24
Freezing Problem
vdPvdTvv
dPP
vdT
T
vvv
vv
vvx
vxvxv
if
TPif
ff
fi
ffi
)1(
Thermo & Stat Mech - Spring 2006 Class 9
25
Freezing Problem
/kgm 10
/kgm 1005.9
Pa 1035.1 Pa 1012
K 1 K 1067
33
35
71-11
1-6
v
vv
P
T
vv
PvTv
vv
vvx
ff
ffff
fi