thermodynamics and statistical mechanics equations of state
TRANSCRIPT
Thermodynamics and Statistical Mechanics
Equations of State
Thermo & Stat Mech - Spring 2006 Class 2
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Thermodynamic quantities
Internal energy (U): the energy of atoms or molecules that does not give macroscopic motion.
Temperature (T): a measure of the internal energy of a system.
Heat (Q): a way to change internal energy, besides work. (Energy in transit.)
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Laws of Thermodynamics
First law: đQ – đW = dU
Q – W = U
Energy is conserved
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Work done by a gas
f
i
V
VPdVW
PdVdW
AdsAF
dW
FdsdW
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Work done by a gas
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Work done by a gas
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Configuration Work
Product of intensive variable times corresponding extensive variable:
đW = xdY
Gas, Liquid, Solid: PdV
Magnetic Material: BdM
Dielectric Material: EdP
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Equation of State
For an ideal gas: PV = nRT
P = pressure (N/m²)(or Pa)
V = volume (m³)
n = number of moles
T = temperature (K)
R = gas constant (8.31 J/(K·mole))
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Ideal gas law
Ideal gas law:PV = nRT
In terms of molar volume, v = V/n, this becomes:
Pv = RT, or P = RT/v
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Real Substance
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Real Substance
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van der Waals equation of state
v cannot be decreased indefinitely, so replace v by v – b. Then,
Next account for intermolecular attraction which will reduce pressure as molecules are forced closer together. This term is proportional to v-2
bvRT
P
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van der Waals equation of state
RTbvva
P
va
bvRT
P
2
2 or , Then,
This equation has a critical value of T which suggests a phase change. The next slide shows graphs for several values of T .
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van der Waals equation of state
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Critical Values
227
278
3
ba
P
Rba
T
bv
C
C
C
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van der Waals equation of state
'38
31
''3
'
Then,
' ,' ,'
2 Tvv
P
TTTPPPvvv CCC
This can be expressed in term of dimensionlesscoordinates, P', v', and T ' with the following Substitutions:
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van der Waals equation of state
This can also be written,
2'3
1'3'8
'vv
TP
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Thermal Expansion
Expansivity or Coefficient of Volume Expansion, .
TVTTV
V
PT
Tv
vTV
V
P
PP
),(
11
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Thermal Expansion
Usually, is positive.
An exception is water in the temperature range between 0° C and 4° C.
Range of is about:
10-3 for gasses.
10-5 for solids.
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Linear Expansion
Coefficient of Linear Expansion, .
TXTTX
X
pT
TX
X
P
P
),(
1
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Relationship Between and
3
)31()1(
)1()1()1('
)1('
3
TVTXYZ
TZTYTXV
XYZV
TVTVVVVV
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Compressibility
Volume also depends on pressure.
Isothermal Compressibility:
PVPPV
V
PTPV
V
T
T
),( 1
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Bulk Modulus
TVP
V
1
ModulusBulk
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A Little Calculus
0
0 const, If,
),( Consider,
VT
VP
TP
dPPV
dTTV
dVV
dPPV
dTTV
dV
PTV
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Cyclical Relation
1
0
PVT
VTP
VTP
VT
TP
PV
TP
PV
TV
TP
PV
TV
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Application
Suppose you need:VT
P
1
PVT VT
TP
PV
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Application
T
P
T
P
PT
V
PV
V
TV
V
PVTV
VT
PVT
P1
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