物理化学英文课件
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1000
500
y'
0
y"
-500
-10
-5
0
5
10
x
Inflection Points
At the inflection point x0
dy 0 dx xx0
and
d2y dx 2
xx0
0
Carbon Dioxide Critical Isotherm
600
304 K
500
400
300 Ideal Gas
• molar volume
Vm= V/n
1m
1m 1m
• Force/Area
Pressure
• units
– Pascal = Newton/m2 = Joule/m3
h
– atmosphere = 101325 Pa
– bar = 100000 Pa
– mm Hg
– torr
Temperature
• Advantages
– fits gas data as accurately as desired. – uses the ideal gas law as a base.
• Disadvantages
– infinite number of terms. – virial coefficients are temperature dependent.
a (dm6 atm mole-1) b (dm mole-1)
He
0.034
0.0237
Ar
1.345
0.0322
N2
1.390
O2
1.360
CO2 3.592
0.0391 0.0318 0.0427
Successive Approximation
Vm
RT p a
b
V2 m
• “Solve” the van der Waals equation for V.
-2
-3
1.5
2.0
attractive
2.5
3.0
3.5
4.0
distance
Taylor Series
f(
x)
=
f(
0)
+
f
(0)x
+
1 2
f
(0)x2
where f (0) = df and f (0) = dx x0
d2 f dx 2
x0
Taylor Series Example
f(x) 1 f(0) 1 1 x
Compressibility
• The compressibility of a gas is defined by
Z pVm RT
• If the gas behaves ideally, then Z=1 at all pressures and temperatures.
• For real gases, however, Z varies with pressure, and deviates from its ideal value
pAV nART
• pA = partial pressure of gas A • V = total volume • nA = moles of gas A • T = temperature • R = universal gas contant = 0.08206 L-atm/mol-K
Dalton’s Law
• The total pressure is the sum of all the partial pressure.
J
pJ
J
nJ RT RT VV
J
nJ
nRT p V
Ideal Gas Model
• Molecules may be treated as point masses relative to the volume of the system.
The Properties of Gases
Equations of State
Bulk Variables
• Volume - m3 • Pressure - Pa • Temperature - K • Composition - moles
• length3
Volume
• units
– m3 or cm3 – liter = 1000 cm3
• thermometry T ~ Volume
• only for an ideal gas
• or real gases in the low pressure limit
Composition
• moles: ni S ni = n
• mole fraction: xi S xi = 1
• partial pressure: pi S pi = p
van der Waals Equation
p a / Vm2 Vm b nRT
repulsion attraction
Vm,eff Vm b peff p a / Vm2
van der Waals Equation
p
RT Vm
b
a Vm2
van der Waals constants
Virial Expansion
Z( p) Z(0) Bp Cp2
pVm RT 1 Bp Cp2
where B second virial coefficient C third virial coefficient
Virial Expansion
Z (1/Vm ) Z (0) B(1/Vm ) C(1/Vm )2
p V T c (atm)
m,c (cm3)
c (K)
He 2.26 57.76 5.2
Ar 48.00 75.25 150.7
N2 33.54 O2 50.14 CO2 72.85
90.10 126.3 78.00 154.8 94.0 304.2
Inflection Points
1500
y (x)
pVm RT 1 B /Vm C /Vm2
where B = second virial coefficien t C = third virial coefficien t
Relation of p and 1/V Expansions
B B RT
C
C B2
RT 2
Second Virial Coefficients B
Argon Compressibility 273 K
2.5
Z
2.0
Z = pVm/RT
1.5
1.0
attractive
0.5
repulsive
0.0
0
200
400
600
800
1000
pressure (atm)
Intermolecular Forces
4 3
2 1
Energy
0
-1
repulsive
f (x)
1
1 x2
f (0)
1
f (x)
2
1 x3
f (0)
2
1 1 x x2 x3 1- x
Convergence of (1-x)-1
x 1+x 1+x+x2 1+x+x2+x3 (1-x)-1 0.01 1.010 1.010 1.010 1.010 0.10 1.100 1.110 1.111 1.111 0.20 1.200 1.240 1.248 1.250 0.30 1.300 1.390 1.417 1.429
• Molecular collisions are elastic, i.e. kinetic energy is conserved.
• Intermolecular forces of attraction and repulsion have negligible on the molecular motion.
Definition of Physical Chemistry
Physical chemistry stands in the same relation to the subdivisions of chemistry in which philosophy stands toward all the sciences. Its main object is to unify thought within the science of chemistry; therefore, it might well be named, the “philosophy of chemistry” - S.L. Bigelow, 1912
200
150
4000K
100
2000K
50
1000K
500K
100K
0
0.0
1.0
2.0
3.0
4.0
5.0
Vm/L
van der Waals Isotherms - Ar
200
500K
150
200K
100
150K
50
0
100K
-50
-100
0.00
0.10
0.20
0.30
0.40
Vm/L
Critical Constants
van der Waals Virial Expansion
1 Vm
b
1 Vm
1
b Vm
b2 Vm2
p
RT Vm
1
b
a/ Vm
RT
b2 Vm2
100 K 273 K 373 K 600 K
He 11.4 12.0 11.3 10.4
Ar -187.0 -21.7 -4.2 11.9
N2 -160.0 -10.5 6.2 21.7
O2 -197.5 -22.0 -3.7 12.9
CO2
-149.7 -72.2 -12.4
cm3/mole
Virial Expansion
Equations of State
• P,V,T, and n are not independent. • Any three will determine the fourth. • An equation of state is an equation that
relates P,V,T, and n for a given substance. • Gases have the simplest equations of state. • The simplest equation of state is the ideal
• Use an intial estimate to evaluate the right hand side.
• Use this calculated value of V as a better estimate.
• Repeat till converged.
Ideal Gas Isotherms
200Real Gas源自10000.0
0.1
0.2
0.3
0.4
Vm /L
van der Waals Inflection Point
dp dVm
RT (Vm b)2
2a Vm3
0
d2p dVm2
2RT (Vm b)3
6a Vm4
0
solution: Vm,c 3b and pc a / 27b2
gas law, pV = nRT
Ideal Gas Law
pV nRT
• p = pressure • V = volume • n = moles • T = temperature • R = universal gas contant = 0.08206 L-atm/mol-K
Partial Pressure
500
y'
0
y"
-500
-10
-5
0
5
10
x
Inflection Points
At the inflection point x0
dy 0 dx xx0
and
d2y dx 2
xx0
0
Carbon Dioxide Critical Isotherm
600
304 K
500
400
300 Ideal Gas
• molar volume
Vm= V/n
1m
1m 1m
• Force/Area
Pressure
• units
– Pascal = Newton/m2 = Joule/m3
h
– atmosphere = 101325 Pa
– bar = 100000 Pa
– mm Hg
– torr
Temperature
• Advantages
– fits gas data as accurately as desired. – uses the ideal gas law as a base.
• Disadvantages
– infinite number of terms. – virial coefficients are temperature dependent.
a (dm6 atm mole-1) b (dm mole-1)
He
0.034
0.0237
Ar
1.345
0.0322
N2
1.390
O2
1.360
CO2 3.592
0.0391 0.0318 0.0427
Successive Approximation
Vm
RT p a
b
V2 m
• “Solve” the van der Waals equation for V.
-2
-3
1.5
2.0
attractive
2.5
3.0
3.5
4.0
distance
Taylor Series
f(
x)
=
f(
0)
+
f
(0)x
+
1 2
f
(0)x2
where f (0) = df and f (0) = dx x0
d2 f dx 2
x0
Taylor Series Example
f(x) 1 f(0) 1 1 x
Compressibility
• The compressibility of a gas is defined by
Z pVm RT
• If the gas behaves ideally, then Z=1 at all pressures and temperatures.
• For real gases, however, Z varies with pressure, and deviates from its ideal value
pAV nART
• pA = partial pressure of gas A • V = total volume • nA = moles of gas A • T = temperature • R = universal gas contant = 0.08206 L-atm/mol-K
Dalton’s Law
• The total pressure is the sum of all the partial pressure.
J
pJ
J
nJ RT RT VV
J
nJ
nRT p V
Ideal Gas Model
• Molecules may be treated as point masses relative to the volume of the system.
The Properties of Gases
Equations of State
Bulk Variables
• Volume - m3 • Pressure - Pa • Temperature - K • Composition - moles
• length3
Volume
• units
– m3 or cm3 – liter = 1000 cm3
• thermometry T ~ Volume
• only for an ideal gas
• or real gases in the low pressure limit
Composition
• moles: ni S ni = n
• mole fraction: xi S xi = 1
• partial pressure: pi S pi = p
van der Waals Equation
p a / Vm2 Vm b nRT
repulsion attraction
Vm,eff Vm b peff p a / Vm2
van der Waals Equation
p
RT Vm
b
a Vm2
van der Waals constants
Virial Expansion
Z( p) Z(0) Bp Cp2
pVm RT 1 Bp Cp2
where B second virial coefficient C third virial coefficient
Virial Expansion
Z (1/Vm ) Z (0) B(1/Vm ) C(1/Vm )2
p V T c (atm)
m,c (cm3)
c (K)
He 2.26 57.76 5.2
Ar 48.00 75.25 150.7
N2 33.54 O2 50.14 CO2 72.85
90.10 126.3 78.00 154.8 94.0 304.2
Inflection Points
1500
y (x)
pVm RT 1 B /Vm C /Vm2
where B = second virial coefficien t C = third virial coefficien t
Relation of p and 1/V Expansions
B B RT
C
C B2
RT 2
Second Virial Coefficients B
Argon Compressibility 273 K
2.5
Z
2.0
Z = pVm/RT
1.5
1.0
attractive
0.5
repulsive
0.0
0
200
400
600
800
1000
pressure (atm)
Intermolecular Forces
4 3
2 1
Energy
0
-1
repulsive
f (x)
1
1 x2
f (0)
1
f (x)
2
1 x3
f (0)
2
1 1 x x2 x3 1- x
Convergence of (1-x)-1
x 1+x 1+x+x2 1+x+x2+x3 (1-x)-1 0.01 1.010 1.010 1.010 1.010 0.10 1.100 1.110 1.111 1.111 0.20 1.200 1.240 1.248 1.250 0.30 1.300 1.390 1.417 1.429
• Molecular collisions are elastic, i.e. kinetic energy is conserved.
• Intermolecular forces of attraction and repulsion have negligible on the molecular motion.
Definition of Physical Chemistry
Physical chemistry stands in the same relation to the subdivisions of chemistry in which philosophy stands toward all the sciences. Its main object is to unify thought within the science of chemistry; therefore, it might well be named, the “philosophy of chemistry” - S.L. Bigelow, 1912
200
150
4000K
100
2000K
50
1000K
500K
100K
0
0.0
1.0
2.0
3.0
4.0
5.0
Vm/L
van der Waals Isotherms - Ar
200
500K
150
200K
100
150K
50
0
100K
-50
-100
0.00
0.10
0.20
0.30
0.40
Vm/L
Critical Constants
van der Waals Virial Expansion
1 Vm
b
1 Vm
1
b Vm
b2 Vm2
p
RT Vm
1
b
a/ Vm
RT
b2 Vm2
100 K 273 K 373 K 600 K
He 11.4 12.0 11.3 10.4
Ar -187.0 -21.7 -4.2 11.9
N2 -160.0 -10.5 6.2 21.7
O2 -197.5 -22.0 -3.7 12.9
CO2
-149.7 -72.2 -12.4
cm3/mole
Virial Expansion
Equations of State
• P,V,T, and n are not independent. • Any three will determine the fourth. • An equation of state is an equation that
relates P,V,T, and n for a given substance. • Gases have the simplest equations of state. • The simplest equation of state is the ideal
• Use an intial estimate to evaluate the right hand side.
• Use this calculated value of V as a better estimate.
• Repeat till converged.
Ideal Gas Isotherms
200Real Gas源自10000.0
0.1
0.2
0.3
0.4
Vm /L
van der Waals Inflection Point
dp dVm
RT (Vm b)2
2a Vm3
0
d2p dVm2
2RT (Vm b)3
6a Vm4
0
solution: Vm,c 3b and pc a / 27b2
gas law, pV = nRT
Ideal Gas Law
pV nRT
• p = pressure • V = volume • n = moles • T = temperature • R = universal gas contant = 0.08206 L-atm/mol-K
Partial Pressure