Part 1. Thermodynamics
Part 1. Thermodynamics
Chapter 1 The Properties of Gases
1A The Perfect Gas
1B The Kinetic Model
1C Real Gases
Part 1. Thermodynamics
1A The Perfect Gas
Ch. 1. The Properties of gases
Gas: the simplest model of matter
state
p, V, T, n
General form of an Equation of state
ex)
for perfect gas
p, V, T, n
p, T, V/n : Intensive property
not independent each other
(a) Pressure: P = F/A, 1Pa = 1N m-2
Mechanical
equilibrium
Standard pressure: 1 bar = 105 Pa
1A. Perfect gas
(b) Temperature: Property indicating direction of the flow of thermal energy
In thermal equilibrium:
The zeroth law of thermodynamics
Ex) Thermometer
Celsius scale: T/K =
+ 273.15
Perfect gas temperature scale
T = 273.16 p/p3
Thermodynamic temperature scale
T = (1- )Th
: heat efficiency
Perfect gas
The gas law: empirical basis
pV = constant at constant T, n
2)
law
pV = kNT = nRT
p1V1
n1T1
p2V2
n2T2
R
V=
at constant p, n
V=
at constant p, T
(limiting law) (N = nNA)
R: ideal gas constant
Standard condition:
Standard ambient condition:
0 C, 1 atm, Vm = 22.414 L mol-1
298.15 K, 1bar, Vm = 24.789 L mol-1
Perfect gas
Ex 1A.3 The mass percentage composition of dry air at sea level is approximately N2: 75.5; O2: 23.2;
Ar: 1.3. What is the partial pressure of each component when the total pressure is 1.20 atm?
wt(%)
mole
mole fraction( )
partial pressure
1B. The kinetic model
1B The Kinetic model of gases.
1. gas
molecules of mass m, in ceaseless random motion.
d
2. The size of molecules is negligible.
r << d
3. molecules
elastic collision
r
Total translational kinetic energy is conserved.
(a) Pressure and Molecular Speeds
·
(velocity) :
·
(speed) :
·
·
·
=(
(momentum) :
(force) :
(pressure) : p =
m =m
The kinetic model
pV =
: root-mean-square speed
(microscopic/molecular level)
(macroscopic/bulk properties) (p, T
Reaction
)
collision between molecules
Collision frequency, Z (collision # of unit time)
-
Newton
(distribution of molecular velocities)
.
:
- Distribution function
.
- Collision frequency
(reaction rate)
.
The kinetic model
= m)
-
t
·
=
t·A
·
N=
·
: (
t
·
momentum
=
t·A
The kinetic model
·
(wall)
:
F=
=
·p=
(
·
)
x, y, z
=
<
>=
p=
pV =
= nRT
=
The kinetic model
·
T
·
·
:
pV, T
)
<
>
:
?
The kinetic model
Maxwell- Boltzmann Distribution of speeds
cf)
decaying exponential function
the fraction of molecules with very high speeds
=> very small :
M is large, exp. factor goes most rapidly
towards zero
T is high, exp. factor falls towards zero slowly
as increases
factor in pre-exp.
This factor goes to zero as goes to zero,
=>
whatever their mass.
: normalizing factor
When we sum the fractions over the entire range
of speeds from zero to infinity, then we get 1.
. 1B.4
The kinetic model
The kinetic model
In Text. 1B.2 The Maxwell-Boltzmann distribution of speeds
by Bozmann distribution,
The kinetic model
Maxwell- Boltzmann distribution
isotropic distribution(
)
speed distribution
,
,
)=
d =
spherical polar coordinate
The kinetic model
By normalization condition
4
,
independent)
Maxwell- Boltzmann
Distribution of speeds
cf)
The kinetic model
Maxwell- Boltzmann Distribution of speeds
decaying exponential function
the fraction of molecules with very high speeds
=> very small :
M is large, exp. factor goes most rapidly
towards zero
T is high, exp. factor falls towards zero slowly
as increases
factor in pre-exp.
This factor goes to zero as goes to zero,
=>
whatever their mass.
: normalizing factor
When we sum the fractions over the entire range
of speeds from zero to infinity, then we get 1.
. 1B.4
The kinetic model
*definite integral
n = 1, 3, 5· · · ·
n = 2, 4, 6· · · ·
n=0
n=1
n=2
n=3
n=4
n=5
The kinetic model
Self-test 7 8.2 Evaluate the
root-mean -square speed of
the molecules by integration.
Use mathematical software
or use a standard integral in
the Resource section.
2
The kinetic model
example
Ex 1B.1, ST1B.2
Average molecular speed :
Root-mean-square speed :
Most probable speed
:
The kinetic model
*kinetic energy distribution: Skip
=
=
v=
, vdv =
f( ) d
)=
*< > =
for ideal monoatomic gas
no internal motion
The kinetic model
* For chemical reaction: Skip
>
let
)
erfc(
+ erfc(
)
The kinetic model
Collision diameter
2. Collision frequency : upper limit of reaction rate
: collision cross section, d: collision diameter
t
=
(B)
z : collision frequency
= N
A, B
(Fig. 1B. 8)
:
V=N
t
The kinetic model
: reduced mass
if, A=B
d=2
z
z
=
= 7.7
mean free path,
vrel
z
for N 2
: fleight length per collision
BI 1B.3, 1B.4
kT
p
1
p
9.5 10 8 m at 1 atm
The kinetic model
If T change in a vessel of constant V,
=?
The kinetic model
1C. Real Gases
deviation from the ideal gas law at high p and low T
Repulsive force
more expand in high p
Attractive force
more compress at low T
The compression factor, Z: to measure imperfection of gas
For ideal gas
Repulsion dominant at high p
Attraction dominant at low T
BI 1C.1
500 K, 100 bar,
Z=?
Real Gases
Virial equation of state
Z
Vm
Vmo
pVm
RT
1 B ' p C ' p2
1
B, C , D
B
Vm
C
Vm2
virial coefficients, function of T
Virial coefficients
to correct the small deviation from ideal gas
p
1/V
power series
https://blog.naver.com/PostView.nhn?isHttpsRe
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218876337
Table 1C.1
BI 1C.2
Real Gases
calculated pressure as 385 kPa for perfect gas
Real Gases
Virial coefficients:
to correct the small deviation from ideal gas
power series
p
1/V
Z
Vm
Vmo
pVm
RT
1 B ' p C ' p2
1
B, C , D
Table 1C.1
BI 1C.2
B
Vm
C
Vm2
virial equation of state
virial coefficients, function of T
C
Real Gases
for ideal gas,
dZ/dp=0 in all pressure range
Real Gases
Condensation of real gas
Real gas
Ideal gas
T2
T1
As p increase
A
B
C:
phase separation, L + V two phases
C D E
E:
E
no pressure change, vapor
one phase
F:
liquid
entirely liquid
no volume change with increase of p
Real Gases
Critical point(
T1
T2:
)
CDE
TC :
phase
Liq.
(meniscus)
.
.
gas phase vs vapor
vapor
T > TC: liquid
Critical point: TC, pC, VC
Super critical fluid(
) : dense fluid in T > TC, not liquid
Real Gases
Van der Waals Equation: simple physical approach
1) Effect of molecular size, b
: Vm- b
p(Vm- b) = RT
Vm = RT/p + b
1 + bp/RT > 1
2) Attractive interaction : two body interaction
N / V particle
Z
N
,
p a(
n2
) V
V2
pVm / RT
1
p
1
N(N 1) pair interaction
2V 2
1
1
N(N 1)
N 2 (n2 / V 2 )
2
2
2V
2V
a
RTVm
RT ,
p
RT / Vm a / Vm2
1
Real Gases
b: excluded volume
Real Gases
Vm3
V
b
RT
Vm2
p
p
a
Vm
p
RT
(Vm
ab
p
0
3
a
b) Vm2
1) at high T , large Vm (low p)
Vm
b,
p
RT / Vm
2) at low T
p
a / Vm2
RT /(Vm b)
min/max
Vm
unrealistic behavior
liquid vapor
van der waals loop => equal area
3) at TC
inflection point ==> critical point
Real Gases
(3) at
inflection point,
,
·······
at
·······
·······
Z at critical point
Real Gases
2(C) The principles of corresponding states
for general relation of v, d, w
molecule
a, b
critical point
: reduced variable
v, d, w
[
reduced variable
+3
,
][3(
spherical, non-polar molecule
principles of corresponding states
Real Gases
Real Gases
Real Gases