Lecture 1: Gasses
The Ideal Gas Equation
An ideal gas is a gas with no intermolecular interaction and with negligible volume.
P = pressure (Pa)
PV =nRT V = volume (m3)
n = amount of moles
Slide 16
RT R = gas constant (8.3145
v m=
P J/mol/K)
T = temperature (K)
The ideal gas equation is a combination of three different laws. The Boyle’s law is the
first law which states: At a constant temperature, the pressure of a fixed amount of
gas inversely proportional to its volume. The second law is Charles’s law which
states: At constant pressure, the volume of a fixed amount of gas is proportional to
the absolute temperature. The third feature is Avogadro’s principle which states: At a
given temperature and pressure, qual volumes of gas contain the same number of
molecules.
Density
nM pM M
ρ= = molar volume V m =
V RT ρ
Pressure in liquid (hydrostatic)
- p= ρ∙ g ∙h
Dalton’s Law: Partial pressures
When w/w% is given: calculate according to Lesson 5, exercise 5. Take any amount in
grams and divide with MW.
When v/v% is given: multiply the percentage factor with the density to get the
amount in kg L-1. If your total volume is a bottle of 1 L for example, then you know
your weight in grams and divide it with the MW to get the amount of moles.
Ptot = total pressure (Pa)
Ptot =P A + P B +etc PA,B,ETC = partial pressure (Pa) Slide 20
P A =Ptot ∙ X A PA = partial pressure (Pa)
Xa = mol fraction of component a
na
X A= na = amount molecules a Book page 11
ntot ntot = total amount molecules
Boyl e ' s Law at constant T :
p1 V 1=p 2 V 2=nRT
p1 V 2 V V
; = ; p2= p1 ∙ 1 ∨ p1 /( 2 )
p2 V 1 V2 V1
'
Boyl e s Law bij niet constante T :
1
, T2
p1 V 1 p2 V 2 /V
p1 V 1=nR T 1 p2 V 2=nR T 2 n= = p2 T1 2
R T1 R T2 =
p1 V1
Real gasses
Compressibility factor z=1 for ideal gasses
P = pressure (Pa)
V = volume (m3)
PV z = empirical factor
Z= Slide 24
nRT R = gas constant (8.3145
J/mol/K)
T = temperature (K)
Virial coefficient + Van der Waals equation
a = attractive forces between
2
n molecules ((Pa*m6)/mol2)
( ( ))
p+ a
V
( V −nb )=nRT b = finite size of molecules
(m3/mol)
Slide 25 – 28
Gas kinetica
In the gas kinetic model theory we assume that molecules only interact during elastic
collisions.
Ekin = kinetic energy of a mole (J)
mi = mass particle (kg)
3 vi = speed particle (m/s)
Slide 45
Ekin =0.5 ∙m i ∙ v i= k b T kb = Boltzmann constant (1,38 *
2 Slide 58
10-23 J/K)
T = temperature (K)
F = force (N)
F=m∙ a m = mass (kg) Slide 45
a = acceleration (m/s2)
8 RT 1 /2
v mean=v = ( )
πM M = mol mass (kg/mol)( 10-3
Slide 55
3 RT 1/ 2 g/mol)
v rms= ( )
M vmean = speed (m/s)
RT = 8,314 * temperature (K)
Slide 58
1 2
PV = nM v rms Slide 58
3
Collision Cross Selection
2 σ = cross colission section (m2)
σ =π d Slide 61
d = diameter molecule (m)
V coll =σ ∙ √ 2∙ v Vcoll = speed at collision (m/s0 Slide 62
p∙ V coll v ∙σ ∙ p ∙ √ 2 vmean = speed (m/s)
z= = p = pressure (Pa) Slide 63
k bT k b ∙T
kb = Boltzmann constant (1,38 *
v k ∙T 10-23 J/K)
λ= = b Slide 63
z p ∙ σ ∙ √2 λ = mean free path (m)
Diffusion
2
, 1 λ = mean free path (m)
D i= ∙ v ∙ λ vmean = speed (m/s) Slide 74
3
λ decreases with P Diffusion is slower at higher P
λ and vmean decreases with size Diffusion is slower for large molecules
vmean increases with T1/2 diffusion is faster with T
Flux J in gassen (transport 3 soorten)
1) Matter (molecular diffusion)
1
Di= λ∙ v mean met Di=diffusion coefficient ∈m 2 s−1
3
2) Heat conduction (energy)
kb
κ= ∙ v mean met κ=thermal conductivity ∈J K −1 m −1 s−1
2 √2 σ
3) Momentum (viscosity of gasses)
1
η= λ ∙ v mean ∙ ρ met η=viscosity ∈Pa ∙ s
3
Carnot Cycle
Purpose: Proofs the impossibility of any system to transform the heat in work in order
to achieve an efficiency.
Efficiency of a Tc
η=1−
system Th η = efficiency
Tc Tc = cooling
Efficiency of a
η= temperature (K)
refrigerator T h−T c Th = heat
Efficiency of a Tc temperature (K)
η=
heat pump T h−T c
3
, Lecture 2: The First Law of Thermodynamics
The first law of thermodynamics: The internal energy of an isolated system is
constant.
Internal energy (U)
The capacity of a system to do work or to transfer heat to the surroundings. Grand
total of all the kinetic and potential energies.
Constant T, perfect gas:
Closed system: ΔUsys = -ΔUsur
ΔUsys = q + w
Isolated system: ΔUsys = 0
q = -w
Constant V, no non-expansion work:
If a reaction is carried out in a container of constant volume, the system can do no
expansion work.
Therefore, w=0. Therefore,
ΔU=q
Work
The mode of transfer of energy that achieves or utilizes uniform motion in the
surroundings.
Expansion work
w=( pext ∙ A ) ⋅ h=−P ext ∙ ∆ V
Pext = External pressure (Pa)
h = distance (height)
A = Area
Pext**A=force
A*h=∆ V
∆ V = volume change (m3)
- Free expansion: When the external pressure is zero, then w=0. The system
does no work as it expands.
- At constant volume, w=0 (because ΔU=q)
- Maximum work is done when the external pressure is only infinitesimally less
than the pressure of the gas in the system (=mechanical equilibrium,
maximum expansion work).
Work of reversible isothermal expansion (Perfect gas, constant T)
V2
w=−nRT ln ( )
V1
Work Energy System Volume
Negative Lost Work done by Expansion
Positive Gained Work done on Compression
4
The Ideal Gas Equation
An ideal gas is a gas with no intermolecular interaction and with negligible volume.
P = pressure (Pa)
PV =nRT V = volume (m3)
n = amount of moles
Slide 16
RT R = gas constant (8.3145
v m=
P J/mol/K)
T = temperature (K)
The ideal gas equation is a combination of three different laws. The Boyle’s law is the
first law which states: At a constant temperature, the pressure of a fixed amount of
gas inversely proportional to its volume. The second law is Charles’s law which
states: At constant pressure, the volume of a fixed amount of gas is proportional to
the absolute temperature. The third feature is Avogadro’s principle which states: At a
given temperature and pressure, qual volumes of gas contain the same number of
molecules.
Density
nM pM M
ρ= = molar volume V m =
V RT ρ
Pressure in liquid (hydrostatic)
- p= ρ∙ g ∙h
Dalton’s Law: Partial pressures
When w/w% is given: calculate according to Lesson 5, exercise 5. Take any amount in
grams and divide with MW.
When v/v% is given: multiply the percentage factor with the density to get the
amount in kg L-1. If your total volume is a bottle of 1 L for example, then you know
your weight in grams and divide it with the MW to get the amount of moles.
Ptot = total pressure (Pa)
Ptot =P A + P B +etc PA,B,ETC = partial pressure (Pa) Slide 20
P A =Ptot ∙ X A PA = partial pressure (Pa)
Xa = mol fraction of component a
na
X A= na = amount molecules a Book page 11
ntot ntot = total amount molecules
Boyl e ' s Law at constant T :
p1 V 1=p 2 V 2=nRT
p1 V 2 V V
; = ; p2= p1 ∙ 1 ∨ p1 /( 2 )
p2 V 1 V2 V1
'
Boyl e s Law bij niet constante T :
1
, T2
p1 V 1 p2 V 2 /V
p1 V 1=nR T 1 p2 V 2=nR T 2 n= = p2 T1 2
R T1 R T2 =
p1 V1
Real gasses
Compressibility factor z=1 for ideal gasses
P = pressure (Pa)
V = volume (m3)
PV z = empirical factor
Z= Slide 24
nRT R = gas constant (8.3145
J/mol/K)
T = temperature (K)
Virial coefficient + Van der Waals equation
a = attractive forces between
2
n molecules ((Pa*m6)/mol2)
( ( ))
p+ a
V
( V −nb )=nRT b = finite size of molecules
(m3/mol)
Slide 25 – 28
Gas kinetica
In the gas kinetic model theory we assume that molecules only interact during elastic
collisions.
Ekin = kinetic energy of a mole (J)
mi = mass particle (kg)
3 vi = speed particle (m/s)
Slide 45
Ekin =0.5 ∙m i ∙ v i= k b T kb = Boltzmann constant (1,38 *
2 Slide 58
10-23 J/K)
T = temperature (K)
F = force (N)
F=m∙ a m = mass (kg) Slide 45
a = acceleration (m/s2)
8 RT 1 /2
v mean=v = ( )
πM M = mol mass (kg/mol)( 10-3
Slide 55
3 RT 1/ 2 g/mol)
v rms= ( )
M vmean = speed (m/s)
RT = 8,314 * temperature (K)
Slide 58
1 2
PV = nM v rms Slide 58
3
Collision Cross Selection
2 σ = cross colission section (m2)
σ =π d Slide 61
d = diameter molecule (m)
V coll =σ ∙ √ 2∙ v Vcoll = speed at collision (m/s0 Slide 62
p∙ V coll v ∙σ ∙ p ∙ √ 2 vmean = speed (m/s)
z= = p = pressure (Pa) Slide 63
k bT k b ∙T
kb = Boltzmann constant (1,38 *
v k ∙T 10-23 J/K)
λ= = b Slide 63
z p ∙ σ ∙ √2 λ = mean free path (m)
Diffusion
2
, 1 λ = mean free path (m)
D i= ∙ v ∙ λ vmean = speed (m/s) Slide 74
3
λ decreases with P Diffusion is slower at higher P
λ and vmean decreases with size Diffusion is slower for large molecules
vmean increases with T1/2 diffusion is faster with T
Flux J in gassen (transport 3 soorten)
1) Matter (molecular diffusion)
1
Di= λ∙ v mean met Di=diffusion coefficient ∈m 2 s−1
3
2) Heat conduction (energy)
kb
κ= ∙ v mean met κ=thermal conductivity ∈J K −1 m −1 s−1
2 √2 σ
3) Momentum (viscosity of gasses)
1
η= λ ∙ v mean ∙ ρ met η=viscosity ∈Pa ∙ s
3
Carnot Cycle
Purpose: Proofs the impossibility of any system to transform the heat in work in order
to achieve an efficiency.
Efficiency of a Tc
η=1−
system Th η = efficiency
Tc Tc = cooling
Efficiency of a
η= temperature (K)
refrigerator T h−T c Th = heat
Efficiency of a Tc temperature (K)
η=
heat pump T h−T c
3
, Lecture 2: The First Law of Thermodynamics
The first law of thermodynamics: The internal energy of an isolated system is
constant.
Internal energy (U)
The capacity of a system to do work or to transfer heat to the surroundings. Grand
total of all the kinetic and potential energies.
Constant T, perfect gas:
Closed system: ΔUsys = -ΔUsur
ΔUsys = q + w
Isolated system: ΔUsys = 0
q = -w
Constant V, no non-expansion work:
If a reaction is carried out in a container of constant volume, the system can do no
expansion work.
Therefore, w=0. Therefore,
ΔU=q
Work
The mode of transfer of energy that achieves or utilizes uniform motion in the
surroundings.
Expansion work
w=( pext ∙ A ) ⋅ h=−P ext ∙ ∆ V
Pext = External pressure (Pa)
h = distance (height)
A = Area
Pext**A=force
A*h=∆ V
∆ V = volume change (m3)
- Free expansion: When the external pressure is zero, then w=0. The system
does no work as it expands.
- At constant volume, w=0 (because ΔU=q)
- Maximum work is done when the external pressure is only infinitesimally less
than the pressure of the gas in the system (=mechanical equilibrium,
maximum expansion work).
Work of reversible isothermal expansion (Perfect gas, constant T)
V2
w=−nRT ln ( )
V1
Work Energy System Volume
Negative Lost Work done by Expansion
Positive Gained Work done on Compression
4
