Kinetic Molecular Theory of Ideal Gases - WordPress.com · 01.07.2010 · Kinetic Molecular Theory...
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Kinetic Molecular Theory of Ideal Gases
Theoretical development of ideal gas laws that were determined empiricallyBernoulli et al. (1738)
Main Postulates1. Gas molecules in ceaseless chaotic motion.
2. Pressure (P = f/A) exerted on the container walls is due to the bombardment of the container by the gas molecules.
3. All molecular collisions are elastic, i.e., no energy loss due to friction.
4. No intermolecular forces.
5. Molecules are “point masses”, i.e., infinitesimally small molecular volumes.
6. Absolute T is proportional to the average kinetic energy of all the molecules.
PV = nRT
Boyle
(1627 – 1691)
PV = k1, [n,T]
Charles
(1746-1823)
V/T = k2, [n,P]
Avogadro
(1776 – 1856)
V/n = k3, [P,T]
Gay-Lussac
(1778 – 1850)
1 © Prof. Zvi C. Koren 21.07.10Gas Problems: Ideal Gases: 1-13.
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N identical gas molecules
m = mass of each molecule
ii
molecule of vector velocity v
kji iziyix
v v v
2222 v v v v iziyixi
.v v v v Velocity SquareMean 2222zyx
N
v
v v Averageor Mean
N2
22
iix
xx
v v v :motion randomFor 222zyx
v3 v Velocity SquareMean 22x
2222v v v v :Note zyx
Mean (or Average) Velocities
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The Model
ℓ x
A
For simplicity, first consider:
1. Only one molecule (i) is present.
2. Molecule’s motion is only in x-direction:
only vix component.
& molecule i will be hitting wall A with velocity vix.
Then, we’ll consider N molecules.
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ℓ x
A
fix = force exerted by molecule i on wall
fix
Note: force leads to a change of velocity (and momentum) upon collision with the wall
Rate of change of momentum
t
p
t
)(mv
t
vm ma ixixix
ix
(Newton)
For each collision cycle:
A
pix = m(+vix) – m(-vix)= 2mvix
t = d/v = 2ℓ/vix
2
ix
ix
ixixix
mv
/v2
2mv
t
p f
So, for one molecule i:
For N molecules (continued):4 © Prof. Zvi C. Koren 21.07.10
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2
ix
ix
ixixix
mv
/v2
2mv
t
p f
For N molecules:
So, for one molecule i (from before):
2
xvNm
vm
f f 2ix
N
iixxtotal,
N
v
v
N
i
2
2
ix
x
22
xv3 v
V
vNm
A
vNm
A
f P
22
31
31
totaltotal
2vNm PV3
1 (continued)
2vNm
3
1 f total
(“correction” of the assumption that
all movement is only in x-direction)
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(continued) 2v m N PV3
1
N = # of molecules = n·NAvogadro, NA = 6.02x1023 molecules/mol
M = molar mass [g/mol] = NA·m
mtotal = N·m = n·M
2v M n3
1 PV
TB
k2
3 KE
,
AN
R
constantBoltzmann Bk
exp KMTKMT
M
3RT v
1/21/22
vrms = root mean square velocity
(Note the units)
Graham’s Lawof Diffusion and Effusion(for 2 gases at the same T):
2/1
1
2
2
1
1
2
M
M
R
R
t
t
nRT PV KE A
nN3
2 PV
R = 8.31 J/molK
totalKE
3
2
3
2
3
1 KE N v m ½)(2 N PV 2
“½kT” = basic unit of molecular energy
for each independent motion
kT2
1 3
(Energy of translation)
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Boyle: PV = k1, [N,T]
Charles & Gay-Lussac: V/T = k2, [N,P]
Avogadro: V/N = k3, [P,T]
T 3
2 KE N PV
From KMT
All the empirical gas laws can be derived:
7 © Prof. Zvi C. Koren 21.07.10
Recall:
At what conditions of T or P, does a real gas behave as if it were ideal?
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Maxwell-Boltzmann Distribution of Molecular Velocities
= f (T, MW)
MW Effects:
> >
(continued)
Why do molecules, all at the same T, have such a wide span of velocities?
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Temperature Effects 1:
(continued)9 © Prof. Zvi C. Koren 21.07.10
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Temperature Effects 2:
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11 © Prof. Zvi C. Koren 21.07.10
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RT/M2 vmp vmp
3RT/M v3/2 2 vv mp
1/2
rms
mpv4/π v
Most-Probable, Mean, and Root-Mean-Square Velocities
v
#
Maxwell’s Distribution of Speeds:
RTMvevRT
Mvf 2/2
2/32
24)(
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The mean speed with which one molecule approaches another
identical molecule (exact derivation is too cumbersome)
v2 vrel rendition:Qualitative
8kT/ vrel
For two dissimilar molecules approaching each other:
mm
mm μ
BA
BA
Relative Mean Speed:
reduced mass
:vrel
Oneextreme Typical
Anotherextreme
13 © Prof. Zvi C. Koren 21.07.10
v from before
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Collision Frequency (z) & Collision Diameter (d)
z = Average # of collisions per second made by one molecule in a
system of N molecules in a volume V:
AR/N k :recall gas), idealan (for kTP/relv z
At constant T, z P. Logical?
Collision Cross-Sections
/(nm)2Gas
0.88C6H6
0.52CO2
0.21He
0.43N2
Example:
For an N2 molecule at 1 atm and 25oC,
z 7 x 109 s-1
= ·d2 = collision cross-section(target area that a molecule presents to an incoming molecule)
d
For a sample held at constant volume, as T increases, z ______________
N/Vv z rel
inc. bec. vrel inc.
A “hit” occurs when the centers of two molecules come within a
distance “d” of each other, where “d” is the diameter of impenetrable
hard sphere molecules.
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Mean Free Path, = The average distance a molecule travels between collisions
P)σ2kT/( /zv λ
For N2 at 1 atm: = 70 nm 103 molecular diameters
In a container of fixed volume, is dependent on T?
Summary
• A typical ideal gas molecule (N2 or O2) at 1 atm and 25oC travels at a
mean speed of 350 m/s;
• Each molecule collides within 1 ns,
• Between collisions it travels 102 – 103 molecular diameters.
• If d << , gas is nearly ideal. Why?
tfree = Time in free flight between collisions = 1/z, z = collision frequency
= Average distance traveled in free flight = freetv
15 © Prof. Zvi C. Koren 21.07.10
Gas Problems: KMT: 24, 26-31.
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λ/dλ
/(nm)
tfree
/(ns)
z
/(s–1)/(m/s)
/(nm)2d
/(nm)vrms
/(m/s)/(m/s)
vmp
/(m/s)Gas
C6H6
CO2
He
N2
16 © Prof. Zvi C. Koren 21.07.10
Lecture Problem #1 (to hand in next week to the Recitation instructor):
Fill in the table below at 1 atm and 25 oC:
(see next slide)
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17 © Prof. Zvi C. Koren 21.07.10
Give the full reference for “d”
(author’s last name & first name initial, book title, publisher, city of
publication, year, page number; and exact website address, if from internet).
Explain all the values and interpret them.
Careful with all the units.
Don’t forget to convert “atm” to “Pascal” and that (nm)2 is 10–18 m2.
:בשני הספרים האלה dהנתונים לגבי קוטר מולקולרי http://books.google.com/books?id=vYyjL3MJq2YC&pg=PA202&lpg=PA202&dq=benzene+collision+diameter&sourc
e=bl&ots=_jAAirau-
o&sig=9y8rVIdx9n7F5EfgkhLpFAzjpso#v=onepage&q=benzene%20collision%20diameter&f=false
Bלפי פאראמטר )יש גם טבלה בספר הקלאסי הבא שחישב את הקוטר המולקולרי לפי שתי שיטות
שהיא σאבל תזהרו מהטבלה מכיוון שהוא מסמל את הקוטר עם אות , )של הגז )η)ולפי הצמיגות
.תראו תנסו ותהנו .משהו אחר במשוואות שלנוhttp://www.kayelaby.npl.co.uk/general_physics/2_2/2_2_4.html
Notes About the Previous Table: