Kinetic Model for Ideal Gas
Three assumptions:
•(I) The gas consists of molecules of mass m in
ceaseless random motion.
•(ii) The size of the molecules is negligible, in the sense
that their diameters are much smaller than the average
distance traveled between collisions.
•(iii) The molecules do not interact, except that they
make perfect elastic collisions when they are in contact.
Kinetic Models of Gas, Properties of
Real Gas
Application questions: how fast do you think
a gas molecule in the room is traveling? How
often do they colloid with each other?
Will my use of ideal gas law be valid to gases
in the gas cylinder?
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Derivation of ideal gas law
• Consider a container of Volume V,
at T, contains n mole gas.
• Pressure is a result of molecular
collisions on the wall.
• Every collision, the momentum
change is 2mvx,
• No. of molecules that will collide in
time ∆t= (1/2)(particles in greenshaded volume).
• Total momentum change/∆t=force
• Pressure = Force/Area
PV = (1/3)nM c2
C =<s2>1/2 ,root-meansquare speed
Maxwell Distribution of Speed
• The speed of molecules in gas actually obey Maxwell
distribution of speed: f (s)
⎛ M ⎞
f ( s ) = 4π ⎜
⎟
⎝ 2πRT ⎠
3/ 2
(
s 2 exp − Ms 2 / 2 RT
)
• f(s)ds: is the fraction of molecules that have speeds in
the range of s to s+ds.
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Maxwell Distribution of Speed
Given the Maxwell Distribution of
speed, we can obtain
•Average of speed or mean
speed
c =∫
∞
0
1/ 2
⎛ 8 RT ⎞
sf ( s )dv = ⎜
⎟
⎝ πM ⎠
•Root-mean-square of speed
c = <s2>1/2 =(3RT/M)1/2
Final result--Ideal Gas law
• Kinetic model leads to PV = (1/3)nM <s2>
• The Maxwell distribution of speed gives c = <s2>1/2
=(3RT/M)1/2
• Combine these two lead to PV = nRT
Note:
• Pressure is determined by the temperature---the
higher the temperature, the larger the pressure
• Pressure is also determined by the number density,
n/V,---the higher the number density, the higher the
pressure.
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More about root-mean-square speed
• The root-mean-square of speed or the average speed
is proportional to (T/M)1/2. ---lighter gas molecules
have higher average speed
• However, the mean kinetic energy of molecules,
(1/2)M<v2>, only depend on temperature, not the
molecular masses.
• It can be argued that thermal equilibrium implies that
the mean kinetic energy of molecules must be equal
regardless if it is in solid, liquid or gas
Real Application questions
• Let’s estimate how fast H2 and N2 are traveling in the
air.
• Can we estimate how often the gas molecules will
collide with each other? (concept of collision
frequency, mean free path).
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Real gases
• Real gases show deviations from
the perfect gas law because of
molecular interaction
• Repulsive forces assist
expansion
• Attractive forces assist
compression
Figure: A typical intermolecular interaction
potential
•
Repulsive forces dominant only when
molecules are close together on average.
Typically this would mean a high pressure.
•
Attractions will be significant when
molecules are relatively close but not too
close. Or at low temperature when
molecules do not move fast enough and
they can capture each other. Typically
moderate pressure, low temperature.
Compression factor
• We can use compression factor Z to quantify the
deviations of real gas from ideal gas law
• Z=PVm/RT or Z = Vm/Vm0 where Vm0 is the molar
volume of the ideal gas (i.e,: Vm0 =RT/P).
• According to this definition:
– Z=1 if the gas obeys ideal gas law
– Z >1 implying the gas are more difficult to compress than
ideal gas (larger molar volume than the ideal gas Vm0)--repulsive forces dominant.
– Z < 1 implying the gas are more compressible than ideal
gas (less molar volume than Vm0) ---attractive forces
dominant.
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Some examples of compression factor
Graph on the left shows how Z
varies with pressure at 0ºC for a
few real gases. Observe:
1. At low pressure, all Z ~ 1. Gas
behave ideally.
2. At very high pressure, all Z >1.
Repulsive forces dominant.
3. At intermediate pressure, most
gases have a range Z <1,
attractive forces dominant. Some
may not, but will have Z<1 range
at lower temperature.
Compression factor at different temperature
•
•
Graph on the left shows
how Z would vary with
pressure at different
temperature T for a given
type of gas.
Boyle temperature is the
temperature at which dZ/dP
= B’(T)=0 (when P →0)
or dZ/d(1/Vm)=B(T) → 0.
(see the discussion on the
virial expansion)
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Virial Expansion
• Virial expansion is another way to describe how the
real gas behavior deviate from the ideal gas law.
• One incorporate the deviation by including higher
order terms which are absent in the idea-gas law.
pVm=RT(1+B’P+C’P2+…)…the coefficients, B’, C’, ..are
called virial coefficients. They themselves are functions of
temperature.
According to this, Z=1+B’P+C’P2+….
• Another way is to write the virial equation is in the order of
(1/Vm)
pVm = RT (1 + VBm + VC2 + K)
m
Isotherm of Real Gases
• Graph on the left is the
experimental isotherms of CO2
gas at several temperature.
• At 20°C, there is a discontinuity
in isotherm at point C-D-E. This
is the gas-liquid phase transition.
• The critical temperature Tc is the
point below which there is liquidgas phase transition, above which
there is not.
• The critical point is marked by the
*. The two-coexisting phases (like
C,D merge together to one point
at Tc that gives the critical point.
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Van-der Walls equation
• In 1873, J. H. van der Walls proposed a general
equation that can fit many of experimental observed
equation of states of real gases.
P = nRT/(V-nb) - a (n/V)2
or :
P = RT/(Vm-b) - a/Vm2
• a, b are called van-der Walls coefficients. They are
independent of T, P or V, but are characteristic of
molecular nature of the gases.
• (a/Vm2) corrects for the attraction.
• b corrects for repulsion, can be related to the volume of
the molecular spheres, b~ Na (1/6)π d3.
Table 1.5: Van der Walls coefficients
a /(atm L2/mol-2)
b /(L mol-1)
Ar
1.337
0.0320
CO2
3.610
0.0429
He
0.0341
0.0238
Xe
4.137
0.0516
Exercise: from Van-der-Walls equation, obtain second-virial
coefficient B, determine Boyle temperature for Ar.
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Features of VDW equation
Isotherms from VDW equation
• VDW equation predicts
isotherms very similar to that
of observed experimental
isotherms
• At high Temperature,
isotherms are like ideal gas
law. P~RT/(Vm-b).
• At lower temperature, it has
van-der-walls loop, signify
the liquid-gas phase transition
Van-der-walls loop
The critical point in VDW equation
• The critical temperature Tc is the temperature below
which the van-der-walls develops. It is a well-defined
point with specific Tc, Pc and Vc in terms of van-derwalls constants a and b.
• The critical point is defined by the following two
conditions
2a
dP
RT
=
+ 3 = 0 which leads to: Vc=3b,
2
dVm (Vm − b) Vm
Pc=a/27b2, Tc=8a/27Rb
2 RT
6a
d 2P
=
−
=0
dVm2 (Vm − b) 3 Vm 4
so Zc=PcVc/RTc=3/8.
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The principle of corresponding states
• Molecular characteristics are reflected in van-der-walls
constants a, and b, which in turn determines Pc, Tc, Vc.
• Let’s define Pr=P/Pc, Tr=T/Tc, Vr=V/Vc
• The observation that the real gases at the same reduced
volume and reduced temperature exerts the same
reduced pressure is called “the principle of
corresponding states”.
• Experiments confirm such truth to certain extents. –
most gases with spherical molecules obey the
corresponding states well, but not non-spherical or
highly polar molecules.
Examples of experimental data confirm
principles of corresponding state
This figure shows how Z for different gases at same reduced
temperature form a common curve.
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Other examples
• There are many examples where such plot with
“reduced variables” produce a single curve regardless
of “molecular nature”.
Ref: Wang & Teraoka, Macromolecules 30, 8473 (1997)
Use of Model Systems
• The principle of corresponding states reflects the fact
that many physical laws are not governed by the
molecular characteristics, but are governed by some
other physical principles. ---”statistical principles”.
• This justify the use of “model fluids”, “model
polymer chains” where these model systems do not
need to include chemical identify.
• One model fluid that help to understand the liquid-gas
phase transition is the Lennard-Jones fluids.
• One example of model of polymer chains is the selfavoiding walks on the lattice.
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Hard sphere fluid vs. LJ fluid
• Many research are done with molecules modeled as
purely repulsive hard spheres (no attractive
interactions).
• For this type of fluid, there is no liquid-gas transition.
The collections of spheres either are in gas phase, or
are in solid phase directly.
• Adding an attractive term in molecular interactions
turns on the liquid-gas phase transition. Study of
these model fluid through computer simulations help
to illustrate the phase transition greatly.
Summary
• Simple Kinetic model of gas can lead to the ideal gas
law. The major assumption in simple kinetic model is
that molecules do not attract or repel each other
except making perfect collision.
• Kinetic model also shows that average speed of gas
molecule is proportional to (T/M)1/2.
• Real gas molecules however attract or repel each
other at appropriate conditions. This makes the
properties of many real gas to deviate from ideal gas
law behavior.
• Compression factor Z can be used to check how the
real gas deviates from ideal gas law.
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Sample problem Solving
• At what pressure does the mean free path of argon at
25C become comparable to the diameter of a
spherical vessel of volume 1.0L that contains it? Take
σ=0.36nm2
• How does the mean free path in a sample of gas vary
with temperature in a constant-volume container?
• Express the van-der-walls equation of state as a virial
expansion in powers of 1/Vm, and obtain expression
of B and C in terms of a and b.
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