Chapter 1 : Slide 1

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Chapter 1 The Properties of Gases
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PHYSICAL CHEMISTRY We shall apply some of the
methods of physics to chemical problems, in order
to understand the theoretical principles behind
chemical phenomena, and to be as quantitative as
possible when we describe them.
We will focus first on THERMODYNAMICS, a
description of systems at equilibrium and their
energies.
Then we will look at KINETICS, and study how
systems change and how fast that happens.
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PRESSURE is force per unit area. 1 N m-2 1
Pa. 1 bar 105 Pa exactly and is p?, the
standard pressure. Other units 1 atm 760 mmHg
or 760 Torr 101325 Pa (and is roughly 1 bar).
Pressures may be measured e.g. with a manometer.
Suppose p1 gt p2 and the difference supports a
column of liquid of height h, with density ?, in
the tube radius r.
Weight of liquid vol. density g
(gravitational acceleration, 9.81 m s-2) so p1 -
p2
?gh
Independent of r
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TEMPERATURE When a hot body A and a cold body B
are put together, heat flows from A to B until
the temperatures are equal, and THERMAL
EQUILIBRIUM is attained. (Note Heat is not the
same as temperature). "Zeroth Law of
Thermodynamics" If objects A and B are in thermal
equilibrium, and B and C are in equilibrium, then
so are A and C. Note a scientific 'law'
summarizes experience The (old) definition of
the Celsius or Centigrade scale, ? assume
mercury expands linearly with temperature, and
divide the difference between two fixed points
into 100 equal divisions.
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The STATE of a system (whatever we are looking
at) is uniquely defined in terms of a few
variables e.g. for a gas amount of substance
(n), temperature (T) and pressure (p).
Alternatively, mass (m), volume (V) and p or m,
V T etc. More than three variables are
unnecessary for a gas because they are not all
independent they are linked by an EQUATION OF
STATE. A system is in thermodynamic
EQUILIBRIUM if there is no change with time in
any of the system's macroscopic properties.
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GAS LAWS (1) Boyle's Law p ? 1/V or pV
constant (at constant T). This works well
at moderate p (lt 1 atm). (2) Charles' Law ?
temp. in oC V0 gas volume when ? 0. He
found V(?) const (? 273 oC) We can
define a new scale of ABSOLUTE temperature, T
273 ? The intervals are the same the origin
is shifted. Then on THIS scale, known as the
thermodynamic temperature scale, V ? T
(constant p). This rule works best as p tends to
zero.
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(3) Avogadro proposed that equal volumes of gas
contain equal numbers of particles i.e. V ? n
(constant p and T). We can combine
Boyle's and Charles' laws and Avogadros
hypothesis into a single expression pV
constant nT A gas which obeys this relation at
any p is called IDEAL or PERFECT.
The PERFECT GAS EQUATION OF STATE is pV
nRT R is the gas constant (8.314 J K-1 mol-1).
EXAMPLE What is the molar volume Vm for an ideal
gas at SATP?
V nRT/p 0.0248 m3 24.8 dm3 24.8 L 24800
cm3
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ISOTHERMS and ISOBARS
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Daltons Law Suppose we have two gases in a
container nA moles of gas A and nB moles of gas
B. We can define individual partial pressures
pA nART/V and pB nBRT/V . Daltons Law is
that the measured total pressure p is the sum of
the partial pressures of all the components p
pApB (nAnB)RT/V. Mole fractions define
xJ for species J as nJ/n where n (nA
nB). Then, xA xB 1 and pJ p xJ
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REAL GASES
Deviations from ideality can be described by the
COMPRESSION FACTOR, Z (sometimes called the
compressibility). Z RT/pVm Vm/Vmp For ideal
gases Z 1 always.
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Microscopic interpretation
When p is very high, r is small so short-range
repulsions are important. The gas is more
difficult to compress than an ideal gas, so Z gt
1. When p is very low, r is large and
intermolecular forces are negligible, so the gas
acts close to ideally and Z ? 1. At intermediate
pressures attractive forces are important and
often Z lt 1.
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VIRIAL COEFFICIENTS We can consider the perfect
gas law as the first term of a general
expression p Vm RT (1 Bp Cp2
...) i.e. p Vm RT (1 B/Vm C/Vm2
...) This is the virial equation of state and B
and C are the second and third virial
coefficients. The first is 1. B and C are
themselves functions of temperature, B(T) and
C(T). Usually B/Vm gtgt C/Vm2
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Consider dZ/dp. This is zero for a perfect gas
since Z is constant. dZ/dp B 2pC .... In
the limit as p tends to zero, dZ/dp B which is
zero only when B 0. The temperature at which
this occurs is the Boyle temperature, TB, and
then the gas behaves ideally over a wider range
of p than at other temperatures. Each gas has a
characteristic TB, e.g. 23 K for He, 347 K for
air, 715 K for CO2.
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CONDENSATION or LIQUEFACTION This demonstrates
that there are attractive forces between gas
molecules, if they are pushed close enough
together. E.G. CO2 liquefies under pressure at
room temperature. Above 31 0C no amount of
pressure will liquefy CO2 this is the CRITICAL
TEMPERATURE, Tc.
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A typical pV diagram for a real gas
p
Tc, pc and Vm,c are the critical constants for
the gas.
V
The isotherm at Tc has a horizontal inflection at
the critical point dp/dV 0 and d2p/dV2
0. At the critical temperature the densities of
the liquid and gas become equal - the boundary
disappears. The material will fill the container
so it is like a gas, but may be much denser than
a typical gas, and is called a 'supercritical
fluid'.
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REAL GASES - THE VAN DER WAALS APPROXIMATION 1)
Real molecules have non-zero volume n b (b is a
constant depending on the type of gas, the
'excluded volume'). The molecules have less free
space to move around in, so replace V in the
ideal gas equation by V-nb. Very roughly, b ?
4/3 ?r3 where r is the molecular radius. 2)
There are attractive forces between real
molecules, which reduce the pressure p ? wall
collision frequency and p ? change in momentum
at each collision. Both factors are proportional
to concentration, n/V, and p is reduced by an
amount a(n/V)2, where a depends on the type of
gas. Note a/V2 is called the internal pressure
of the gas - see later.
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The Van der Waals equation of state " p nRT/V
" becomes p nRT/(V - nb) -
a(n/V)2 i.e. p an2/V2 V - nb
nRT When V or T are very large i.e. at low p or
high T, then this van der Waals equation of state
becomes equivalent to the ideal equation. a and
b are empirical Van der Waals constants. It is
easy to solve for p given V. To find V given p
you need to solve a cubic equation (with VmV/n)
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Consider 1 mol of gas, with molar volume V, at
the critical point (Tc, pc, Vc) 0 dp/dV
-RTc(Vc-b)-2 2aVc-3 0 d2p/dV2
2RTc(Vc-b)-3 - 6aVc-4 The solution is Vc 3b,
pc a/(27b2), Tc 8a/(27Rb).
THE PRINCIPLE OF CORRESPONDING STATES Define
reduced variables pr p/pc Tr
T/Tc Vr Vm/Vm,c Van der Waals hoped that
different gases confined to the same Vr at the
same Tr would have the same pr.
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Proof rewrite Van der Waals equation for 1 mol
of gas, p RT/(V-b)-a/V2, in terms of reduced
variables
Substitute for the critical values
Thus
Thus all gases have the same reduced equation of
state (within the Van der Waals approximation).
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With reduced variables, different gases fall on
the same curves.
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There are many other equations of state for real
gases, including 1) The Berthelot Equation.
Replace Van der Waals' "a" with a temperature
dependent term, a/T p a/(Vm2T) Vm - b
RT 2) The Dieterici Equation. p exp(a/VmRT)
Vm - b RT Several others are listed in Table
1.6 of the text.
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BAROMETRIC DISTRIBUTION (discussion question 1)
p as a function of height. Consider the
manometer we analyzed earlier, but now allow ? to
vary with p, i.e. deal with a compressible gas
rather than an (essentially) incompressible
liquid. Consider a column of gas with unit cross
sectional area.
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Boundary condition ground level pressure is p0
so that p p0 exp(-Mgh/RT) An exponential
decrease of p with height. Equal ?h's always give
the same proportional change in p. Note the
assumptions 1) Ideal gas behavior 2) Constant
g 3) Isothermal atmosphere Mgh is the
gravitational potential energy. We will often see
properties varying in proportion to exp(-E/RT)
exp(-?/kBT) where E is a form of molar energy
(? is a molecular energy) because these are
examples of "Boltzmann distributions".
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Numerical Example Take T 298 K and M 0.028
kg mol-1. Mg/RT 1.1 x 10-4 m-1 1/9000
m. We can talk about a SCALE HEIGHT for the
atmosphere, Hs, of about 9000 m and write p
p0 exp(-h/Hs)
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• SUMMARY
• The STATE of a system is specified by a few
  variables, that may be linked by an EQUATION OF
  STATE e.g. for IDEAL GASES pV nRT. ISOTHERMS
  and ISOBARS.
• Dalton's Law of partial pressures.
• The VIRIAL EQUATION and the BOYLE TEMPERATURE.
• REAL GASES the COMPRESSION FACTOR and
  INTERMOLECULAR FORCES. pV diagrams LIQUEFACTION
  and the CRITICAL POINT.

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• The VAN DER WAALS approximate equation of state p
  RT/(Vm-b) - a/Vm2 is more realistic for REAL
  GASES. There are other equations of state which
  work well.
• REDUCED VARIABLES and the PRINCIPLE OF
  CORRESPONDING STATES.
• Worked example BAROMETRIC DISTRIBUTION.

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Homework Exercises 1.19 to 1.26 parts (a)
only Problems 1.11, 1.12 and 1.20