Transcript Topic 3.2 Thermal Physics

Thermal Physics
3.2 Modelling a gas
Understanding
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Pressure
Equation of state for an ideal gas
Kinetic model of an ideal gas
Mole, molar mass, and the Avogadro
constant
Differences between real and ideal gases
Applications and Skills
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Equation of state for an ideal gas
Kinetic model of an ideal gas
Boltzmann equation
Mole, molar mass, and the Avogadro
constant
Differences between real and ideal gases
Equations
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Pressure p = F/A
# moles of a gas as
ratio of # molecules to
Avogadro’s constant n
= N/NA
Equation of state for an
ideal gas pV=nRT
Pressure and mean
square velocity of an
ideal gas
1 2
p  c
3
Understand the proof for the
formula
1 2
p  c
3
Equations
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Mean kinetic energy of ideal gas molecules
Ek(mean) = (3/2)kBT = (3/2)(R/NA)T
The Gas Laws (1)
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Developed independently & experimentally between mid 17th and
start of 19th centuries
Ideal gases defined as those which obey the gas laws under all
conditions i.e. no intermolecular interactions between molecules
and only exert forces when colliding. Real gases only
approximate ideal gases as long as pressures are slightly greater
than normal atmospheric pressure
Boyle showed that p α 1/V or pV = k(at const temp)
pV graphs aka isothermal curves
Charles, around 1787 confirmed that all gases expanded by
equal amounts when subjected to equal pressure. The volume
changed by 1/273 of the volume at zero. At -273 °C volume
becomes zero.
For a fixed mass at constant pressure, volume directly
proportional to absolute temperature V α T (const pressure)
V/T = constant (at constant pressure)
The Gas laws (2)
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Third gas law, for a gas of fixed mass and
volume, the pressure is directly proportional
to the absolute temperature
p α T (const volume) p/T = constant (at const
volume)
Avogadro stated that the number of particles
in a gas at const temp and pressure is
directly proportional to the volume of the gas
n α V n/V = constant
Gas laws (3)
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Combining the four equations and four
constants gives pV/nT = R or
pV = nRT
R = 8.31 JK-1mol-1 when p in pascals , V in
m3, n = # moles of gas
The mole and Avogadro’s constant
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The mole (mol) measures the amount of
substance something has and is one of he
seven base SI units.
Defined as the amount of substance having
the same number of particles as there are
neutral atoms in 12 grams of carbon – 12
One mole of gas contains 6.02 x 1023 atoms or
molecules (Avogadro’s constant NA ). So 2
moles of oxygen gas contains 12.04 x 1023
molecules.
Molar mass
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Since diatomic gases have two atoms per molecule a
mole of a diatomic gas will have 6.02 x 1023 molecules
but 12.04 x 1023 atoms.
One mole of oxygen atoms has approximate mass 16.0
g so a mole of oxygen molecules will have mass 32.0 g
i.e. its molar mass.
Consider one mole of CO2 (g) which contains one mole
of carbon atoms has mass 12.0 g and one mole of
oxygen molecules 32.0 g. The molar mass of CO2 is
44.0 g mol-1
Example 1
Molar mass of Oxygen is 32 x10-3 kg mol-1
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If I have 20g of Oxygen, how many moles do I
have and how many molecules?
Molar mass of Oxygen gas is
20 x 10-3 kg / 32 x10-3 kg mol-1
 0.625 mol
 0.625 mol x 6.02 x 1023 molecules
 3.7625 x 1023 molecules
Example 2
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Calculate the percentage change in the
volume of a fixed mass of an ideal gas when
its pressure is increased by a factor of 3 and
its temperature increases from 40.° C to
100.° C.
n is constant so (p1V1/T1) = (p2V2/T2)
V2/V1 = [(p1T2 )/(p2T1) = 373/(3 x 313) ≈0.40
i.e. a 60% reduction in volume of gas
Kinetic Model of Ideal gasesKey assumptions
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Gas consists of large number of identical tiny particles- molecules in
constant random motion
Statistical averages of this number can be made
Each molecule’s volume is negligible when compared to the volume of the
whole gas
At any instant as many molecules are moving in one direction as any other
direction
Molecules undergo perfectly elastic collisions between each other and with
the walls of the container; momentum is reversed during collision
No intermolecular forces between molecules between collisions i.e. energy
is completely kinetic
Duration of collision negligible compared with the time between collisions
Each molecule produces a force on the wall of the container
The forces of individual molecules will average out to produce a uniform
pressure throughout the gas- ignoring the effect of gravity
Developing a relation between
pressure and density (1)
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Refer to text p. 107 Figure 7
Consider one molecule which has momentum
and collides elastically with the right side of
the box of length L
∆p = -2mcx F= ∆p/∆t and ∆t =2L/cx so
F=(-mcx2/L) is force of box on molecule
Newton’s III states molecule exerts an equal
and opposite force F=mcx2/L on box
Developing a relation between
pressure and density (2)
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N molecules exert a total force
Fx= (m/L)(cx12 + cx22+ cx32+ … cxN2)
The forces average out giving a constant force with
so many molecules
The mean value of the square of the velocities c 2 
(cx12 + cx22+ cx32+ … cxN2)/N
The total force on right hand wall is Fx= (m/L) c 2
Developing a relation between
pressure and density (3)
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From Pythagoras,
c 2  c x2  c y2  c z2 and on average
there is an equal likelihood of molecule
moving in any direction so
c 2  3c x2 or c x2 
1 2
c now
3
F
1 Nm 2
c and since p x  x & A  L2
A
3 L
1 Nm 2
1 Nm 2
c but Nm is total mass

c
px 
3 V
3 L3
1
c 2
p
3
F 
Molecular interpretation of
temperature
1 2
p  c
3
p
Nm 2
c
3V
Nm 2 3
3 Nm 2
1 2
pV 
c  pV  (
c )  N  mc
3
2
2 3
2
3 nRT 1 2
N
3 RT 1 2
 mc but  N A so
 mc
2 N
2
n
2 NA 2
Boltzmann Constant
If a new constant is defined as kB  R / NA, then
3
1 2
k BT  m c
2
2
R
8.3Jmol 1 K 1
 23
1
kB 


1
.
38
x
10
JK
23
1
N A 6.02 x10 mol
Linking temperature with
energy
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The kinetic theory now links temperature with
the microscopic energies of the gas
molecules
The equation resembles the kinetic energy
formula.
Adjusting for N molecules gives 3/2 NkBT
This represents the total internal energy of an
ideal gas (only considering translational
motion of molecules of monoatomic gases)
Alternative equation of state
for ideal gas
pV  nRT , but pV  Nk BT
so nRT  Nk BT , and nR  Nk B
if n  1 mol, then N  N A  6.02 x10
23
Real Gases vs Ideal Gases (1)
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Since an ideal gas obeys the ideal gas laws
under all conditions, ideal gases cannot be
liquefied
In 1863, Andrews experiments showed a
deviation from Boyles’ pV curves for CO2 at high
pressures and low temperatures
Later experimentation showed that real gases
do not behave like ideal gases and that all gases
can be liquefied at high pressures and low
temperatures
Real Gases vs Ideal gases (2)
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The ideal gas law describes the behaviour of all gases at
relatively low pressures and high temperatures.
The ideal gas law fails when the main assumptions of
the Kinetic Theory are invalid i.e molecular volumes and
intermolecular forces are negligible.
Real gases can only be compressed so far indicating
that molecules occupy a non negligible volume and weak
attractive forces exist between the molecules of real
gases just as those between molecules of liquids
Deviation of a real gas from an ideal gas
Real Gases vs Ideal gases (3)
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Source Google images
Real Gases vs Ideal gases (4)
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Short range repulsive forces act between gas molecules
when they approach each other reducing the distance
they can effectively move i.e. reducing gas volume below
the V used to develop ideal gas law.
At slightly greater distances molecules attract each
other slightly forming small groups and reducing the
effective number of particles. This slightly reduces the
pressure.
Enter van der Waals who modified the ideal gas law
introducing two new constants.
No single simple equation of state applying to all gases
has been found to this date