Lecture 19: Molecular Speeds

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Lecture 19 Molecular Speeds
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• Newtons second law
• Pressure in an ideal gas

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Quiz test your intuition
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T 20 C
CO2 molecules in atmosphere
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• CO2 molecules in the bottle move
• Faster than those in air
• Slower than those in air
• At the same speed as those in air
• Faster than the speed of sound (330 m/s)
• Slower than the speed of sound

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Newtons second law
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v
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m
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area O
Large force on O
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Small force on O
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Pressure P in ideal gas
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vy
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vx
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Wire enclosing area O
vxt
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Number of molecules flying through O in t seconds
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Impuls per molecule mvx , so the pressure Px
exerted by the flying molecules is
Gas
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O
Px
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Px
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Net pressure on O is zero
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Repeat pressure calculation for other velocity
components
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Pressure everywhere the same

P
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Ideal gas law
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The root-mean-square molecular speed
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For an ideal gas the two pressures are identical
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Thus the root-mean-square (rms) molecular speed
equals
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This result is also valid for non-ideal
molecules, see lecture 20.
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RMS speeds at T 298 K
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CO2 M 44 x 10-3 kg mol-1

1480 km/hr
N2 515 m/sec T 298 K
H2 1928 m/sec
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Speed of sound 330 m/sec
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Molecules translate at supersonic speeds
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(Sound 330 m/s)

rotations
vibrations
Poisonous gas
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YES! Molecules collide
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CO2 molecules travel 411 meter in 1 second
without collisions
0.5 cm
collisions
diffusion
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Lecture 20
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• Maxwell-Boltzmann distribution
• Average molecular kinetic energy

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Gigantic Numbers of Molecules
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18 mL water contains 3 x 18 x 6 x 1023 velocity
components
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Introduction Maxwell-Boltzmann Distribution
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molecular speeds are distributed
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Distribution
Probability to find molecule with speed v
depends on
number of molecules
speed (m/s)
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How to handle these distributions and to
calculate, for example, the average speed?
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Discrete distribution ni molecules have speed
vi
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Average speed for a total of N molecules
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For continuous distribution
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P(v)dv probability to find
speed in the interval v ,v dv
P(v)
v
v
v dv
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normalisationfactor
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Normalised probability
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Continuous distribution (II)
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Generalise
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Maxwell Boltzmann distribution function for speed
vx in the x-direction
follows from normalisation
See Ball 19.32
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Rms-speed from Maxwell-Boltzmann distribution
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We find
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In x, y, z direction
rms-speed
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Average kinetic energy per molecule
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Average kinetic energy per mole
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Kinetic energy also follows from pressure
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For ideal gas1)
PV nRT
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Putting both pressures equal we find
1) Despite of this restriction in this derivation
the result for also applies to non-ideal
molecules