20 B Week II Chapters 9 -10) Macroscopic Pressure Microscopic pressure( the kinetic theory of gases:...
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Transcript of 20 B Week II Chapters 9 -10) Macroscopic Pressure Microscopic pressure( the kinetic theory of gases:...
20 B Week II Chapters 9 -10)• Macroscopic Pressure
•Microscopic pressure( the kinetic theory of gases: no potential energy)
• Real Gases: van der Waals Equation of State
London Dispersion Forces: Lennard-Jones V(R )
and physical bonds
Chapter 10
• 3 Phases of Matter: Solid, Liquid and Gas of a
single component system( just one type of molecule, no solutions)
Phase Transitions:
A(s) A(g) Sublimation/Deposition
A(s) A(l) Melting/Freezing
A(l) A(g) Evaporation/Condensation
For a fixed mass( of gas, e.g., Air) how does the Volume occupied by the gas change when the gas is cooled or heated?
Lets do an experiment!
• Define a Thermodynamic Temperature scale to make a thermometer( to measure T)
Assumes we know nothing about the Boltzmann distributionso T is a parameter
• Charles’ Law V/T = const
Quantifying the temperature scale requires some materialProperty that changes if the material is heated or cooled.
The thermal expansion coefficient = (1/V)(V/T)
assume fractional rate of change of V with respect to T is ideal.
The temperature scale should be material independent(Universal)
appears to be Universal for low pressure gases
Over the T range where water freezes and boils, Charles observed that = (1/V)(V/T) ~const for low pressure gases.
= (1/V)(V/T)=(1/V0){V-V0}/{T-T0}
V0 and T0, are the initial volume and Temperature, thermometer material and V and T, , are the final volume and Temperature, respectively!
The Experiment: we need 3 T’s to be accurateThe material for the experiment is air
20±?Room Temp!
Assigned values
= (1/V0){V-V0}/{T-T0} Thermal expansion
Let V0 be the gas Volume at T0 = 0
The Freezing point of water!
T = (1/{(V/V0) – 1}
Re-arrange to V=V0T}
If T has units of °C (Celsius), the boiling point of water must be set to 100 °C!
All measurements are for air at low pressure
V=V0T} solve for T at V=0 gives T=C
Fitting the V vs T the equation to the data, we can extrapolate to absolute zero!
V0= 1.5 L
V0= 1.0 L
V=V0T} so as V goes to zero T goes to The absolute T=0 KC
However, the mass of gas is fixed!So V can approach zero but cannot be zero!
V0= 1.5 L
V0= 1.0 L
In terms of an absolute temperature T=-273.15 C= 0 K (Kelvin)
V~T since as V approaches 0, T 0 V = TConst.
This is most useful form of Charles’ Law
V~ N (the number of gas atoms/Molecules) (Since the more molecules the greater the volume)
V~1/P from Boyle’s Law
Therefore V~ NT/P or V=kNT/P: PV=kNT
The is Ideal or Perfect Gas Law where k = Boltzmann Constant
: PV=kNT
PV=kNT: The Idea Gas Law! Applied two different gases, e.g., N2 and O2
(Or even better! The Law describing the behavior of aTheoretically Perfect/Ideal Gas)! Can be used for all gases at Low Pressure and high enough temperature
~ ultrahigh vacuumultralow pressures!
Effusion Cell
~ very small hole or area A.If P=PO2 + PN2(partial pressures)P=NO2kT/V + NN2kT/VP = (NO2 + NN2 )kT/V=NkT/VN=NO2 + NN2 which is true
Fig. 9-9, p. 377
+ =
Partial Pressures add. P= P1 + P2 + P3 + etc
When dealing with the behavior of large numbers N~NA
NA=6.022 x 1023 mole-1 then we must average over the behavior of the individual atom/molecule in the system
For example.
If i is the total energy of the ith atom/molecule, Kinetic + Potential, then the average energy per atom/molecule
is <>= (1 + 2 + 3 ………. + N )/N=∑i/NThis is the behavior we observe and is the domain ofStatistical Mechanics: the science that describes the behavior of a system with large numbers of atoms/molecules based on the behavior of individual atoms/molecules, e.g.,Gases!
The Kinetic Theory of Gases uses this idea to describe the
Behavior of an ideal gas!
Think about this for a min!
The average behavior can be determined by watching one
particle for a very long time(infinitely long in principle),
or a large number of particles (infinitely large in principle)
for a short time, i.e., snapshot or an instant
<(vx)2
The Kinetic Theory
Nanoscopic theory of gas pressure
watch the average behavior of one particle and describe the behavior of a system with a large number of
particles
L
The Kinetic Theory
Nanoscopic theory of gas pressure
watch the average behavior of one particle and describe the behavior of a system with a large number of
particles
- FA =F = ma=d(mv)/dt
mv = linear momentum(mv)=(mvx)f - (mvx)i
(mv)=-mvx - mvx=-2mvx
t=2L/vx
L
-FA =F = ma=d(mv)/dt ~ -2mvx/(2L/vx)= - m(vx)2/L
FA=2m(vx)2/L; <FA>=m<(vx)2>/L average force on the wall
Px=P= N<FA>/A=Nm<(vx)2>/AL=(N/V) m<(vx)2>
The Kinetic Theory
Microscopic pressure
one particle at a time
-FA = F = ma=d(mv)/dt force atom/molecule
mv = linear momentum(mv)=(mvx)f - (mvx)i
(mv)=-mvx - mvx=-2mvx
t=2L/vxL
V=AL
Fig. 9-11, p. 379
v2 = (vx)2 + (vy)2 +(vz)2 =u2 then
< v2 >=<u2>=<(vx)2 + (vy)2 +(vz)2>=3<(vx)2> or <(vx)2> = <u 2 >/3
The average of speed in all directions must be the same <(vx)2>=<(vy)2>=<(vz)2> for the random motion of an atom/molecule
P= (N/V) m<(vx)2>= (N/3V) m< u2 > but <KE>= (m/2)< u2 >
So P=(N/V)<KE>(2/3) recall that PV=NkT so <KE>=(3/2)kT
Since <>=<PE> + <KE> = <KE> for a prefect gas,
that is a gas described by PV=NkT
Motion in all directions x,y, and z are equally likely: <vx>=<vy>=<vz>
Fig. 9-9, p. 377
+ =
Partial Pressures add. P= P1 + P2 + P3 + etc
20 B Week II Chapters 9 -10)• Macroscopic Pressure
•Microscopic pressure( the kinetic theory of gases: no potential energy)
• Real Gases: van der Waals Equation of State
London Dispersion Forces: Lennard-Jones V(R )
and physical bonds
Chapter 10
• 3 Phases of Matter: Solid, Liquid and Gas of a
single component system( just one type of molecule, no solutions)
Phase Transitions:
A(s) A(g) Sublimation/Deposition
A(s) A(l) Melting/Freezing
A(l) A(g) Evaporation/Condensation
Fig. 9-18, p. 392
A AR
<V(R )> = 0
For RVery LargeDensity N/V is lowThereforeP=(N/V)kT is low
Real Gases and Intermolecular Forces
well depth ~ Ze or Mass but it’s the # of e
Lennard-Jones Potential
V(R ) = 4{(R/)12 -(R/)6}
Ar+ ArkT >>
Real Gases and Intermolecular Forces
well depthBond DissociationD0= h
~ hard sphere diameter
Lennard-Jones Potential
V(R ) = 4{(R/)12 -(R/)6}
The London Dispersion
or Induced Dipole Induced Dipole forces
Weakest of the Physical Bonds but is always present!