Transcript 1-3 - University of Reading

Dr Roger Bennett
[email protected]
Rm. 23 Xtn. 8559
Lecture 1
Overview
• This module will introduce the key concepts
that form a cornerstone of modern physics.
• We will develop an understanding of the
generic properties of matter - Thermodynamics
• We will relate atomic scale events to
macroscopic phenomena – Statistical Mechanics
• We will use all of our problem solving tools to
extract real world information – it is not a
purely theoretical subject but a methodology.
Books – recommended texts
• There is a need to read around this topic and the
module is designed to encourage this.
• Carrington Basic Thermodynamics Oxford
– Most closely follows the module in the earlier
stages covering only Thermodynamics but at
exactly the right level. Good worked examples and
questions.
• F. Mandl Statistical Physics Wiley ~£25
– Most closely follows the module in the latter (more
difficult) stages. Best overall buy as it combines
Thermodynamics and Statistical Mechanics from
the outset.
Books – reference texts
• R. Bowley and M. Sanchez Introductory Statistical
Mechanics Oxford Science Publications
– A good all rounder with a Thermodynamics
introduction to the Statistical Physics. Many
examples.
• C. Kittel and H. Kroemer Thermal Physics Freeman
~£lots
– Detailed and comprehensive but probably a little
too advanced. A good reference book to turn to.
• D.S. Betts and R.E. Turner Introductory Statistical
Physics Addison Wesley
– A paperback on the Statistical Physics only. A bit
mathematical but has good introduction to topic.
Books – reference texts
• Also don’t forget:• Feynman, Leighton and Sands The Feynman
Lectures on Physics vol I Addison Wesley
which as a series are always readable and
informative, especially at potential stumbling
points.
• Thermodynamics is often covered at a good
level in most general physics textbooks. The
Statistical Physics aspects, however, often
prove to be the most problematic.
Books – A Warning
• Many books adopt subtly different symbols for the
same quantity. Take care when looking at different
sources.
• I will standardise in questions posed and our
discussions and provide a definitive list.
• There is much terminology to get to grips with. It is
probably worth making your own list as you proceed.
• There are many simple equations – some are always
true and some only true under certain conditions.
Don’t just rely on remembering them.
The Module
• 2 lectures and 1 workshops per week for first 7
weeks of each term.
• Assessed problems in week 3 of each term. 10%
each term
• Directed Reading and Independent Learning weeks
8-10 each term. 1 Summary lecture (Autumn term
only). Exam question based on this task!
• Departmental Test week 8 of Spring Term. 20%
• 2 Hour end of year exam. 60%
What's it all about?
• We need to understand the properties of
matter.
• It is far too complicated to start from classical
mechanics – there are too many atoms
involved in even the simplest of systems.
• We have to take averages and understand
what the majority are doing. In essence we
start by sacrificing detailed knowledge at the
atomic scale to understand the macroscopic
properties of the system – Thermodynamics.
Some famous quotes
• “A theory is the more impressive the greater
the simplicity of its premises, the more
different kinds of things it relates, and the
more extended its area of applicability.
Therefore the deep impression classical
thermodynamics made upon me. It is the
only physical theory of universal content
which I am convinced will never be
overthrown, within the framework of
applicability of its basic concepts.”
Albert Einstein
Some famous quotes
• “But although, as a matter of history,
statistical mechanics owes its origin to
investigations in thermodynamics, it seems
eminently worthy of an independent
development, both on account of the
elegance and simplicity of its principles, and
because it yields new results and places old
truths in a new light in departments quite
outside thermodynamics.”
J.W. Gibbs
Some famous quotes
• “The Physics of desperate men.”
Unknown, Blackett Laboratory,
Imperial College
Some famous quotes
• “It’s a funny subject. The first time you go
through it, you don’t understand it all. The
second time you go through it, you think you
understand it, except for one or two small
points. The third time you go through it, you
know you don’t understand it, but by that
time you are so used to it, it doesn’t bother
you any more.”
Sommerfield
Dr Roger Bennett
[email protected]
Rm. 23 Xtn. 8559
Lecture 2
Why does matter heat up when
compressed? A taster.
• Let us take the simplest case and investigate a gas
and determine what is meant by pressure.
• Imagine a piston of volume V and cross sectional
area A containing a monatomic gas (Ar, He etc).
V
A
F
Vacuum
x
dx
Why does matter heat up when
compressed? A taster.
• To stop the piston being ejected we have to hold it in
– i.e. apply force F
• The magnitude of the force depends on the area –
we define pressure P = F / A
V
A
F
Vacuum
x
dx
Why does matter heat up when
compressed? A taster.
• We compress the gas by pushing the piston through
an elemental distance -dx.
• The work done on the gas is therefore:-
dw = F (-dx) = -PA dx = -PdV
(the area times the distance is the volume change dV)
V
A
F
Vacuum
x
dx
Why does matter heat up when
compressed? A taster.
dw = -PdV
• How is pressure described microscopically?
• The force on the piston is due to reflection of atoms
as they scatter off of the piston - they impart
momentum to the piston.
• The Force on the piston is the amount of momentum
transferred per second, by definition.
V
A
F
Vacuum
x
dx
Why does matter heat up when
compressed? A taster.
dw = -PdV
• Split the problem into two parts:
– What is the momentum imparted per
collision?
– How many collisions do we get per
second?
V
A
F
Vacuum
x
dx
Why does matter heat up when
compressed? A taster.
dw = -PdV
• We must assume the piston reflects atoms
perfectly – why?
• If v is the velocity of the atom of mass m, vx
is the velocity in towards the piston, mvx is
the momentum towards and away (perfect
reflector) from the piston.
• The momentum transferred per collision is
therefore:
2mvx
Why does matter heat up when
compressed? A taster.
dw = -PdV
• In time t only molecules within vxt of the
piston will hit it.
• Let us suppose there are n atoms in our
volume V so the density is  = n / V.
• So the number of collisions in t is the number
of atoms in volume Avxt which is:
• No. of collisions = Avxt (in time t)
• No. of collisions per second = Avx
• Therefore force on piston F = Avx 2mvx
Why does matter heat up when
compressed? A taster.
dw = -PdV
• Pressure P = F / A = 2 mvx2
• Uh Oh! Duh!
• We have assumed all atoms have same velocity! –
need to take averages of the velocity. P = m<vx2>
where is the two?
•
•
•
•
•
•
What’s so special about x direction? Nothing!
<vx2> = <vy2> = <vz2>
<vx2> = 1/3 <vx2 + vy2 + vz2> = <c2>/3
c is speed
P = 1/3 m <c2> = 2/3 <mc2/2>
PV = (2/3) n<mc2/2>
Why does matter heat up when
compressed? A taster.
dw = -PdV
PV = (2/3)U
• PV = (2/3) n<mc2/2>
• PV= (2/3)U where U is the internal energy of
the entire system.
• We now know how much work we do on the
gas by compressing it a little and the
relationship between volume, pressure and
energy. We can link the two by considering
how much work we do on the gas goes into
changing its internal energy.
Why does matter heat up when
compressed? A taster.
dw = -PdV
PV = (2/3)U
• We shall assume that on compressing the gas all the
work done on the gas goes into internal energy. This
means there is no leakage of “heat”.
• Such a compression (or expansion) where there is no
flow of heat through the walls of the piston or vessel
is called adiabatic. From the Greek a (not) dia
(through) bainein (to go).
• For generality with other systems:
– PV = (2/3)U is more commonly written as
– PV = (-1)U so  = 5/3 in this example.
Why does matter heat up when
compressed? A taster.
dw = -PdV
PV = (-1)U
• For the adiabatic compression only all the work goes
into internal energy so
dU = dW = -PdV
U = PV / (-1)
So by product rule dU = (PdV + VdP) / (-1)
Hence, PdV = - (PdV +VdP) / (-1)
Grouping terms gives:
dV / V = -dP / P
Which we can all integrate – hopefully!???
Why does matter heat up when
compressed? A taster.
• For the adiabatic compression
dw = -PdV
PV = (-1)U

PV = C
( / V) dV = (-1/P) dP
( / V ) dV =  (-1/P) dP
 ln(V) = -ln(P) + ln(C)

PV = C
• This is our result. It tells us that under adiabatic
conditions the pressure times the volume to the
power 5/3 (for our example) is constant.
• We discovered this without knowing anything about
our gas – it must be true in general for monatomic
gases or more specifically a “perfect” or “ideal” gas.
Dr Roger Bennett
[email protected]
Rm. 23 Xtn. 8559
Lecture 3
Temperature
• We can relate internal energy to pressure and volume.
How are these related to temperature?
• Common sense tells us that when two bodies at the
differing temperatures are placed next to each other (in
thermal contact) the temperatures rise and fall until
both bodies reach the same temperature. When at the
same temperature they are in thermal equilibrium.
• This is commonly referred to as the 0th Law of
Thermodynamics:– “If two bodies A and B are in thermal equilibrium
with a third body C then A and B are in thermal
equilibrium with each other.”
• When in thermal equilibrium there is no net energy flow
from one body to the other.
Temperature Scales
• To measure temperature in general we need a
property that varies with temperature- X(T).
– length of mercury in a capillary
– Resistance of a wire
– Pressure of a gas at constant volume
– Volume of a gas at constant pressure
• We need a reference point which is taken to be
the triple point of water. By definition set to
273.16K in 1954.
Temperature Scales
• Then:
Tx = 273.16 (X / Xtp)
• But different
methods of X give
differing values.
Worse still differing
gases give different
results in constant
volume gas
thermometer!
Apply
pressure
to fix
volume
Hg
Height of Hg = h
Pressure = gh
gas
System to be
measured
Temperature Scales
• However, works in the limit of low gas density
for all gases.
Tcvgt = 273.16 limlow  (P / Ptp)
Tcpgt = 273.16 limlow  (V / Vtp)
• Works best when X = PV product for a gas
T = 273.16 limlow  ((PV) / (PV)tp)
• This defines the Ideal Gas Temperature Scale
limlow  (PV) = (limlow  (PV)tp / 273.16) T
PV = NRT= nkT
This is the ideal gas law
The Ideal Gas Law PV = NRT = nkT
• N is the number of moles of gas atoms or
molecules.
• One mole is 6.02 × 1023 entities, this is
Avagadro’s number N0
• R is the molar gas constant 8.31 J mol-1 K-1
• n is the number of atoms
• k is Boltzmann’s constant 1.381 × 10-23 J K-1
• Chemists like to use N and R, Physicists tend
to use n and k. You will see both in your
reading.
The Ideal Gas Law PV = NRT = nkT
• The fundamental assumption here is:– The gas behaves as “A non-interacting assembly
of point masses”
• This is increasingly realistic of the nature of gasses as
T increases and or P (or density) decreases.
• It has been experimentally confirmed.
• PV = nkT is an example of an equation of state. P, V
and T are state variables or thermodynamic
coordinates.
• Other equations of states can be defined to fit nonideal gas behaviour for example: van der Waals
equation of state (P + aN2/V2)(V-Nb) = NRT where a
and b are constants correcting for potential energy
and excluded volume of gas molecules respectively.
Temperature on the atomic scale
• We have found that
– PV = nkT = (2/3) U = (2/3) n<mc2/2>
– Average energy per molecule = 3/2kT
• How is this energy distributed in the gas?
– We should attempt to find the distribution
of velocities in the gas. This means finding
a result for of the order of N0 atoms.