Lecture 25
Goals:
• Chapter 18
 Understand the molecular basis for pressure and the idealgas law.
 Predict the molar specific heats of gases and solids.
 Understand how heat is transferred via molecular collisions
and how thermally interacting systems reach equilibrium.
 Obtain a qualitative understanding of entropy, the 2nd law
of thermodynamics
• Assignment
 HW12, Due Tuesday, May 4th
 For this Tuesday, Read through all of Chapter 19
Physics 207: Lecture 26, Pg 1
Macro-micro connection
Mean Free Path
If a molecule, with radius r, averages n collisions as it
travels distance L, then the average distance between
collisions is L/n, and is called the mean free path λ
The mean free path is independent of temperature
The mean time between collisions is temperature
dependent
Physics 207: Lecture 26, Pg 2
The mean free path is…
 Some typical numbers
Vacuum
Pressure (Pa)
Molecules / cm3
Molecules/ m3
mean free path
Ambient
pressure
Medium
vacuum
105
2.7*1019
2.7*1025
68 x 10-9 m
100-10-1
1016 – 1013 1022-1019
Ultra
High
vacuum
10-5-10-10
109 – 104
0.1 - 100 mm
1015 – 1011 1-105 km
Physics 207: Lecture 26, Pg 3
Distribution of Molecular Speeds
A “Maxwell-Boltzmann” Distribution
O2 at 25°C
1.4
# Molecules
1.2
1.0
O2 at 1000°C
0.8
Fermi Chopper
1
0.6
2
0.4
0
1 Most probable
2 Mean (Average)
200
600
1000
1400
1800
Molecular Speed (m/s)
Physics 207: Lecture 26, Pg 4
Macro-micro connection
 Assumptions for ideal gas:
 # of molecules N is large
 They obey Newton’s laws
 Short-range interactions
with elastic collisions
 Elastic collisions with walls
(an impulse…..pressure)
 What we call temperature T is a
direct measure of the average
translational kinetic energy
 What we call pressure p is a
direct measure of the number
density of molecules, and how
fast they are moving (vrms)
Relationship between Average Energy per Molecule
& Temperature
Physics 207: Lecture 26, Pg 5
Macro-micro connection
 One new relationship
3
k BT   avg
2
2
T
 avg
3k B
2N
p
 avg
3V
vrm s  (v ) avg
2
3k BT

m
Physics 207: Lecture 26, Pg 6
Exercise
 Consider a fixed volume of ideal gas. When N or T is
doubled the pressure increases by a factor of 2.
1 2 3
mv  kB T
2
2
pV  N kBT
1. If T is doubled, what happens to the rate at which a single
molecule in the gas has a wall bounce (i.e., how does v vary)?
(A) x1.4
(B) x2
(C) x4
2. If N is doubled, what happens to the rate at which a
single molecule in the gas has a wall bounce?
(A) x1
(B) x1.4
(C) x2
Physics 207: Lecture 26, Pg 8





A macroscopic “example” of the equipartition theorem
Imagine a cylinder with a piston held in place by a spring.
Inside the piston is an ideal gas a 0 K.
What is the pressure? What is the volume?
Let Uspring=0 (at equilibrium distance)
What will happen if I have thermal energy transfer?
 The gas will expand (pV = nRT)
 The gas will do work on the spring
+Q
Conservation of energy
 Q = ½ k x2 + 3/2 n R T (spring & gas)
 and Newton
S Fpiston= 0 = pA – kx  kx =pA
 Q = ½ (pA) x + 3/2 n RT
 Q = ½ p V + 3/2 n RT (but pV = nRT)
 Q = ½ nRT + 3/2 n RT (25% of Q went to the spring)
½ nRT per “degree of freedom”
Physics 207: Lecture 26, Pg 9
Degrees of freedom or “modes”
 Degrees of freedom or “modes of energy storage in the system” can
be: Translational for a monoatomic gas (translation along x, y, z axes,
energy stored is only kinetic) NO potential energy
 Rotational for a diatomic gas (rotation about x, y, z axes, energy
stored is only kinetic)
 Vibrational for a diatomic gas (two atoms
joined by a spring-like molecular bond
vibrate back and forth, both potential and
kinetic energy are stored in this vibration)
 In a solid, each atom has microscopic
translational kinetic energy and microscopic
potential energy along all three axes.
Physics 207: Lecture 26, Pg 10
Degrees of freedom or “modes”
 A monoatomic gas only has 3 degrees of freedom
(x, y, z to give K, kinetic)
 A typical diatomic gas has 5 accessible degrees of
freedom at room temperature, 3 translational (K) and
2 rotational (K)
At high temperatures there are two more, vibrational
with K and U to give 7 total
 A monomolecular solid has 6 degrees of freedom
3 translational (K), 3 vibrational (U)
Physics 207: Lecture 26, Pg 11
The Equipartition Theorem
 The equipartition theorem tells us how collisions distribute the energy
in the system. Energy is stored equally in each degree of
freedom of the system.
 The thermal energy of each degree of freedom is:
Eth = ½ NkBT = ½ nRT
 A monoatomic gas has 3 degrees of freedom
5
 A diatomic gas has 5 degrees of freedom E  nRT
th
2
 A solid has 6 degrees of freedom E  3nRT
th
3
Eth  nRT
2
 Molar specific heats can be predicted from the thermal energy,
because
Eth  nCT
Monoatomic gas Diatomic gas Elemental solid
3
CV  nRT
2
CV  3nRT
5
CV  nRT
2
Physics 207: Lecture 26, Pg 12
Exercise
 A gas at temperature T is an equal mixture of hydrogen
and helium gas.
Which atoms have more KE (on average)?
(A) H
(B) He
(C) Both have same KE
 How many degrees of freedom in a 1D simple harmonic
oscillator?
(A) 1
(B) 2 (C) 3 (D) 4 (E) Some other number
Physics 207: Lecture 26, Pg 13
The need for something else: Entropy
V1
You have an ideal gas in a box of volume
V1. Suddenly you remove the
partition and the gas now occupies a
larger volume V2.
P
(1) How much work was done by the
system?
P
(2) What is the final temperature (T2)?
V2
(3) Can the partition be reinstalled with all
of the gas molecules back in V1?
Physics 207: Lecture 26, Pg 14
Free Expansion and Entropy
V1
You have an ideal gas in a box of
volume V1. Suddenly you remove
the partition and the gas now
occupies a larger volume V2.
P
(3) Can the partition be reinstalled with
all of the gas molecules back in V1
P
V2
(4) What is the minimum process
necessary to put it back?
Physics 207: Lecture 26, Pg 17
Free Expansion and Entropy
V1
You have an ideal gas in a box of
volume V1. Suddenly you remove
the partition and the gas now
occupies a larger volume V2.
P
(4) What is the minimum energy
process necessary to put it back?
P
Example processes:
V2
A. Adiabatic Compression followed by
Thermal Energy Transfer
B. Cooling to 0 K, Compression,
Heating back to original T
Physics 207: Lecture 26, Pg 18
Exercises
Free Expansion and the 2nd Law
What is the minimum energy process
necessary to put it back?
V1
P
Try:
B. Cooling to 0 K, Compression,
Heating back to original T
Q1 = n Cv T out and put it where…???
P
V2
Need to store it in a low T reservoir and
0 K doesn’t exist
Need to extract it later…from where???
Key point: Where Q goes & where it
comes from are important as well.
Physics 207: Lecture 26, Pg 19
Modeling entropy
 I have a two boxes. One with fifty pennies. The other has none.
I flip each penny and, if the coin toss yields heads it stays put. If
the toss is “tails” the penny moves to the next box.
 On average how many pennies will move to the empty box?
Physics 207: Lecture 26, Pg 20
Modeling entropy
 I have a two boxes, with 25 pennies in each. I flip each penny
and, if the coin toss yields heads it stays put. If the toss is “tails”
the penny moves to the next box.
 On average how many pennies will move to the other box?
 What are the chances that all of the pennies will wind up in
one box?
Physics 207: Lecture 26, Pg 21
2nd Law of Thermodynamics
 Second law: “The entropy of an isolated system never decreases. It
can only increase, or, in equilibrium, remain constant.”
Increasing
Entropy
Entropy measures the probability
that a macroscopic state will
occur or, equivalently, it
measures the amount of
disorder in a system
 The 2nd Law tells us how collisions move a system toward
equilibrium.
 Order turns into disorder and randomness.
 With time thermal energy will always transfer from the hotter to the
colder system, never from colder to hotter.
 The laws of probability dictate that a system will evolve towards the
most probable and most random macroscopic state
Physics 207: Lecture 26, Pg 22
Entropy
 Two identical boxes each contain 1,000,000 molecules.
In box A, 750,000 molecules happen to be in the left half of the
box while 250,000 are in the right half.
In box B, 499,900 molecules happen to be in the left half of the
box while 500,100 are in the right half.
 At this instant of time:
 The entropy of box A is larger than the entropy of box B.
 The entropy of box A is equal to the entropy of box B.
 The entropy of box A is smaller than the entropy of box B.
Physics 207: Lecture 26, Pg 23
Entropy
 Two identical boxes each contain 1,000,000 molecules.
In box A, 750,000 molecules happen to be in the left half of the
box while 250,000 are in the right half.
In box B, 499,900 molecules happen to be in the left half of the
box while 500,100 are in the right half.
 At this instant of time:
 The entropy of box A is larger than the entropy of box B.
 The entropy of box A is equal to the entropy of box B.
 The entropy of box A is smaller than the entropy of box B.
Physics 207: Lecture 26, Pg 24
Reversible vs Irreversible
 The following conditions should be met to make a process
perfectly reversible:
1. Any mechanical interactions taking place in the process
should be frictionless.
2. Any thermal interactions taking place in the process should
occur across infinitesimal temperature or pressure gradients
(i.e. the system should always be close to equilibrium.)
 Based on the above answers, which of the following processes
are not reversible?
1. Melting of ice in an insulated (adiabatic) ice-water mixture at
0°C.
2. Lowering a frictionless piston in a cylinder by placing a bag of
sand on top of the piston.
3. Lifting the piston described in the previous statement by
slowly removing one molecule at a time.
4. Freezing water originally at 5°C.
Physics 207: Lecture 26, Pg 25
Reversible vs Irreversible
 The following conditions should be met to make a process
perfectly reversible:
1. Any mechanical interactions taking place in the process
should be frictionless.
2. Any thermal interactions taking place in the process should
occur across infinitesimal temperature or pressure gradients
(i.e. the system should always be close to equilibrium.)
 Based on the above answers, which of the following processes
are not reversible?
1. Melting of ice in an insulated (adiabatic) ice-water mixture at
0°C.
2. Lowering a frictionless piston in a cylinder by placing a bag of
sand on top of the piston.
3. Lifting the piston described in the previous statement by
removing one grain of sand at a time.
4. Freezing water originally at 5°C.
Physics 207: Lecture 26, Pg 26
Lecture 26
• To recap:
 HW12, Due Tuesday May 4th
For this Tuesday, read through all of Chapter 19!
Physics 207: Lecture 26, Pg 30