Goals:

1
Lecture 25

• Goals

• Chapter 18
• Understand the molecular basis for pressure and
  the ideal-gas 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

2
Macro-micro connectionMean 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 ?
The mean free path is independent of
temperature The mean time between collisions is
temperature dependent
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The mean free path is

• Some typical numbers

Vacuum Pressure (Pa) Molecules / cm3 Molecules/ m3 mean free path
Ambient pressure 105 2.71019 2.71025 68 x 10-9 m
Medium vacuum 100-10-1 1016 1013 1022-1019 0.1 - 100 mm
Ultra High vacuum 10-5-10-10 109 104 1015 1011 1-105 km
4
Distribution of Molecular SpeedsA
Maxwell-Boltzmann Distribution
Fermi Chopper
1 Most probable 2 Mean (Average)
5
Macro-micro connection

• Assumptions for ideal gas
• of molecules N is large
• They obey Newtons 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
6
Macro-micro connection

• One new relationship

pV NkBT
E ½ m v 2
7
Kinetic energy of a gas

• The average kinetic energy of the molecules of an
  ideal gas at 10C has the value K1. At what
  temperature T1 (in degrees Celsius) will the
  average kinetic energy of the same gas be twice
  this value, 2K1?
• (A) T1 20C
• (B) T1 293C
• (C) T1 100C
• The molecules in an ideal gas at 10C have a
  root-mean-square (rms) speed vrms.
• At what temperature T2 (in degrees Celsius) will
  the molecules have twice the rms speed, 2vrms?
• (A) T2 859C
• (B) T2 20C
• (C) T2 786C

8
Exercise

• Consider a fixed volume of ideal gas. When N or
  T is doubled the pressure increases by a factor
  of 2.

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)?
(B) x2
(A) x1.4
(C) x4
2. If N is doubled, what happens to the rate at
which a single molecule in the gas has a wall
bounce?
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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 Uspring0 (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
• 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)

Q
½ nRT per degree of freedom
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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.

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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)

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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
• A diatomic gas has 5 degrees of freedom
• A solid has 6 degrees of freedom
• Molar specific heats can be predicted from the
  thermal energy, because

13
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

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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.
• How much work was done by the system?
• (2) What is the final temperature (T2)?
• (3) Can the partition be reinstalled with all of
  the gas molecules back in V1?

P
P
V2
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Exercises 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.
• How much work was done by the system?

P
P
V2
(A) W gt 0 (B) W 0 (C) W lt 0
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Exercises 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. (2) What is the
final temperature (T2)?
P
P
V2
(A) T2 gt T1 (B) T2 T1 (C) T2 lt T1
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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. (3) Can the
partition be reinstalled with all of the gas
molecules back in V1 (4) What is the minimum
process necessary to put it back?
P
P
V2
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Free Expansion and Entropy
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. (4) What is the
minimum energy process necessary to put it
back? Example processes A. Adiabatic Compression
followed by Thermal Energy Transfer B. Cooling to
0 K, Compression, Heating back to original T
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Exercises Free Expansion and the 2nd Law
What is the minimum energy process necessary to
put it back? Try B. Cooling to 0 K,
Compression, Heating back to original T Q1 n Cv
DT out and put it where??? Need to store it in
a low T reservoir and 0 K doesnt exist Need to
extract it laterfrom where??? Key point Where
Q goes where it comes from are important as
well.
V1
P
P
V2
20
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?

21
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?

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2nd Law of Thermodynamics

• Second law The entropy of an isolated system
  never decreases. It can only increase, or, in
  equilibrium, remain constant.
• 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

Entropy measures the probability that a
macroscopic state will occur or, equivalently, it
measures the amount of disorder in a system
Increasing Entropy
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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.

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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.

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 0C.
• 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 5C.

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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 0C.
• 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 5C.

27
Exercise

• A piston contains two chambers with an
  impermeable but movable barrier between them. On
  the left is 1 mole of an ideal gas at 200 K and
  1 atm of pressure. On the right is 2 moles of
  another ideal gas at 400 K and 2 atm of pressure.
  The barrier is free to move and heat can be
  conducted through the barrier. If this system is
  well insulated (isolated from the outside world)
  what will the temperature and pressure be at
  equilibrium?

p,T,VR
p,T,VL
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Exercise

• If this system is well insulated (isolated from
  the outside world) what will the temperature and
  pressure be at equilibrium?
• At equilibrium both temperature and pressure are
  the same on both sides.
• DETh(Left) DETh(Right) 0
• 1 x 3/2 R (T-200 K) 2 x 3/2 R (T-400 K) 0
• (T-200 K) 2 (T-400 K) 0
• 3T 1000 K
• T333 K
• Now for p.notice p/T const. n R / V
• nL R / VL nR R / VR
• nL VR nR VL
• VR 2 VL

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Exercise

• If this a system is well insulated (isolated from
  the outside world) what will the temperature and
  pressure be at equilibrium?
• VR 2 VL
• and
• VR VL Vinitial (1 x 8.3 x 200 / 105 2 x
  8.3 x 400 / 2x105 )
• Vinitial 0.050 m3
• VR 0.033 m3 VL 0.017 m3
• PR nR RT / VR 2 x 8.3 x 333 / 0.033 1.7
  atm
• Pl nL RT / Vl 1 x 8.3 x 333 / 0.017 1.7
  atm

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Lecture 26

• To recap
• HW12, Due Tuesday May 4th
• For this Tuesday, read through all of Chapter 19!