Goals:

1
Lecture 26

• Goals

• Chapters 18, entropy and second law of
  thermodynamics
• Chapter 19, heat engines and refrigerators

• No lab this week.

2
Equipartition theorem

• Things are more complicated when energy can be
  stored in other degrees of freedom of the system.

monatomic gas translation solids
translationpotential energy diatomic molecules
translationvibrationsrotations
3
Equipartition theorem

• The thermal energy is equally divided among all
  possible energy modes (degrees of freedom). The
  average thermal energy is (1/2)kBT for each
  degree of freedom.

eavg(3/2) kBT (monatomic gas) eavg(6/2) kBT
(solids) eavg(5/2) kBT (diatomic molecules)

• Note that if we have N particles

Eth(3/2)N kBT (3/2)nRT (monatomic
gas) Eth(6/2)N kBT (6/2)nRT (solids) Eth(5/2)N
kBT (5/2)nRT (diatomic molecules)
4
Specific heat

• Molar specific heats can be directly inferred
  from the thermal energy.

Eth(6/2)N kBT (6/2)nRT (solid) ?Eth(6/2)nR?TnC
?T C3R (solid)

• The specific heat for a diatomic gas will be
  larger than the specific heat of a monatomic gas

CdiatomicCmonatomicR
5
Second Law and Entropy

• A perfume bottle breaks in the corner of a room.
  After some time, what would you expect?

B)
A)
6
very unlikely

• The probability for each particle to be on the
  left half is ½.

probability(1/2)N
7
Second Law of thermodynamics

• The entropy of an isolated system never
  decreases. It can only increase, or in
  equilibrium, remain constant.
• The laws of probability dictate that a system
  will evolve towards the most probable and most
  random macroscopic state
• Thermal energy is spontaneously transferred from
  a hotter system to a colder system.

8
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.)

9
Reversible vs Irreversible

• Based on the above comments, which of the
  following processes is not reversible?
• A. Lowering a frictionless piston in a cylinder
  by placing a bag of sand on top of the piston.
• B. Lifting the piston described in the previous
  statement by removing one tiny grain of sand at a
  time.

10
Heat Engines

• Turning heat into work Industrial revolution.

11
Key concepts

• Work done by the system

Wsystem-Wexternal

• Energy reservoir An object that interacts with
  the system that is sufficiently large such that
  its temperature is almost constant.

QH The amount of heat transferred to/from hot
reservoir QC The amount of heat transferred
to/from cold reservoir
12
Energy-transfer diagram
Hot reservoir
QH
cyclic system ?Esystem0
WoutQH-QC
Wout
QC
Cold reservoir
13
Thermal efficiency
For practical reasons, we would like an engine to
do the maximum amount of work with the minimum
amount of fuel. We can measure the performance of
a heat engine in terms of its thermal efficiency
? (lowercase Greek eta), defined as
We can also write the thermal efficiency as
14

• What is the largest thermal efficiency that a
  heat engine can have?

C) ?1/2
A) ?2
B) ?1
D) ?0
15
Refrigerators

• Devices that uses work to transfer heat from a
  colder object to a hotter object.

Hot reservoir
QH
WinQCQH
Win
KQC/Win
QC
Cold reservoir
16
Is perfect engine possible?
Hot reservoir
QH1
QH2
QH

Wout
Win
QC
QC
Cold reservoir
17
Turbines Brayton Cycle
18

• Which of the following processes would have the
  largest work output per cycle?

A)
B)
C)
P
P
P
V
V
V
19
Internal combustion engine gasoline engine

• A gasoline engine utilizes the Otto cycle, in
  which fuel and air are mixed before entering the
  combustion chamber and are then ignited by a
  spark plug.

Otto Cycle
(Adiabats)
20
The best thermal engine ever, the Carnot engine

• A perfectly reversible engine (a Carnot engine)
  can be operated either as a heat engine or a
  refrigerator between the same two energy
  reservoirs, by reversing the cycle and with no
  other changes.

21
The Carnot Engine

• Carnot showed that the thermal efficiency of a
  Carnot engine is

• All real engines are less efficient than the
  Carnot engine because they operate irreversibly
  due to the path and friction as they complete a
  cycle in a brief time period.