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Temperature and Thermal Equilibrium
Thermodynamics
From the Greek thermos meaning heat and dynamis
meaning power is a branch of physics that studies the
effects of changes in temperature, pressure, and
volume on physical systems at the macroscopic scale
by analyzing the collective motion of their particles
using statistics.
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Temperature, pressure, and volume quantitatively define the
state of a gas
Temperature, pressure, and volume are state variables
We want to determine the relationship (i.e., find some
equations) between these state variables and more
importantly, relationships between changes in these
state variables.
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The zeroth law of thermodynamics:
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If two systems are each in thermal equilibrium with a
third, then they are in thermal equilibrium with one
another.
If two thermal systems are in thermal equilibrium with
one another, then they have the same temperature.
Temperature is a way of determining
(measuring?) thermal equilibrium
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Two systems have the same temperature
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They are in thermal equilibrium
Two systems have different temperatures
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The are NOT in thermal equilibrium
Wikipedia: http://en.wikipedia.org/wiki/Thermodynamics
Temperature Scales
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tF = 32° at freezing point of water
tF = 212° at boiling point
Solids expand when temperature increases; describe by
coefficient of thermal expansion (α):
α=
Celsius:
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tC = 0° at freezing point of water
tC = 100° at boiling point
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T (° F ) = T (°C ) + 32
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Thermal Expansion and Stress
Fahrenheit:
Kelvin:
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tK = 0° at absolute zero
ΔtK = ΔtC
T (K ) = T (°C ) + 273.15
T (°C ) = T (K ) − 273.15
T (°C ) =
„
5
[T (°F ) − 32]
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OR ΔL = LoαΔT
For fluids, use coefficient of volume expansion (β) instead
(as length is not well-defined):
β=
„
1 ΔL
Lo ΔT
1 ΔV
Vo ΔT
OR ΔV = Vo βΔT
Induced stress when the material does not freely expand or
contract due to a temperature change
‰
The material is “restricted” in some manner
∴ σ=
F
L αΔT
=E o
= EαΔT
A
Lo
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Ideal Gas Law
Kinetic Theory of Gases
Using moles
Using molecules
PV = nRT
PV = NkT
n = number of moles
N = number of molecules
R = universal gas constant
k = Boltzmann’s constant
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1
2 ⎛1
⎞
PV = Nmv 2 = N ⎜ mv 2 ⎟
3
3 ⎝2
⎠
= 1.38 x 10-23 J / K
= 8.315 J / (mol-K)
•Dilute, Volume of Molecules is ~ Zero
•No Attractions Between Molecules
•Temperature must be in Absolute Units, K
“Real” gases do not follow the ideal gas law precisely. However, at low
pressure and temperatures “not too close” to the liquefaction point, the
ideal gas law is quite accurate and useful for “real” gases.
Molecular Speeds
1
3
K = mv 2 = kT
2
2
⇒
1
1
3
K = mv 2
and PV = NkT
∴ K = mv 2 = kT
2
2
2
Temperature is a measure of the average
kinetic energy (internal energy?) of the gas.
For constant volume, pressure increases
directly proportional to an increase in
average kinetic energy (temperature) AND
an increase in the number of molecules.
But
This is called the Equation of State. Why “Ideal”?
Since
Connect microscopic properties (kinetic
energy and momentum) of molecules to
macroscopic “state” properties of a gas
(temperature and pressure).
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Real Gasses
v2 =
3kT
= vrms
m
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Distribution of molecular speeds,
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The Maxwell-Boltzmann distribution
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„
mv 2
⎛ m ⎞ 2 2 − 2 kt
f (v) = 4πN ⎜
⎟ v e
⎝ 2πkT ⎠
∞
∫0
Note
v=
vrms =
‰
1 ∞ 2
3kT
v f (v)dv =
∫
0
N
m
„
vp ⇒
2kT
df (v)
= 0 ⇒ vp =
m
dv
Pressure not too high
Not near liquefaction point
Why the “breakdown”?
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f (v)dv = N
1 ∞
8kT
∫ vf (v)dv = πm
N 0
Reasonable
Near liquefaction
Intermolecular forces matter
P-V Diagram
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Each line is at a constant temperature
Solid = Real, Dashed = Ideal
Point c is the critical point, curve C is critical temp.
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Real Gasses - Better Approximations
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Ideal Gas
P=
nRT
V
Vapor Pressure
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Since the liquid has a distribution of
molecular speeds
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First Order
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Clausius Equation of State (EoS)
P=
„
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Van der Waals
P=
Vapor pressure “explains” boiling
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Liquid temperature = SVP temperature
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The total pressure of a mixture of gasses = the
sum of vapor pressure of the constituent gasses
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Boiling occurs
Law of partial pressures
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RH =
partial pressure of H 2 O
× 100%
SVP of H 2O
Dew Point: Temperature at which unsaturated air
will become saturated.
Some molecules will be “recaptured”
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Evaporation
Since the gas has a distribution of
molecular speeds
Condensation
The gas will exert a “vapor” pressure
‰
At equilibrium: saturated vapor
pressure (SVP)
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This is temperature dependent.
Mean Free Path
Molecular concentration
Assume the other molecules are not moving,
then the number of molecules in the cylinder is
NC =
PT = P1 + P2 + …
Relative Humidity
‰
„
nRT
a
−
V − nb (V / n )2
Vapor Pressure
„
„
nRT
V − nb
Second Order
Some molecules will “escape”
N
N
Vcyl = π (2r ) 2 v Δt
V
V
Mean Free Path is average
distance between collisions
lM =
d
v Δt
1
=
=
NC N π (2r ) 2 v Δt N 4πr 2
V
V
If you account for the movement of
the other molecules
lM =
1
N
4 2πr 2
V
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Heat
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Work
What is heat?
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Heat will only flow from the system with the higher
temperature to the system with the lower temperature
Heat will only flow from the system with the higher average
internal energy to the system with the lower average
internal energy
Total internal energy does not matter.
First Law of Thermodynamics
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When temperature changes, internal energy has changed –
may happen through heat transfer or through mechanical work
First law is a statement of conservation of energy
Change in internal energy of system equals the difference
between the heat added to the system and the work done by the
system
ΔU = Q − W
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1
Heat (Q) is the “flow” or “transfer” of energy from
one system to another
Often referred to as “heat flow” or “heat transfer”
Requires that one system must be at a higher
temperature than the other
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V
W = ∫V 2 PdV
dU = dQ − dW
Differential form
Heat added +, heat lost -, work done by system +, work done on system –
Internal Energy U is a state property
Work W and heat Q are not
But work and heat are involved in thermodynamic processes that change
the state of the system
Molar Specific Heats for Gasses
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Molar specific heats for gasses are different if heat is
added at constant pressure vs. constant volume
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Isobaric, ΔP = 0
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QP = nCPΔT
QV = nCVΔT
W = PΔV
QP = ΔU + PΔV
Isochoric, ΔV = 0
‰
W = 0 ⇒ QV = ΔU
If the two processes result in the same temperature change, ΔU is the same.
⇒ QV = QP − PΔV
⇒ nC P ΔT − nCV ΔT = nRΔT
⇒ C P − CV = R
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Types of Transformations
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Isothermal, ΔT = 0
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ΔU = 0, ⇒ W = Q
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V
V
A
‰
A
⎛V ⎞
nRT
dV =nRT ln⎜⎜ B ⎟⎟
V
⎝ VA ⎠
V
A
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Work done by the system lowers the
internal energy of the system by an
equal amount
Temperature can change only if work
is done.
C
PV = constant, where γ = P
CV
γ
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Results from molecular
interactions
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Energy is transferred
through interaction
Convection
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Collisions?
P1V1 − P2V2
γ −1
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ΔU calculated from 1st law
The change in internal energy of
the system equals the heat added
Reversible & Irreversible Processes
T −T
T −T
ΔQ
= kA 1 2 = A 1 2
l
Δt
R
l
R-Value: R =
k
dQ
dT
= −kA
dt
dx
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Example of a Reversible Process:
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Results from the mass
transfer of material
Think fluid flow
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ΔQ
= eσAT 4
Net heat flow
Energy transferred by Δt
between two objects
electromagnetic
0
≤
e
≤
1
radiation (waves)
ΔQ
4
4
σ = 5.67 × 10−8 W / m 2 ⋅ K 4 Δt = eσA T1 − T2
Does not require a
“medium”
Radiation
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A
W = 0 ⇒ ΔU = Q
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Wadiabatic =
V
Isochoric, ΔV = 0
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Heat Transfer
Conduction
Work = Pressure*Change in Vol
W = ∫V B PdV =P ∫V B dV =P(VB − VA )
⇒ ΔU = -W
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W = PΔV
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Adiabatic, Q = 0
‰
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Isobaric, ΔP = 0
Work done by the system equals the
heat added to the system
W = ∫V B PdV = ∫V B
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Types of Transformations
(
)
Cylinder must be pulled or pushed slowly
enough (quasistatically) that the system
remains in thermal equilibrium (isothermal).
Change where system is always in
thermal equilibrium: reversible process
Change where system is not always in
thermal equilibrium: irreversible process
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Examples of irreversible processes:
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Free expansion of a gas
Melting of ice in warmer liquid
Frictional heating
Anything that is real All real
processes
are irreversible!
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Heat Engine
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An engine is a device that cyclically
transforms thermal energy (heat?)
into mechanical energy (useful work).
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Efficiency: Fraction of heat flow becomes mechanical work:
e=
„
W QH − QL
Q
=
= 1− L
QH
QH
QH
„
Heat pumps, refrigerators, and air conditions
are engines run in reverse:
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A minimal version of an engine has two reservoirs
at different temperatures TH and TL, and follows a
idealized reversible cycle known as the Carnot
cycle.
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Heat Pumps, Refrigerators, and
Air Conditioners
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Efficiency of the Carnot cycle
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Realistically, What is TL?
What is a reasonable eC?
eC =
W
Q
T
= 1− L = 1− L
QH
QH
TH
Heat Pumps and Refrigerators
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Since the (idealized) Carnot engine is the
most efficient heat engine, the Carnot
refrigerator is the most efficient refrigerator.
Coefficient of Performance:
Q
QL
TL
CP = L =
=
W QH − QL TH − TL
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Refrigerator and air conditions remove heat from
the cold reservoir and put it into the surroundings
(hot reservoir), keeping the food/room cold.
A heat pump takes energy from the cold reservoir
and puts it into a room or house (hot reservoir),
thereby warming it.
In either case, energy must be added!
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Work must be performed ON the system!
Carnot Example
Temperature differences in the ocean have been proposed as a possible
energy source in the tropics. Surface water at 29oC could act as a hot
reservoir, and deep water at 3.0oC could serve as a cold reservoir.
Ammonia gas could be a working fluid in a heat engine that runs between
the two thermal reservoirs. What is the maximum efficiency of such an
engine?
Heat Pumps work similarly but have a
different objective, namely warm the house.
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Coefficient of Performance:
CP =
QH
QH
TH
=
=
W QH − QL TH − TL
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The Second Law of Thermodynamics
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There are many ways of expressing the
second law of thermodynamics; here are two:
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The Clausius form: It is impossible to construct a
cyclic engine whose only effect is to transfer
thermal energy from a colder body to a hotter body.
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Spontaneous heat flow always goes from the highertemperature body to the lower-temperature one.
The Kelvin form: It is impossible to construct a
cyclic engine that converts thermal energy from a
body into an equivalent amount of mechanical work
without a further change in its surroundings.
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Entropy and the Second Law
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Entropy is a measure of disorder.
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Entropy is a measure of the energy
unavailable to do work.
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Thermal energy cannot be entirely converted to work.
A 100% efficient engine is impossible.
There is some controversy about this!
The process of creating disorder (as well as order)
increases entropy.
It is a measure of the dispersal of energy.
Energy is dispersed (used up?) in processes that
create both order and disorder.
These definitions are incomplete!
Entropy and the Second Law
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The entropy of an isolated system never decreases;
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The Laws of Thermodynamics Simplified
spontaneous (irreversible) processes always increase
entropy.
All the consequences of the second law of
thermodynamics follow from the treatment of
entropy as a measure of disorder. (?)
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Making engines that would convert mechanical energy
entirely to work would require entropy to decrease in
isolated system – can’t happen.
Many familiar processes increase entropy – shuffling cards,
breaking eggs, and so on.
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We never see these processes spontaneously happening in
reverse – a movie played backwards looks silly. This
directionality is referred to as the arrow of time.
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Zeroth: “You must play the game.”
First:
“You can’t win.”
Second: “You can’t break even.”
Third: “You can’t quit the game.”
Thermodynamics at www.wikipedia.org.
So, to what state is the universe heading?
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