Section 3.5: Temperature
Temperature
• Temperature  The property of an object that
determines the DIRECTION OF HEAT
energy (Q) TRANSFER to or from other objects.
Temperature Scales
• Three Common Scales are used to
measure temperature:
1. Fahrenheit Scale (°F)
2. Celsius (Centigrade) Scale (°C)
3. Kelvin Scale (K)
Temperature Scales
3 Common Temperature Scales
Fahrenheit Scale (°F)
• Used widely in the U.S. Divides the difference between
freezing & boiling point of water at sea level into 180 steps.
Celsius (Centigrade) Scale (°C)
• Used almost everywhere else in the world. Divides the
freezing to boiling continuum into 100 equal steps.
Kelvin Scale (K)
• Used by scientists. Created by Lord Kelvin. Starts with
T = 0 K  “Absolute Zero”.
• 3 Common Scales are used to measure temperature.
• However, historically, there have also been many other
temperature scales used in the past! Among these are:
1. Rankine Scale (°Ra). 2. Réaumur Scale (°Ré)
3. Newton Scale (°N). 4. Delisle Scale (°D).
5. Rømer Scale. (°Rø).
Some Conversions:
Temperature Scale Comparisons
• Boiling Point of Water
212°F = 100°C = 373.15 K
• Melting Point of Ice
32°F = 0°C = 273.15 K
• “Absolute Zero”
-459.67°F = -273.15°C = 0 K
• Average Human Body Temperature:
98.6°F = 37°C = 310.16 K
• Average Room Temperature:
68°F = 20°C = 293.16 K
Common Conversions
Celsius to Fahrenheit:
F° = (9/5)C° + 32°
Fahrenheit to Celsius:
C° = (5/9)(F° - 32°)
The Kelvin Scale
Sometimes Called the Thermodynamic Scale
• The Kelvin Scale was created by Lord
Kelvin to eliminate the need for negative
numbers in temperature calculations.
The Kelvin Scale is DEFINED as follows:
1. The degree size is IDENTICAL to that on
the Celsius scale.
2. The temperature in Kelvin degrees at the triple
point of water is DEFINED to be Exactly
273.16 K
How is Temperature Measured?
• Of course, temperature is measured using a
Thermometer!
• Thermometer  Any object that has a
property characterized by a
Thermometric Parameter
• Thermometric Parameter  Any
parameter X, that varies in a known (calibrated!)
way with temperature. Measure the value of X at
TWO fixed points of temperature & interpolate &
extrapolate as needed.
• Thermometric Parameter  Any parameter X, that
varies in a known (calibrated!) way with temperature.
Measure the value of X at TWO fixed points of
temperature & interpolate & extrapolate as needed.
Two (or more)
reference points can
result in errors when
extrapolating outside
of their range!!
Xm
X
•
X2
X1 •
FP1
Error!
FP2
T
Ranges of Various Types of Thermometers
V
P or V
n.b.p.  “normal boiling point”
Reference Points for Temperature Scales
Some Brief History.
Daniel Fahrenheit (1724)
• Ice, water & ammonium chloride mixture = 0 °F
• Human body = 96 °F (now taken as 98.6 °F)
Anders Celsius (1742)
• Originally: Boiling point of water = 0 ºC!
Melting point of ice = 100 ºC!
• The Scale was later reversed.
This scale was originally
called “centigrade”
Pt & RuO2 Resistance Thermometers
2
Rt = R0 (1 + At + Bt )
tT
For 0 ºC < T < 850 ºC
Blundell & Blundell,
Concepts in Thermal
Physics (2006)
Spectral Distribution of Thermal Radiation
(Planck Distribution Law)
Radiation
Energy
Density
UV-Visible
Infrared
Fixed Temperature Reference Points
Melting points of metals and alloys
Reports on Progress in Physics,
vol. 68 (2005) pp. 1043–1094
Temperature Scale (with Single Fixed Point)
• Defining a temperature scale with a single fixed point
requires a linear (monotonic) relationship between a
Thermometric Parameter X & the
Temperature Tx: X = cTx, (c is a constant)
• By international agreement in 1954,
The Kelvin or Thermodynamic
Temperature Scale
uses the triple point (TP) of water as the fixed point. There,
The temperature is DEFINED
(NOT measured!) to be EXACTLY 273.16 K
The Triple Point of Water
•
At the triple point of water: gas, solid & liquid
all co-exist at a pressure of 0.0006 atm.
Temperature Scale with a Single
Fixed Point
• For Thermometric Parameter X at
any temperature Tx:
cTx
Tx
X
=
=
X TP cTTP 273.16 K
X
So, Tx = 273.16(
)
X TP
• What variable should be measured to use the
thermodynamic temperature scale?
The Ideal Gas Temperature Scale
The Ideal Gas Law:
Hold V & n constant!
nR )
(
P = V T = cT
Gas
P, V
Unknown T
TP = 273.16K
P
T
P
=
⇒T = 273.16 K
PTP 273.16 K
PTP
A Constant-Volume Gas Thermometer
Defining the Kelvin & Celsius Scales
• “One Kelvin degree is (1/273.16) of the
temperature of the triple point of water.”
• Named after William Thompson (Lord Kelvin).
Relationship between °C and K
°C = K - 273.15
• Note that careful measurements find that at
1 atm. water boils at 99.97 K above the
melting point of ice (i.e. at 373.12 K) so
1 K is not exactly equal to 1° Celsius!
Comparison of Temperature Scales
Comparison of temperature scales
Comment
Rankine
Delisle
Newton
Réaumur
Rømer
0.00
559.73
−90.14
−218.52
−135.9
0
−128
331
284
−29
−71
−39
−17.78
0.00
459.67
176.67
−5.87
−14.22
−1.83
273.15
0.00
32.00
491.67
150.00
0.00
0.00
7.50
273.16
0.01
32.018
491.688
149.985
0.0033
0.008
7.50525
Ave. surface temp on Earth
288
15
59
519
128
5
12
15
Ave. human body temp.*
310
37
98
558
95
12
29
27
Highest recorded surface
temperature o Earth ('Aziziya,
331
58
136
596
63
19
46
38
Water boils (standard pressure)
373.1
5
100.00
211.97
671.64
0.00
33.00
80.00
60.00
Titanium melts
1941
1668
3034
3494
−2352
550
1334
883
The surface of the Sun
5800
5500
9900
10400
−8100
1800
4400
2900
Kelvin
Celsius
Fahrenheit
Absolute zero
0.00
−273.15
−459.67
Lowest recorded surface
temperature on Earth (Vostok,
184
−89
Fahrenheit's ice/salt mixture
255.37
Ice melts (standard pressure)
Triple point of water
Antarctica - July 21, 1983)
Libya - September 13, 1922) But
that reading is questioned
Section 3.6: Heat Reservoirs
nd
2
The
Law Tells Us That:
Heat flows from objects at
high temperature to objects at
low temperature
because this process increases disorder & thus
it increases the entropy
of the system.
Heat Reservoirs
• The following discussion is similar to Sect. 3.3, where
the Energy Distribution Between Systems in
Equilibrium was discussed & the conditions for
equilibrium were derived.
A1
A2
• Recall: We considered 2 macroscopic
E2 =
systems A1, A2, interacting & in
E1
E - E1
equilibrium. The combined system
A0 = A1 + A2, was isolated.
• Then, we found the most probable energy of system A1,
using the fact that the probability finding of A1 with a
particular energy E1 is proportional to the product of the
number of accessible states of A1 times the number of
accessible states of A2, Consistent with Energy
Conservation:
E = E1 + E 2
• The probability finding of A1 with a particular energy E1 is
proportional to the number of accessible states of A1 times the
number of accessible states of A2,
Consistent with Energy Conservation: E = E1 + E2.
• That is, it is proportional to
  E1 , E  E1   1  E1   2  E  E1 
• Using differential calculus to find the E1 that maximizes
Ω(E1, E – E1) resulted in statistical definitions of both the
Entropy S & the Temperature Parameter :
S  kB ln   E 
• It also resulted in the fact that the
equilibrium condition for A1 & A2 is
that the two temperatures are equal!
  ln   E  
 

E

 N ,V
1   2
• Consider a special case of the situation just reviewed. A1
& A2 are interacting & in equilibrium. But,
A2 is a Heat Reservoir or Heat Bath for A1.
• Conditions for A2 to be a Heat Reservoir for A1:
E1 <<< E2,
f1 <<< f2
Reif’s Terminology: A2 is “large” compared to A1
• Assume that A2 absorbs a small amount of heat energy Q2 from A1.
Q2 = E2  E1
• The change in A2’s entropy in this process is
S2 = kB[lnΩ(E2 + Q2) – lnΩ(E2)]
• Expand S2 in a Taylor’s Series for
small Q2 & keep only the lowest order
term. Also use the temperature
parameter definition.
  ln   E  
 


E

 N ,V
S2 = kB[lnΩ(E2 + Q2) – lnΩ(E2)]
• Expand S2 in a Taylor’s Series for small Q2 & keep only the
lowest order term. Use the temperature parameter definition &
connection with absolute temperature T:
  ln   E  
 

E

 N ,V
  ln   E  
1
 kB 

T
E

 N ,V
l
• This results in S2  kBQ2. Also note that since
the two systems are in equilibrium, T2 = T1 T so:
S2  [Q2/T]
• In Reif’s notation this is:
S'  [Q'/T]
Summary: Heat Reservoirs
• For a system interacting with a heat reservoir at
temperature T & giving heat Q' to the
reservoir, the change in the entropy of the
reservoir is:
S'  [Q'/T]
• For an infinitesimal amount of heat đQ
exchanged, the differential change in the
entropy is:
dS = [đQ/T]
The
nd
2
Law of Thermodynamics:
• Heat flows from high temperature objects to low
temperature objects because this increases the disorder
& thus the entropy of the system. We’ve shown that,
For a system interacting with a heat
reservoir at temperature T & exchanging
heat Q with it, the entropy change is:
Q
S  .
T
Section 3.8: Equations of State
External Parameter Dependence of Ω
• The following is similar to Sect. 3.3, where the
Energy Distribution Between Systems in
Equilibrium was discussed & the conditions for
equilibrium were derived.
• Recall: We considered 2 macroscopic systems
A1, A2, interacting & in equilibrium. The combined
system A0 = A1 + A2, was isolated.
A1
x1
A2
x2
E2 =
E - E1
• Now: Consider the case in which A1
& A2 are also characterized by
E1
external parameters x1 & x2.
• As discussed earlier, corresponding to x1 & x2, there
are generalized forces X1 & X2.
• In earlier discussion, we found the most probable energy
of system A1, using the fact that the probability finding of A1
with energy E1 is proportional to the product of the number of
accessible states of A1 times the number of accessible states of
A2, Consistent with Energy Conservation: E = E1 + E2
• That is, it is proportional to
  E1 , E  E1   1  E1   2  E  E1 
• Using calculus to find E1 that maximizes Ω(E1, E – E1)
resulted in statistical definitions of the Entropy S & the
Temperature
  ln   E  
 

S  kB ln   E 
Parameter .
E

 N ,V
• Another result is that the equilibrium condition
for A1 & A2 is that the temperatures are equal!
1   2
• When external parameters are present, the
number of accessible states Ω depends on
them & on energy E.
Ω = Ω(E,x)
• In analogy with the energy dependence
discussion, the probability finding of A1 with a
particular external parameter x1 is proportional
to the number of accessible states of A1 times
the number of accessible states of A2.
• That is, it is proportional to
Ω(E1,x1;E2,x2) = Ω(E1,x1)Ω(E - E1,x2)
• The probability finding of A1 with a particular external
parameter x1 is proportional to the number of accessible
states of A1 times the number of accessible states of A2.
Ω(E1,x1;E2,x2) = Ω(E1,x1)Ω(E - E1,x2)
• Using differential calculus to find the x1 that maximizes
Ω(E1,x1;E2,x2) results in a statistical definition of
The Mean Generalized Force <X>
or
<X>  ∂ln[Ω(E,x)]/∂x
<X> = (kBT)∂ln[Ω(E,x)]/∂x
(1)
(2)
<X> = T[∂S(E,x)]/∂x
(3)
• In terms of Entropy S:
• (1) ((2) or (3)) is called an Equation of State for system
A1. Note that there is an Equation of State for each
different external parameter x.
Summary
• For interacting systems with an external
parameter x, at equilibrium
The Mean Generalized Force <X> is
Or
<X>  ∂ln[Ω(E,x)]/∂x
<X> = (kBT)∂ln[Ω(E,x)]/∂x
<X> = T ∂S(E,x)]/∂x
(1)
(2)
(3)
• (1) ((2) or (3)) is an Equation of State for
system A1. There is an Equation of State
for each different external parameter x.