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:

Fahrenheit Scale (°F)Celsius (Centigrade) Scale (°C)

Kelvin Scale (K)

Temperature Scales• 3 Common Scales used to measure temperature:

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 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 ScaleSometimes 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.

X

T

•

•

FP2FP1

X1

X2

Error!

Xm

• 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!!

n.b.p. normal boiling point

Ranges of Various Types of Thermometers

P or V

V

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”

Reference Points for Temperature Scales& Some Brief History.

Pt & RuO2 Resistance Thermometers

Blundell and Blundell, Concepts in Thermal

Physics (2006)

)++1(= 20 BtAtRRt

t T For 0 ºC < T < 850 ºC

Radiation Energy Density

InfraredUV-Visible

Spectral Distribution of Thermal Radiation(Planck Distribution Law)

Reports on Progress in Physics, vol. 68 (2005) pp. 1043–1094

Fixed Temperature Reference Points

Melting points of metals and alloys

Temperature Scale with a Single Fixed Point

• Defining a temperature scale with a single fixed point requires a linear (monotonic) relationship between a

Thermometric Parameter X & theTemperature Tx: X = cT, is a constant

• By international agreement in 1954,

The Kelvin or ThermodynamicTemperature 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.

K

T

cT

cT

XX x

TP

x

TP 16.273==

)(16.273=TP

x XX

T

•What variable should be measured to use the

thermodynamic temperature scale?

So,

Temperature Scale with a Single Fixed Point

•For Thermometric Parameter X atany temperature Tx:

( ) cTTP VnR ==

TPTP PP

KTK

TPP

16.273=⇒16.273

=

The Ideal Gas Temperature Scale

The Ideal Gas Law:Hold V & n constant!

TP = 273.16KUnknown T

Gas

P, V

A Constant-Volume Gas Thermometer

Defining the Kelvin & Celsius Scales•“One Kelvin degree is (1/273.16) of thetemperature 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 at1 atm. water boils at 99.97 K above themelting point of ice (i.e. at 373.12 K) so1 K is not exactly equal to 1° Celsius!

Comparison of temperature scales

Comment Kelvin Celsius Fahrenheit Rankine Delisle Newton Réaumur Rømer

Absolute zero 0.00 −273.15 −459.67 0.00 559.73 −90.14 −218.52−135.9

0

Lowest recorded surface temperature on Earth (Vostok, Antarctica - July 21, 1983)

184 −89 −128 331 284 −29 −71 −39

Fahrenheit's ice/salt mixture 255.37 −17.78 0.00 459.67 176.67 −5.87 −14.22 −1.83

Ice melts (standard pressure) 273.15 0.00 32.00 491.67 150.00 0.00 0.00 7.50

Triple point of water 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, Libya - September 13, 1922) But that reading is questioned

331 58 136 596 63 19 46 38

Water boils (standard pressure)373.1

5100.00 211.97 671.64 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

Comparison of temperature scales

Section 3.6: Heat Reservoirs

The 2The 2ndnd Law Tells Us Law Tells Us::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.

E1 E2 = E - E1

• Recall: We considered 2 macroscopic systems A1, A2, interacting & in equilibrium. The combined system A0 = A1 + A2, was isolated.

A1 A2

• 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 + E2

1 1 1 1 2 1,E E E E E E

• 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 :

• 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

lnBS k E ,

ln

N V

E

E

1 2 • It also resulted in the fact that the

equilibrium condition for A1 & A2 is that the two temperatures are equal!

• 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

• Suppose that A2 absorbs a small about 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

N V

E

E

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:

• This results in S2 kBQ2. Also noting that since the two systems are in equilibrium, T2 = T1 T gives:

S2 [Q2/T]• In Reif’s notation this is:

S' [Q'/T]

,

ln

N V

E

E

,

ln1B

N V

Ek

T E

Summary•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 2The 2ndnd Law Law::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:

.T

QS

Section 3.8: Equations of State

Dependence of Ω on External Parameters• 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.• Now: Consider the case in which

A1 & A2 are also characterized by external parameters x1 & x2.

E1 E2 = E - E1

x2 A2

x1 A1

• 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 1 1 1 1 2 1,E E E E E E

• Using calculus to find E1 that maximizes Ω(E1, E – E1) resulted in statistical definitions of the Entropy S & the Temperature Parameter :

lnBS k E

,

ln

N V

E

E

• 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><X> ∂ln[Ω(E,x)]/∂x (1)

Or <X> = (kBT)∂ln[Ω(E,x)]/∂x (2)

In terms of Entropy S:

<X> = T ∂S(E,x)]/∂x (3)• (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 equilibriumThe Mean Generalized Force <X> is

<X> ∂ln[Ω(E,x)]/∂x (1)

Or <X> = (kBT)∂ln[Ω(E,x)]/∂x (2)

<X> = T ∂S(E,x)]/∂x (3)

• (1) ((2) or (3)) is an Equation of State for system A1.

• Note that there is an Equation of State for each different external parameter x.