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 = K Melting Point of Ice 32°F = 0°C = K “Absolute Zero” °F = °C = 0 K Average Human Body Temperature: 98.6°F = 37°C = K Average Room Temperature: 68°F = 20°C = 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 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 FP 2 FP 1 X1X1 X2X2 Error! XmXm 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 & RuO 2 Resistance Thermometers Blundell and Blundell, Concepts in Thermal Physics (2006) t  Tt  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 & the Temperature T x : X = cT, 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 K.

The Triple Point of Water At the triple point of water: gas, solid & liquid all co-exist at a pressure of atm.

What variable should be measured to use the thermodynamic temperature scale? So, Temperature Scale with a Single Fixed Point For Thermometric Parameter X at any temperature T x :

The Ideal Gas Temperature Scale The Ideal Gas Law: Hold V & n constant! T P = KUnknown T Gas P, V

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 Note that careful measurements find that at 1 atm. water boils at K above the melting point of ice (i.e. at K) so 1 K is not exactly equal to 1° Celsius!

Comparison of temperature scales Comment Kelvin Celsius Fahrenheit Rankine Delisle NewtonRéaumur Rømer Absolute zero 0.00 − − −90.14− − Lowest recorded surface temperature on Earth (Vostok, Antarctica - July 21, 1983) 184−89− −29−71−39 Fahrenheit's ice/salt mixture − −5.87−14.22−1.83 Ice melts (standard pressure) Triple point of water Ave. surface temp on Earth Ave. human body temp.* Highest recorded surface temperature o Earth ( 'Aziziya, Libya - September 13, 1922) But that reading is questioned Water boils (standard pressure) Titanium melts − The surface of the Sun − Comparison of temperature scales

Section 3.6: Heat Reservoirs

The 2 nd 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. E1E1 E 2 = E - E 1 Recall: We considered 2 macroscopic systems A 1, A 2, interacting & in equilibrium. The combined system A 0 = A 1 + A 2, was isolated. A1A1 A2A2 Then, we found the most probable energy of system A 1, using the fact that the probability finding of A 1 with a particular energy E 1 is proportional to the product of the number of accessible states of A 1 times the number of accessible states of A 2, Consistent with Energy Conservation: E = E 1 + E 2

Using differential calculus to find the E 1 that maximizes Ω(E 1, E – E 1 ) resulted in statistical definitions of both the Entropy S & the Temperature Parameter  : The probability finding of A 1 with a particular energy E 1 is proportional to the number of accessible states of A 1 times the number of accessible states of A 2, Consistent with Energy Conservation: E = E 1 + E 2. That is, it is proportional to It also resulted in the fact that the equilibrium condition for A 1 & A 2 is that the two temperatures are equal!

Consider a special case of the situation just reviewed. A 1 & A 2 are interacting & in equilibrium. But, A 2 is a Heat Reservoir or Heat Bath for A 1. Conditions for A 2 to be a Heat Reservoir for A 1 : E 1 <<< E 2, f 1 <<< f 2 Reif’s Terminology: A 2 is “large” compared to A 1 Suppose that A 2 absorbs a small about of heat energy Q 2 from A 1. Q 2 =  E 2  E 1 The change in A 2 ’s entropy in this process is  S 2 = k B [lnΩ(E 2 + Q 2 ) – lnΩ(E 2 )] Expand  S 2 in a Taylor’s Series for small Q 2 & keep only the lowest order term. Also use the temperature parameter definition :

 S 2 = k B [lnΩ(E 2 + Q 2 ) – lnΩ(E 2 )] Expand  S 2 in a Taylor’s Series for small Q 2 & keep only the lowest order term. Use the temperature parameter definition & connection with absolute temperature T: This results in  S 2  k B  Q 2. Also noting that since the two systems are in equilibrium, T 2 = T 1  T gives:  S 2  [Q 2 /T] In Reif’s notation this is:  S'  [Q'/T]

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 2 nd 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:

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 A 1, A 2, interacting & in equilibrium. The combined system A 0 = A 1 + A 2, was isolated. Now: Consider the case in which A 1 & A 2 are also characterized by external parameters x 1 & x 2. E1E1 E 2 = E - E 1 x2x2 A2A2 x1x1 A1A1 As discussed earlier, corresponding to x 1 & x 2, there are generalized forces X 1 & X 2.

In earlier discussion, we found the most probable energy of system A 1, using the fact that the probability finding of A 1 with energy E 1 is proportional to the product of the number of accessible states of A 1 times the number of accessible states of A 2, Consistent with Energy Conservation: E = E 1 + E 2 That is, it is proportional to Using calculus to find E 1 that maximizes Ω(E 1, E – E 1 ) resulted in statistical definitions of the Entropy S & the Temperature Parameter  : Another result is that the equilibrium condition for A 1 & A 2 is that the temperatures are equal!

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 A 1 with a particular external parameter x 1 is proportional to the number of accessible states of A 1 times the number of accessible states of A 2. That is, it is proportional to Ω(E 1, x 1 ;E 2,x 2 ) = Ω(E 1,x 1 )Ω(E - E 1,x 2 )

The probability finding of A 1 with a particular external parameter x 1 is proportional to the number of accessible states of A 1 times the number of accessible states of A 2. Ω(E 1, x 1 ;E 2,x 2 ) = Ω(E 1,x 1 )Ω(E - E 1,x 2 ) Using differential calculus to find the x 1 that maximizes Ω(E 1, x 1 ;E 2,x 2 ) results in a statistical definition of The Mean Generalized Force   ∂ln[Ω(E,x)]/∂x (1) Or = (k B T)∂ln[Ω(E,x)]/∂x (2) In terms of Entropy S: = T ∂S(E,x)]/∂x (3) (1) ((2) or (3)) is called an Equation of State for system A 1. 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 is   ∂ln[Ω(E,x)]/∂x (1) Or = (k B T)∂ln[Ω(E,x)]/ ∂x (2) = T ∂S(E,x)]/∂ x (3) (1) ((2) or (3)) is an Equation of State for system A 1. Note that there is an Equation of State for each different external parameter x.