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)

Celsius (Centigrade) Scale (°C) 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) T = 0 K  “Absolute Zero”. Used by scientists. Created by Lord Kelvin. Starts with T = 0 K  “Absolute Zero”.

Some Physics History! (Interesting to me!) 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

Celsius to Fahrenheit: Fahrenheit to Celsius: 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!

How is Temperature Measured? Of course, temperature is measured using a Thermometer! Thermometer  Any object that has a property characterized by a Thermometric Parameter

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. X T • FP2 FP1 X1 X2 Two (or more) reference points can result in errors when extrapolating outside of their range!! Error! Xm

n.b.p.  “normal boiling point” Ranges of Various Types of Thermometers V P or V n.b.p.  “normal boiling point”

This scale was originally called “centigrade” 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 t  T For 0 ºC < T < 850 ºC Blundell & Blundell, Concepts in Thermal Physics (2006)

(Planck Distribution Law) 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

The Kelvin or Thermodynamic 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: So, What variable should be measured to use the thermodynamic temperature scale?

The Ideal Gas Temperature Scale The Ideal Gas Law: Hold V & n constant! Gas P, V TP = 273.16K Unknown T

A Constant-Volume Gas Thermometer

Defining the Kelvin & Celsius Scales Relationship between °C and K “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 Comment Kelvin Celsius Fahrenheit Rankine Delisle Newton Réaumur Rømer Absolute zero 0.00 −273.15 −459.67 559.73 −90.14 −218.52 −135.9 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 −17.78 459.67 176.67 −5.87 −14.22 −1.83 Ice melts (standard pressure) 273.15 32.00 491.67 150.00 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 Ave. human body temp.* 310 37 98 558 95 29 27 Highest recorded surface temperature o Earth ('Aziziya, Libya - September 13, 1922) But that reading is questioned 58 136 596 63 19 46 38 Water boils (standard pressure) 373.15 100.00 211.97 671.64 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

Heat Reservoirs (or “Heat Baths”)

The 2nd Law Tells Us That: it increases the entropy 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 Conservation: E = E1 + E2 The following discussion is similar to that for which the Energy Distribution Between Systems in Equilibrium was discussed & the conditions for equilibrium were derived. A1 A2 Recall: We considered 2 macroscopic systems A1, A2, interacting & in equilibrium. The combined system A0 = A1 + A2, was isolated. E2 = E - E1 E1 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

Consistent with Energy Conservation: E = E1 + E2. 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 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 : It also resulted in the fact that the equilibrium condition for A1 & A2 is that the two temperatures are equal!

A2 is a Heat Reservoir or Heat Bath for A1. 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 Terminology in some books: 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.

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: 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 another 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 2nd 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:

Equations of State

External Parameter Dependence of Ω The following is similar to our previous discussion, 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. A2 A1 Now: Consider the case in which A1 & A2 are also characterized by external parameters x1 & x2. x2 x1 E2 = E - E1 E1 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 Using calculus to find E1 that maximizes Ω(E1, E – E1) resulted in statistical definitions of the Entropy S & the Temperature Parameter . Another result is that the equilibrium condition for A1 & A2 is that the temperatures are equal!

Ω(E1,x1;E2,x2) = Ω(E1,x1)Ω(E - E1,x2) 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)

<X> = T[∂S(E,x)]/∂x (3) 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.

The Mean Generalized Force <X> is Summary For interacting systems with an external parameter x, at equilibrium The 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. There is an Equation of State for each different external parameter x.