Category: Thermodynamics1

Updated:2014Jan6W. Pezzaglia

I. Temperature and the

Zeroth Law of Thermodynamics

Outline

A. Zeroth Law of Thermodynamics

B. Thermal Expansion

C. Kinetic Theory of Gasses

D. References

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A. Zeroth Law of Thermodynamics

1. Temperature Scales

2. Thermal Equilibrium

3. The 0th Law of Thermodynamics

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1. What is “temperature”

a) We directly “sense” hot and cold

b) Intensive (rather than extensive) quantity• Extensive quantity is like “mass”, if you double the

size of system you double the quantity• Intensive is like “density”, its independent of size of

system. Every subpart of the system has same value.

c) Common scales: Fahrenheit, Reaumur, Celsius, Kelvin, Rankine

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1c. Common Temperature Scales

• 1701 Rømer develops first Mercury temperature scale, Fahrenheit visits him in 1708.

• 1709 Fahrenheit invents alcohol thermometer

• 1724 Fahrenheit scale (mercury) 0F freezing point of brine 32F freezing point of water 100F approximate body temp (dog?) 212F boiling point of water

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Daniel Gabriel Fahrenheit(1686 – 1736) 

1c. Common Temperature Scales

• 1730 Reaumur Scale [cheese!] (0Re water freezes, 80 Re it boils)

• 1742 Anders Celsius shows boiling point of water changes with pressure, so calibrates at sea level.

• Centigrade scale (now called “Celsius”)

0C freezing point of water at sea level 100F boiling point of water at sea level

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Ander Celsius(1701 – 1744) 

2. Thermal Equilibrium

a) Equilibrium: system is “balanced” or unchanging in time

b) Thermal Equilibrium: system’s temperature does not change with time

c) A frigorific mixture: is a mixture of two chemicals that reaches an equilibrium temperature independent of the temperatures of the components chemicals

– “ice bath” will stabilize at 0C).– Ammonium chloride and ice stabilizes at -17.8C or

0F

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3. Zeroth Law of Thermodynamics

(a) Rankine wrote in 1853:– Two portions of matter are

said to have equal temperatures, when neither tends to communicate heat to the other

• Note: The laws of thermodynamics are stated differently by various scientists

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William John Macquorn Rankine(1820 – 1872)

3b. Zeroth Law of Thermodynamics

• In 1872 Maxwell wrote: "If when two bodies are placed in thermal communication, one of the two bodies loses heat, and the other gains heat, that body which gives out heat is said to have a higher temperature than that which receives heat from it.”

• "If when two bodies are placed in thermal communication neither of them loses or gains heat, the two bodies are said to have equal temperatures or the same temperature. The two bodies are then said to be in thermal equilibrium."

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James Clerk Maxwell(1831 – 1879)

3c. Zeroth Law of Thermodynamics

• 1897 Max Planck states it as:– "If a body, A, be in thermal

equilibrium with two other bodies, B and C, then B and C are in thermal equilibrium with one another

– i.e. if A=B and A=C, then B=C

• The zeroth law (first called this by Sommerfeld 1951) tells us that the idea of temperature makes sense, i.e. we can use a thermometer to measure temperature because it will be in thermal equilibrium with the system.

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Max Planck(1858 – 1947)

B. Thermal Expansion

1. Law of Thermal Expansion

2. Thermometer Designs

3. Thermal Stress

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Expansion joints in a bridge

1. Laws of Thermal Expansion

Galileo (1593) discovers fluids expand when heated.

a)Linear Expansion L= L T

b)Area Expansion A= A 2 T

c)Volume Expansion V= V 3 T

=Coefficient of linear expansion

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1a Law of Linear Expansion

• The change in length “L” due to change in temperature of a solid

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TLL

Coefficient of linear expansion for steel: =1210-6 C-1

http://www.youtube.com/watch?feature=endscreen&NR=1&v=TDnLbjd429M

1a Coefficients of Linear Expansion (at 20C) 14

1a Antiscalding valve

• If water is too hot, the plunger expands and closes off the flow of water

• Nearly 2/3 of hospital visits are due to scald injuries. (Children under 5 in particular.)

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1b Law of Areal Expansion• The change in area “A=xy”

due to change in temperature of a solid

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Coefficient of linear expansion:

TAA

Ty

y

x

x

A

A

yxxyA

2

2

QuestionWhen heated, the “hole” in a plate will get:

(a) Bigger

(b) Smaller

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http://www.youtube.com/watch?v=MXk57NIM3w8&feature=endscreen&NR=1

1c Law of Volumetric Expansion

• The change in volume “V=xyz” due to change in temperature:

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3

Coefficient of volume expansion related to coefficient of linear expansion:

TVV

Tz

z

y

y

x

x

V

V

zxyyxzxyzV

3

1c Water has strange properties

• Above 4C water expands with increase in temperature as expected (density decreases)

• Below 4C however it shrinks with increasing temperature (density increases).

• This causes lakes to freeze only at the top

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TV

V

2. Thermometer Design

(a) Galilean Thermoscope

• 1593 Galileo discovers that when heated, liquids expand (i.e. density decreases)

• Thermoscope: As temperature rises, the graduated weights will sink, one by one.

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Video: http://www.youtube.com/watch?v=917UC2MZOGU

(b) Liquid Thermometers• The smallest markings easiest to read are about h=1 mm apart.

Suppose we want to measure to nearest T=0.2 C.

• The simplest thermometer would be an open vertical column of fluid in a rigid glass tube. Change in volume pushes fluid up so: h/h=V/V=T.

• Assuming alcohol (=112010-6 C-1) solving for h=4.5 meters! So the device would extend from the first floor to the second floor! Too Big!

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• Instead, put a fat bulb on the end to contain the large volume (and make diameter of tube very small) so that can make convenient size.

(c) Gas Thermometer• At constant pressure, Charles’ law shows that

volume of gas changes linearly with temperature.

• Regardless of the size of sample (or the fixed pressure used), extrapolating the data shows all lines converge at -273C, where the volume would be “zero”.

• Kelvin uses this as a definition of “absolute zero” (Lambert did similar with PT graph in 1779)

• (1848) Kelvin scale: K=C +273.15• (1859) Rankine scale: R=F +459.67

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3 Thermal Stress(a) Bimetal Strips:

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http://www.youtube.com/watch?v=sP5NwEkd3ds

3a Bimetal Strip Applications

• As a switch (e.g. turn on heater when it gets cold):

• As a thermometer gauge

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3b Review: Stress and Young’s Modulus

• Stress on a rod is tension force divided by cross section area

• Units: Pascal=N/m2

• Breaking Stress: maximum stress before fracture.For steel: 400106 Pa

• Strain: relative change in length, is proportional to stress

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A

F

YL

L 1

Young’s Modulus for steel: Y=200109 Pa

3c Thermal Stress

• Equate Thermal expansion to elastic expansion

• Thermal Stress: induced by change in temperature

• Example: if steel cable suspended between fixed points, it will break if you cool it by T=-166 C

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TYA

FA

F

YL

LT

1

A

F

YT

1

C. Kinetic Theory of Temperature

1. Gas Laws

2. Kinetic Theory of Pressure

3. Maxwell Boltzmann Distribution

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1. The Gas Laws

(a) Boyle’s and Charles’ Law

(b) Amonton’s Law and gas thermometer

(c) Avogadro and the Ideal Gas Law

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1a. Early Gas Laws

• Boyle’s Law (1662) at constant temperature, pressure is inversely proportional to volume, or

• Charles Law (1678) at constant pressure, volume is proportional to (absolute) temperature:

• Combined, these tell us for a given sample of gas:

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Robert Boyle(1627 – 1691)

constantT

PV

Jacques(1746 – 1823)

constantPV

TV

1b. Amontons’ Law

• Amontons’ Law (1702) at constant volume, change in pressure is proportional to change in temperature, or

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Guillaume Amontons(1663 – 1705)

TP

• Developed the (constant volume) gas thermometer.

• Lambert (1779) extrapolated this data to propose “absolute zero” where pressure is zero (note this is 69 years earlier than Kelvin)

1c Avogadro and Atomic Theory

• Avogadro (1811): two different types of gas at same P, V, T will contain same number of molecules. Idea not generally accepted.

• Clapeyron (1834) first states the ideal gas law

• Where “n” is the number of moles of molecules and the “gas constant” is given:

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Amedeo Avogadro(1776 – 1856) nRTPV

Benoît Clapeyron(1799 – 1864)

molK

atmLiter8206.0

molK

J8.31R

C. Kinetic Theory of Temperature

1. Gas Laws

2. Kinetic Theory of Pressure

3. Maxwell Boltzmann Distribution

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2a. Atomic Theory of Matter• 1803 Dalton proposes atomic theory

• 1811 Avogadro clarifies distinction between molecules and atoms

• 1827 Brown discovers dust particles (inside floating pollen grains) jiggled about for no apparent reason.

• 1905 Einstein proposes that thermal kinetic energy of molecules is the cause, verified experimentally by Perrin (1908)

• Avogadro’s number is not well determined until 1900s, by Jean Perrin (Nobel Prize 1926).

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231002.6 aNAvogadro’s number

2b. Kinetic Theory of Pressure• 1738 Daniel Bernoulli derives Boyle’s law by assuming

gasses consist of moving molecules, and the impacts with wall causes pressure.

• Impulse by collision

• Time between collisions for box of width “L” knowing x-velocity

• Average Force on wall for 1 molecule

• Average Kinetic Energy in 3D

• Pressure is related to KE

• Boyle’s Law for N molecules

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Daniel Bernoulli (1700 – 1782)

KENPV

KELAA

FPV

mvvvvmmvKE

L

mv

t

pF

v

Lt

mvp

xzyx

x

x

32

32

223222

212

21

2

)(

2

2

2c. Kinetic Theory of Temperature

• Equate Kinetic pressure law with ideal gas law and we find average kinetic energy of a mole of gas is

• 1900(?) Planck writes that the average Kinetic Energy of a single monoatomic gas atom is given by:

• Hence “temperature” is a measure of average kinetic energy of molecules.

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KJ

avg

k

kTK

23-

23

10×1.38

Boltzmann Constant=R/Na

RTK avg 23

3a Molecular Velocities• Equate thermal energy to

kinetic energy

• The “root mean square” velocity (the square root of the average of the squared velocity) is hence:

• At room temperature Oxygen moving around 478 m/s, while hydrogen is close to 1900 m/s

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m

kTv

kTmv

rms

3

232

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3b Measuring molecular speeds• Molecules are NOT all traveling at same speed. Some are

slower, some much faster than average. The speed distribution can be measured with the following experiment (Stern 1920)

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3c Maxwell speed distribution

• From theory, Maxwell (1860) calculates correct speed distribution.

• At higher temperatures, more molecules are moving faster.

• A certain fraction of the molecules in the atmosphere will be moving faster than the escape speed and will leave the planet.

• The moon’s escape speed is so low, than over a short time, it lost all of its atmosphere.

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kT

mvv

kT

mvf

2exp

2)(

22

3

3c Maxwell speed distribution

• Molecules with smaller mass “m” will have higher speeds at the same temperature.

• Hence hydrogen is moving nearly 4x faster than Nitrogen, hence it more quickly escapes from the earth’s atmosphere, while Nitrogen is effectively “trapped”.

• Mars has lost all of its light gasses, only the heavier CO2 gas remains

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The distribution of speeds of three different gases at the same temperature

m

kTvrms

3

Question 40

Demonstration List• Mechanical Universe program: # 45 Temperature and Gas Laws (30 min), link on Goggle videos:

http://video.google.com/videoplay?docid=949035002599580195 • Galilean Thermoscope, do we have one?• Video demo of: http://www.youtube.com/watch?v=917UC2MZOGU

• Gas Law: http://phet.colorado.edu/en/simulation/gas-properties

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References• Galilean Thermoscope: http://en.wikipedia.org/wiki/Galileo_thermometer • Gas Laws:• Charles Law http://en.wikipedia.org/wiki/Charles's_law• J. Perrin’s measurement of Avogadro’s number is described in Oldenberg, Introduction to Atomic Physics

(McGraw-Hill 1949), p.36. O. Sterns measurement of molecular speeds, p. 42

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Notes to Self• Demos: do we have a demonstration thermometer? Gas Volume? Water Column?• Demos: bimetal strips? Ring and ball. Thermal stress?• Demos: Kinetic Gas demo? Van der walls?

• MCAT includes Van der Waals equation.

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