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MSU Physics 231 Fall 2015 1
Physics 231Topic 12: Temperature,
Thermal Expansion, and Ideal Gases
Alex BrownNov 18-23 2015
MSU Physics 231 Fall 2015 2
3rd midterm
final Thursday 8-10 pm
homework
makeup Friday final 9-11 am
MSU Physics 231 Fall 2015 3
Key Concepts: Temperature, Thermal Expansion, and Ideal
GasesTemperature and Thermometers
Thermal Energy & Temperature
Thermal ExpansionCoefficient of thermal expansion
Ideal GasesState Variables
Ideal gas law
Kinetic Theory of Gases
Kinetic & thermal energy
Maxwell distribution
Covers chapter 12 in Rex & Wolfson
MSU Physics 231 Fall 2015 4
Conversions:
Tc = Tk - 273.15
Tf = (9/5)Tc + 32
Helium boils at Tk=4
MSU Physics 231 Fall 2015 5
R
Potential Energy
0
R
2 atom/molecules
-Emin
The curve depends onthe material, e.g. Emin isdifferent for water andiron
Kinetic energy ~ T (temperature)
Binding Forces
MSU Physics 231 Fall 2015 6
Solid (low T)
R
Potential Energy
0
Kinetic energy ~ T
-Emin
The temperature (and thus kinetic energy)is so small that the atoms/molecules can onlyoscillate around a fixed position Rmin
Rmin
MSU Physics 231 Fall 2015 7
Liquid (medium T)
R
Potential Energy
0
Kinetic energy ~ T
-Emin
Rmin
On average, the atoms/molecules like tostick together but sometimes escape andcan travel far.
MSU Physics 231 Fall 2015 8
Gas (high T)
R
Potential Energy
0
Kinetic energy ~ T
-Emin
Rmin
The kinetic energy is much larger thanEmin and the atoms/molecules move aroundrandomly.
MSU Physics 231 Fall 2015 9
What happens if the temperature of a substance is
increased?
R
0
Kinetic energy ~ T
-Emin
Rmin=Rave(T=0)
T=0: Average distance between atoms/molecules: Rmin
MSU Physics 231 Fall 2015 10
What happens if the temperature of a substance is
increased?
R
0
Kinetic energy ~ T
-Emin
Rmin=Rave(T=0)
Rave(T>0) > Rmin
T>To: The average distance between atoms/molecules is larger than Rmin:
the substance expands
MSU Physics 231 Fall 2015 11
Thermal expansionL= Lo T
L0
L
T=T0T=T0+T
A = Ao T = 2
V = Vo T = 3
length
surface
volume
Some examples: = 24x10-6 1/K Aluminum = 1.2x10-4 1/K Alcohol
: coefficient of linear expansion different for each material
MSU Physics 231 Fall 2015 12
MSU Physics 231 Fall 2015 13PHY 231 13
A Heated Ring
A metal ring is heated. What is true:a) The inside and outside radii become largerb) The inside radius becomes larger, the outside
radius becomes smallerc) The inside radius becomes smaller, the outside
radius becomes largerd) The inside and outside radii become smaller
MSU Physics 231 Fall 2015 14
Demo: Bimetallic Strips
Application: contact in a refrigerator
top
bottom
top<bottom if the temperature increases,
The strip curls upward, makes contact and switcheson the cooling.
MSU Physics 231 Fall 2015 15
Water: a special case
Coef. of expansion isnegative: If T dropsthe volume becomeslarger
Coef. Of expansion ispositive: if T drops the volume becomes smaller
Below this ice is formed (it floats on water)
MSU Physics 231 Fall 2015 16
Ice
liquid
ice
(g/cm3)
1
0.917
Phase transformation
Ice takes a larger volume than water!
A frozen bottle of water might explode
MSU Physics 231 Fall 2015 17
Thermal equilibrium
Low temperatureLow kinetic energyParticles move slowly
High temperatureHigh kinetic energyParticles move fast
Thermal contact
Transfer of kinetic energy
Thermal equilibrium: temperature is the same everywhere
MSU Physics 231 Fall 2015 18
Zeroth law of thermodynamics
If objects A and B are both in thermal equilibriumwith an object C, than A and B are also in thermalequilibrium.
There is no transfer of energy between A, B and C
MSU Physics 231 Fall 2015 19
Ideal Gas: properties
Collection of atoms/molecules that
• Exert no force upon each otherThe energy of a system of two atoms/molecules cannot be reduced by bringing them close to each other
• Take no volumeThe volume taken by the atoms/molecules is negligible compared to the volume they are sitting in
MSU Physics 231 Fall 2015 20
R
Potential Energy
0-Emin
Rmin
Ideal gas: we are neglecting the potential energy betweenThe atoms/molecules
R
Potential Energy
0
Kinetic energy
MSU Physics 231 Fall 2015 21
Properties of gases
V = volumeP = pressureT = temperature in K (Kelvin)n = number of moles
Example balloon
MSU Physics 231 Fall 2015 22
Molecular mass
kg 102.00 1002.6
012.0(carbon)
kg 0.0120 g 12.0 (carbon) examplefor
mass (molar)molecular
numbers sAvagodro' 1002.6
molecule)(or atom one of mass
26-23
molar
23
m
M
NmM
N
m
molar
A
A
MSU Physics 231 Fall 2015 23
X
Z
A
Number of electrons
molar mass in grams
Name
MSU Physics 231 Fall 2015 24
Weight of 1 mol of atoms1 mol of atoms weighs A grams (A is the molar mass)
Examples: 1 mol of Hydrogen weighs 1.0 g 1 mol of Carbon weighs 12.0 g1 mol of Oxygen weighs 16.0 g 1 mol of Zinc weighs 65.4 g
What about molecules?H2O 1 mol of water molecules:
2 x 1.0 g (due to Hydrogen)1 x 16.0 g (due to Oxygen)
Total: 18.0 g
MSU Physics 231 Fall 2015 25
molar
molecules)(or atoms all of mass total
molecules)(or atoms contains mole one so
or moles ofnumber
molecules)(or atoms ofnumber total
MnM
NmM
N
NnNN
Nn
N
A
AA
Number of atoms and moles
MSU Physics 231 Fall 2015 26
Example
A cube of Silicon (molar mass 28.1 g) is 250 g.
A) How much Silicon atoms are in the cube?
B) What would be the mass for the same number of gold atoms (molar mass 197 g)
Total number of moles n = M / Mmolar = 250/28.1 = 8.90
N = n NA = (8.9) (6.02x1023) = 5.4x1024 atoms
M = n Mmolar = (8.90) (197 g) = 1750 g
MSU Physics 231 Fall 2015 27
Question
1) 1 mol of CO2 has a larger mass than 1 mol of CH2
2) 1 mol of CO2 contains more molecules than 1 mol of CH2
a) 1) true 2) true
b) 1) true 2) false
c) 1) false 2) true
d) 1) false 2) false
MSU Physics 231 Fall 2015 28
Properties of gases
V = volumeP = pressureT = temperature in K (Kelvin)n = number of moles
Example balloon
MSU Physics 231 Fall 2015 29
Boyle’s Law (fixed n and T)
P0 V0
2P0 ½V0
½P0 2V0
At constant temperature: P ~ 1/V
implies that PV = constant
MSU Physics 231 Fall 2015 30
Charles’ law (fixed n and P)
V0 T0
2V0 2T0
If you want to maintain a constant pressure, the temperature must be increased linearly with the volume V ~ T implies that (V/T) = constant
MSU Physics 231 Fall 2015 31
Gay-Lussac’s law (fixed n and V)
P0 T0 2P0 2T0
If, at constant volume, the temperature is increased,the pressure will increase by the same factor
P ~ T
implies that (P/T) = constant
MSU Physics 231 Fall 2015 32
Brown’s law (fixed T and P)
n0 V0
2n0 2V0
If you double the number of particles the volume doubles n ~ V implies that (V/n) = constant
MSU Physics 231 Fall 2015 33
Boyle & Charles & Gay-LussacIDEAL GAS LAW
n = number of molesR = universal gas constant 8.31 J/mol·K
If the number of moles is fixed
2
22
1
11or constant T
VP
T
VP
T
PV
nRTPV Does not depend on what type or atom or molecule
MSU Physics 231 Fall 2015 34
ExampleAn ideal gas occupies a volume of 1.0 cm3 at 200 C at 1 atm. A) How many atoms are in the volume?
B) If the pressure is reduced to 1.0x10-11 Pa, while thetemperature drops to 00C, how many atoms remainedin the volume?
PV = nRT, so n = PV/(TR) with R=8.31 J/mol K T=200C=293K, P=1atm=1.013x105 Pa, V=1.0cm3=1x10-6m3
n=4.2x10-5 mol N = n NA = (4.2x10-5) NA=2.5x1019
T = 00C = 273K , P = 1.0x10-11 Pa, V = 1x10-6 m3
n=4.4x10-21 mol N=2.6x103 particles (almost vacuum)
MSU Physics 231 Fall 2015 35
And another!An air bubble has a volume of 1.50 cm3 at 950 m depth (T=7oC). What is its volume when it reaches the surface (T=20oC). (water=1.0x103 kg/m3)?
P950m=P0 + water g h = 1.013 x 105 + (1.0x103)(9.8)(950) = 94.2 x 105 Pa
111
2
2
12
2
22
1
11
)046.1)(0.93(
VVT
T
P
PV
T
VP
T
VP
Vsurface=146 cm3
Expanded by a factor of 97
MSU Physics 231 Fall 2015 36
A volume with dimensions L x W x H is kept underpressure P at temperature T. If the temperature israised by a factor of 2, and the height is made5 times smaller, by what factor does the pressure change, i.e. what is P2/P1? No gas leaks or is added.
a) 0.4 b) 1 c) 2.5 d) 5 e) 10
Use the fact PV/T is constant if no gas is added/leakedP1V1 / T1 = P2V2 / T2
P1V1 / T1 = P2 (V1/5) / (2T1)P2 = (5)(2)(P1 ) = 10 P1 a factor of 10.
Quiz
MSU Physics 231 Fall 2015 37
K 273.15 C 0 T
Pa101.013 atm 1
oo
5
P
“Standard temperature and pressure” (STP)
MSU Physics 231 Fall 2015 38
number sAvagadro'1002.6
objects ofnumber total
moles ofnumber
23A
A
N
N
N
Nn
Moles
MSU Physics 231 Fall 2015 39
macroscopic to microscopic
B
AB
B
kNRn
N
Rk
TkNPV
TRnPV
constant) s(Boltzman' (J/K)1038.11002.6
31.8 2323
macroscopic quantities
N = number of atoms or molecules (microscopic)
MSU Physics 231 Fall 2015 40
Quiz
Given P1 = 1 atm P2 = 2 atm V1 = 2 m3 V2 = 10 m3
T1 = 100 K N1 = NA N2 = 10 NA
T2 = ? K
A) 200B) 500C) 2000D) 5000E) 100
MSU Physics 231 Fall 2015 41
ExampleHow many air molecules at in the room with a volume of 1000 m3
(assume only molecular nitrogen is present N2)?
PV = N kB T
T = 293P = 1.013x105 Pa V = 1000 m3
N = 2.5x1028
MSU Physics 231 Fall 2015 42
microscopic description: kinetic theory of gases
1) The number of objects is large (statistical model)2) Their average separation is large 3) The objects follow Newton’s laws4) Any particular object can move in any direction with a distribution of velocities5) The objects undergo elastic collision with each other6) The objects make elastic collisions with the walls7) All objects are of the same type
MSU Physics 231 Fall 2015 43
Movie of gas in two dimensions
MSU Physics 231 Fall 2015 44
mean free pathd = average distance between collisions
air at P = 1 atm d = 68 nm = 68 x 10-9 m
high vacuum P = 10-5 Pa d = 1m
in space P = 10-12 Pa d = 108 m
MSU Physics 231 Fall 2015 45
The Maxwell DistributionHowever we can model the distribution of the velocities (& thus the kinetic energies) of the individual gas molecules. The result is the Maxwell Distribution.
The root-mean-square (rms) velocity is 2vvrms
MSU Physics 231 Fall 2015 46
Energy of one object
2
2
1vmK
Objects inside the container have a distribution of velocitiesaround an average – so each object has an average kinetic energygiven by
average squared velocityaverage translation kinetic energy
mass of the object (atom or molecule)
MSU Physics 231 Fall 2015 47
MSU Physics 231 Fall 2015 48
Relationship to ideal gas law
3
2 KNPV
The objects bounce off of each other and the walls of the container (elastic). One can derive the following result
TkK
TkNPV
B
B
2
3get to
with combine
How the average kinetic energy of oneatom is related to temperature
MSU Physics 231 Fall 2015 49
2
2
1vmK with
2
3 combine TkK B
root-mean-square (rms) velocity for one atom or molecule
molar
Brms M
RT
m
Tkvv
332
MSU Physics 231 Fall 2015 50
ExampleWhat is the rms speed of air at 1 atm and room temperature (293 K)? Assume it consist of molecular Nitrogen only (N2)?
molar
Brms M
RT
m
Tkvv
332
R = 8.31 J/mol K
T = 293 K Mmolar = (2 x 14)x10-3 kg/mol
vrms = 511 m/s = 1140 mph !
MSU Physics 231 Fall 2015 51
d is the number of “degrees of freedom” for the motion
d = 3 for an atom (motion in x, y, z directions) like helium gas
d = 5 for a diatomic molecule (motion in x, y, z and two ways to rotate) like nitrogen molecule N2 or hydrogen molecule H2
) (since 2223
nRNkPVd
nRTd
Tkd
NKNd
E BBth
(Homework question for “one degree of freedom” use d = 1)
Total thermal energy
MSU Physics 231 Fall 2015 52
ExampleWhat is the total thermal kinetic energy of the air molecules in thelecture room (assume only molecular nitrogen is present N2)?
Eth = (d/2) PV = 2.5x108 J
d = 5P = 1.013x105 Pa V = 1000 m3
Using KE = (1/2) mv2 this is equivalent to 1000 carswith m=1000 kg each moving with v = 22.3 m/s (50 mph)
Can we use that energy to do work?
MSU Physics 231 Fall 2015 53
Diffusion