Thermal PhysicsThermal PhysicsIB Physics IB Physics
Topic 3: Ideal gasesTopic 3: Ideal gases
Ideal GasesIdeal Gases
Understand and Understand and Apply the following.Apply the following. Pressure. Pressure. Gas Laws (by name)Gas Laws (by name) PV = nRTPV = nRT Kinetic Molecular Kinetic Molecular
TheoryTheory Explain PressureExplain Pressure
WilliamThompson (Lord Kelvin)
PressurePressure Pressure is defined as Pressure is defined as
force per unit areaforce per unit area Newtons per square metre Newtons per square metre
or N/m or N/m22
1 Nm1 Nm-2-2 = 1 Pa (pascal) = 1 Pa (pascal) The weight of the person The weight of the person
is the force applied to is the force applied to the bed and the small the bed and the small area of each nail tip area of each nail tip combines to make an combines to make an overall large area.overall large area.
The result is a small The result is a small enough pressure which enough pressure which does not puncture the does not puncture the person.person.
A
FP
Click on Me
Atmospheric PressureAtmospheric Pressure
Basically weight of atmosphere!Basically weight of atmosphere! Air molecules are colliding with you Air molecules are colliding with you
right now!right now! Pressure = 1.013 x10Pressure = 1.013 x1055 N/m N/m22 = 14.7 = 14.7
lbs/inlbs/in22!! Example: Sphere w/ r = 0.1 mExample: Sphere w/ r = 0.1 m
DemoDemo
A = 4 A = 4 r r22 = .125 m = .125 m22
F = 12,000 Newtons (over 2,500 lbs)!F = 12,000 Newtons (over 2,500 lbs)!
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Qualitative Demonstration Qualitative Demonstration of Pressureof Pressure
y
t
ymv
typ
yf
force verticalaverage
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Force due to molecules of fluid colliding with Force due to molecules of fluid colliding with container.container. Force Force αα Impulse = Impulse = pp
Average Pressure = F / AAverage Pressure = F / A
PressurePressure Pressure is defined as force per unit areaPressure is defined as force per unit area
Newtons per square metre N/mNewtons per square metre N/m22
The pressure exerted by a gas results from the The pressure exerted by a gas results from the atoms/ molecules “bumping” into the atoms/ molecules “bumping” into the container wallscontainer walls More atoms gives more bumps per second and More atoms gives more bumps per second and
higher pressurehigher pressure Higher temperature means faster atoms and gives Higher temperature means faster atoms and gives
more bumps per second and higher pressuremore bumps per second and higher pressure At sea level and 20°C, normal atmospheric At sea level and 20°C, normal atmospheric
pressure ispressure is 1atm ≈ 1 x 101atm ≈ 1 x 1055 N/m N/m22
GasesGases Gases (as we will see) can Gases (as we will see) can
behave near perfectly.behave near perfectly. NNAA= 6.02 x 10= 6.02 x 102323 molecules mol molecules mol-1-1
Molecules size ~ 10Molecules size ~ 10-8-8 m to 10 m to 10-10-10 mm
Example: How molecules are there in 6 Example: How molecules are there in 6 grams of hydrogen gas?grams of hydrogen gas?
We have 3 moles, HWe have 3 moles, H22 has 2 grams mol has 2 grams mol-1-1
3 x 6.02 x 103 x 6.02 x 102323 = 1.81 x 10 = 1.81 x 102424 molecules.molecules.
ExampleExampleMake a rough estimate of the number of water Make a rough estimate of the number of water
molecules in an ordinary glass of water.molecules in an ordinary glass of water.
A glass contains about 0.3 L of water, which has A glass contains about 0.3 L of water, which has a mass of about 300 g. Since the molar mass of a mass of about 300 g. Since the molar mass of water (Hwater (H22O) is 18 g molO) is 18 g mol-1-1
Hence, 300g/18g molHence, 300g/18g mol-1-1 ~ 17 mol ~ 10 ~ 17 mol ~ 102525 molecules molecules
The Boyle-Mariotte Law The Boyle-Mariotte Law Gases (at Gases (at constant constant
temperaturetemperature) decrease in ) decrease in volume with increasing volume with increasing pressure.pressure.
P =F/AP =F/A V = V = ππrr22 h h
Figure 17-3Figure 17-3Increasing Pressure by Decreasing VolumeIncreasing Pressure by Decreasing Volume
The Boyle-Mariotte Law The Boyle-Mariotte Law Gases (at constant Gases (at constant
temperature) temperature) decrease in volume decrease in volume with increasing with increasing pressure.pressure.
IsothermalIsothermal ProcessProcess
PV = constantPV = constant
PP11VV11 = P = P22VV22
ExampleExample
The pressure of gas is 2 atm and its The pressure of gas is 2 atm and its volume 0.9 L if the pressure is increased volume 0.9 L if the pressure is increased to 6 atm at constant temperature, what is to 6 atm at constant temperature, what is the new volume?the new volume?
Answer:Answer:PP11VV11 = P = P22VV22
2 x 0.9 = 6 x V; hence V = 0.3 L2 x 0.9 = 6 x V; hence V = 0.3 L
The volume-temperature The volume-temperature law law
Charles & Gay-LussacCharles & Gay-Lussac
IsobaricIsobaric Process Process
V/T = constantV/T = constant
VV1 1 // TT11 = V = V2 2 // TT22
At absolute zero a gas would At absolute zero a gas would take up zero volume. In reality take up zero volume. In reality they liquefy when they get really they liquefy when they get really cold!cold!
The pressure-The pressure-temperature Law temperature Law
Gases (at constant Gases (at constant volume) increase volume) increase in temperature in temperature with increasing with increasing pressure.pressure.
Isochoric ProcessIsochoric Process
P/T= constantP/T= constant
PP11/T/T11 = P = P22/T/T22
0 100 200-200 -100
temp. °C
pres
sure
ExampleExample
A bottle of hair A bottle of hair spray is filled to a spray is filled to a pressure of pressure of 1 atm at 20°C1 atm at 20°C
What is the canister What is the canister pressure if it is pressure if it is placed into boiling placed into boiling water, and allowed water, and allowed to reach thermal to reach thermal equilibrium?equilibrium?
P1/ T1 = P2/ T2
orp1 T2 = p2 T1
1 / 293 = p2 / 373 p2 = 373/293p2 = 1.27 atm
Absolute zeroAbsolute zero
Ideal gas has zero Ideal gas has zero volumevolume
Resistance of metal Resistance of metal drops to zero (actually drops to zero (actually superconductivity cuts in superconductivity cuts in above 0K)above 0K)
Brownian motion ceases!Brownian motion ceases!(kinetic energy = 3/2 kT)(kinetic energy = 3/2 kT)
But lowest temperature But lowest temperature attained is ≈ 10attained is ≈ 10-9-9KK
0-273 °C temp. °C
pres
sure
Equations of stateEquations of state
State, identifies whether solid liquid State, identifies whether solid liquid or gasor gas
Key parameters or state variablesKey parameters or state variables Volume, V (mVolume, V (m33)) Pressure, p (N/mPressure, p (N/m22)) Temperature, T (K)Temperature, T (K) Mass, M (kg) or number of moles, Mass, M (kg) or number of moles, nn
Equation of state relates V, p , T, m Equation of state relates V, p , T, m or or nn
Bulk vs moleculesBulk vs molecules
Consider force Consider force between two between two molecules molecules
At absolute zeroAt absolute zero No thermal energyNo thermal energy Molecules sit at rMolecules sit at r00
Above absolute zeroAbove absolute zero Some thermal energySome thermal energy Molecules are at r> rMolecules are at r> r0 0
(thermal expansion)(thermal expansion) At high temperatureAt high temperature
Thermal energy > Thermal energy > binding energybinding energy
Molecules form a gasMolecules form a gas
r
forceenergy
r0repulsion
attractionbindingenergy
thermal energy
Equation of state for a Equation of state for a gasgas
All gases behave nearly the sameAll gases behave nearly the same pV = pV = nnRTRT
R = 8.3 J/(mol K) for R = 8.3 J/(mol K) for allall gases (as long as gases (as long as they remain a gas)they remain a gas)
T is in K!!!!!!T is in K!!!!!!
ExampleExample
What is the mass What is the mass of a cubic metre of of a cubic metre of air?air? Molecular weigh of Molecular weigh of
air ≈ 32gair ≈ 32g
pV = nRT
Atmospheric pressure = 105 N/m2
Atmospheric temp. = 300K
For a volume of 1 m3
n = pV/RT = 105 / (8.3 x 300) = 40 moles
M = 40 x 0.032 = 1.3kg
Constant mass of gasConstant mass of gas
For a fixed amount of gas, its mass or For a fixed amount of gas, its mass or number of moles remains the samenumber of moles remains the same pV/T = pV/T = nnR = constantR = constant
Comparing the same gas under different Comparing the same gas under different conditionsconditions pp11VV11/T/T11 = p = p22VV22/T/T22
Hence can use pressure of a constant volume of gas Hence can use pressure of a constant volume of gas to define temperature (works even if gas is impure - to define temperature (works even if gas is impure - since all gases the same)since all gases the same)
Must use T in K!!!!!!Must use T in K!!!!!!
ExampleExample
A hot air balloon A hot air balloon has a volume of has a volume of 150m150m33
If heated from If heated from 20°C to 60°C how 20°C to 60°C how much lighter does much lighter does it get?it get? Molecular weight of Molecular weight of
air ≈32gair ≈32g
pV/T = nRn = pV/RT
Balloon has constant volume and constant pressure
ncool =105x150 / (8.3 x293) = 61680
nhot =105x150 / (8.3 x333) = 54271 n = 7409 moles
M = 7409 x 0.032 = 237kg
Molecules have finite Molecules have finite sizesize
Cannot reduce volume of gas to zero!Cannot reduce volume of gas to zero! When you try, it becomes a liquidWhen you try, it becomes a liquid Slightly increases the measured volumeSlightly increases the measured volume
Atoms/ molecules always attract each Atoms/ molecules always attract each otherother Slightly reduces the measured pressureSlightly reduces the measured pressure Van de Waals forcesVan de Waals forces
p-V diagrams (for gases)p-V diagrams (for gases) Useful to Useful to
consider the consider the pressure/volume pressure/volume changes at changes at constant constant temperaturetemperature Isotherms are p-V Isotherms are p-V
values for a fixed values for a fixed amount of gas at amount of gas at constant volumeconstant volume
p p 1/V 1/Vvolume
Pre
ssur
e
Increasing temperature
Kinetic theory of Kinetic theory of gasesgases
A gas consists of a large number A gas consists of a large number of molecules.of molecules.
Molecules move randomly with a range Molecules move randomly with a range of speeds. (Maxwell's Distribution)of speeds. (Maxwell's Distribution)
The Volume of the molecule is negligible The Volume of the molecule is negligible compared with the volume of the gas compared with the volume of the gas itself.itself.
Collisions are elastic (KE conserved)Collisions are elastic (KE conserved) No inter-molecular forces.No inter-molecular forces. Collision time with walls are very smal.Collision time with walls are very smal. Molecules obey Newton’s Laws of Molecules obey Newton’s Laws of
Mechanics Mechanics
Molecules in a gasMolecules in a gas Gas atoms/molecules move Gas atoms/molecules move
in a straight linein a straight line velocity due to thermal energyvelocity due to thermal energy
KE = 1/2 m vKE = 1/2 m vxx22 ≈ 3/2 kT ≈ 3/2 kT
KEKEavgavg α absolute temp absolute temp RMS – (Root mean squared)RMS – (Root mean squared)
Pressure results from Pressure results from collisions with the walls of collisions with the walls of the container. the container. (NOT collisions between (NOT collisions between moleculesmolecules FFimpactimpact = = ΔΔp/t = p/t = (2m v(2m vxx)/t)/t
ExampleExample
How fast does a typical gas molecule How fast does a typical gas molecule (travel at room temperature? Lets take (travel at room temperature? Lets take Nitrogen-14!Nitrogen-14!(k = 1.38x10(k = 1.38x10-23-23J/K) J/K) KE = 1/2 mv2 = 3/2 kT
v = (3kT/m)1/2
v = [(3)(1.38x10-23 x 293/(4.65x10-26)]1/2
v = 511 m/sec
ExampleExample If it takes 2 mins If it takes 2 mins
for your kettle to for your kettle to begin boiling how begin boiling how much longer does much longer does it take to boil dry?it take to boil dry? Assume kettle is Assume kettle is
3kW3kW Starting temp of Starting temp of
water 20°Cwater 20°C
Work done by kettle = power x time = 2 x 60 x 3000 = 360 000J
= Work to boil water of mass M = T x M x cwater 360 000J = 80 x M x 4190
Mass of water = 1.07kg
Energy to boil water = M x Lv (water)
= 1.07 x 2256 x103 = 2420 000J
Time required = Energy /power = 2420 000/3000 = 808 s ≈ 13mins