Physics Beyond 2000 Chapter 8 Heat and Gases Intermolecular force Intermolecular force 0 r roro...
-
Upload
bennett-marshall -
Category
Documents
-
view
221 -
download
1
Transcript of Physics Beyond 2000 Chapter 8 Heat and Gases Intermolecular force Intermolecular force 0 r roro...
Physics Beyond 2000
Chapter 8
Heat and Gases
Intermolecular force
Intermolecularforce
0rro
Potential energy
-ε
The intermolecular force between two gas molecules is so small that it is insignificant.
Ideal gas
• The intermolecular force = 0.
• At absolute zero, gas volume = 0.
http://www.colorado.edu/physics/2000/bec/temperature.html
Macroscopic point of view
• The gas laws.
• Pressure, volume, temperature and mass (mole).
http://edie.cprost.sfu.ca/~rhlogan/gas_laws.html
http://library.thinkquest.org/12354/
Boyle’s Law
dry air
mercury
h
rubber tubing
Find the pressure P and the volume V of the dry air.
L
Boyle’s Law
dry air
mercury
h
rubber tubing
V = L.A
L
A
Boyle’s Law
dry air
mercury
h
rubber tubing
V = L.A
L
A
P
P = gh + PoPo
Boyle’s Law• By raising or lowering the open end, you
may change P and V.
dry air
mercury
h
rubber tubing
Boyle’s Law• For a fixed mass of gas at constant
temperature, P.V = constant.
P
0 V
P
0 V
1
Boyle’s Law• What would happen if the temperature
changes?
P
0 V
P
0 V
1
Boyle’s Law• What would happen if the temperature
changes? P.V is equal to a new constant.
P
0 V
T1 T2 T3 P
0 V
1
T3T2
T1
T1 < T2 < T3
Boyle’s Law• The following curve is called an isothermal.
P
0 V
T1 T2 T3 P
0 V
1
T3T2
T1
T1 < T2 < T3
Ideal Gas
• Not all gases obey Boyle’s law.
• Many gases follow the law well at high temperature and low pressure.
• Gases which obey Boyle’s law exactly are called ideal gases.
PV graph for oxygen
P
0V
ideal gas
real gas
high temperature
low temperature
Example 1
• Assume that water temperature does not change.
Boyle’s law calculator:http://www.1728.com/boyle.htm
Temperature and thermometer
• Name different kinds of thermometer and the substances used in the thermometers.
http://www.howstuffworks.com/therm.htm
Temperature Scale
• Choose a property which changes with temperature as the thermometric substance.
• Choose two particular temperatures.– Lower fixed point– Upper fixed point
Temperature Scales
• Fixed points (i.e. some particular temperatures)
The Celsius Temperature
• Upper fixed point: the temperature of boiling pure water at one atmospheric pressure. It is called 100oC.
• Lower fixed point: the temperature of melting pure ice. It is called 0oC.
http://www.santesson.com/engtemp.html
The Celsius Temperature
• Let X100 be the value of the thermometric substance at 100oC.
• Let Xo be the value of the thermometric substance at 0oC.
X100= L100
Xo= L0
The Celsius Temperature
X100= L100
Xo= L0
What is this temperature ?
The Celsius Temperature
X100= L100
Xo= L0
What is this temperature ?
X=L
CXX
XX 0
0100
0 100
Why?
The Celsius Temperature
X100= L100
Xo= L0
X=L
CXX
XX 0
0100
0 100
X100
Xo
X
1000C0oC
The Celsius Temperature
X100= L100
Xo= L0
X=L
X100
Xo
X
1000C0oC
In designing celsius temperature, it is assumed that the thermometric property changes linearly with the temperature.
Is this assumption correct?
Charles’ Law• For a fixed mass of gas at constant pressure,
the gas volume V changes linearly with the temperature t.
V
t/oC0oC
V = m.t + V0
V0
-273.15oC
Charles’ Law• For a fixed mass of gas at constant pressure,
the gas volume V is directly proportional to the temperature T in kelvin scale.
V
t/oC0oC
V = m.T
V0
V
0K T/K
Disagreement in Celsius Temperature
• Celsius temperature shows disagreement between two thermometers using different thermometric substances.
• Example: Length of mercury column
and Resistance.
• Note that both temperature readings are correct according to their own scales.
Constant volume gas thermometer
• Standard thermometer in physics.
• The thermometric substance is the gas pressure.
• %100
0100
0
PP
PP
Ideal Gas Scale Temperature
• The product P.V is chosen as the thermometric substance.
• Lower fixed point: absolute zero. 0 K.
• Upper fixed point: triple point of water.
Triple point of water Ttr
• This is the temperature at which ice, water and water vapour exist in equilibrium in the absence of air.
• The triple point of water is 0.01oC higher than the ice point.
• The triple point of water is 273.16 K.
http://www.physics.reading.ac.uk/units/flap/glossary/ii/idlgstsl.htm
Ideal Gas Scale Temperature
Ptr
273.16K0 K
Ideal gas temperature/K
Pressure of an ideal gas
P
T
trP
PT 16.273
Triple pointAbsolute zero
Ideal Gas Scale Temperature
• Kelvin scale
• T/K = /oC + 273.15/oC
Example 2
• Calculate the unknown temperature by using a constant volume gas thermometer.
General Gas Law
• For a fixed mass of gas,
T
VP.= constant
http://www.jersey.uoregon.edu/vlab/Piston/
Example 3
• Apply general gas law.
Ideal gas calculator:http://library.thinkquest.org/2923/ideal.html
The universal gas constant R
0
Amount n/mol
PV/J
P1.V1
n1
P2.V2
n2
For an ideal gas at constant temperature, P.V is directly proportional the amount of gas.
P.V n
The universal gas constant R• For a fixed mass of gas,
P.V T
• For a gas at constant temperature,
P.V n
• Combining the above two relations,
P.V n.T
or P.V = R.n.T where R is a constant.
The universal gas constant R
• P.V = nRT
• R = 8.31 J K-1 mol-1.
• R is called the universal gas constant.
http://www.treasure-troves.com/physics/UniversalGasConstant.html
Example 4
• standard temperature and pressure (0oC and 1.03 105 Pa).
• Find the gas volume at s.t.p.
Notations
• Avogadro constant NA = 6.02 1023 mol-1
• So one mole of substance contains 6.02 1023 elementary units.
Notations
• In this text,n = quantity in moles
N = number of moleculesN = n.NA
Notations• In this text,
Mm = molar mass
M = mass of the gas m = mass of a gas moleculesM = Nm = n.NA.m
and
Mm = NA.m
Example 5
• Find the mass of gas at s.t.p.
Avogadro’s Law
• With the same temperature, pressure and volume, two ideal gases contain the same number of molecules.
• How to prove it?
http://www.avogadro.co.uk/definitions/avogconst.htm
Example 6
• Use the Avogadro’s law.
Dalton’s law of partial pressure
• In a mixture of ideal gas, the total pressure equals to the sum of individual gas pressures.
+
P1 + P2= P
http://edie.cprost.sfu.ca/~rhlogan/dalton.html
Dalton’s law of partial pressure
+
P1 + P2= P
V
RTnP 1
1 V
RTnP 2
2 V
nRT
V
RTnnP
)( 21
Typical examples
• Mixture of gases from two chambers
P1
V1
T1
P2
V2
T2
Two gases with different pressure, volume and temperature.
Mixture of gases from two chambers
PV1
T1
PV2
T2
When they are mixed, they will have the samepressure (common pressure) P.What is this pressure P?
Each chamber is kept at its own temperature.
Mixture of gases from two chambers
• Find the total number of moles n when they are still separated.
P1.V1=n1RT1P2.V2=n2RT2
21
12221121 TRT
TVPTVPnnn
P1
V1
T1
P2
V2
T2
Mixture of gases from two chambers
• Find the total number of moles n after they are mixed.
P.V1= RT1P.V2= RT2
21
1221'2
'1 TRT
TPVTPVnnn
'1n
'2n
PV1
T1
PV2
T2
Mixture of gases from two chambers
21
1221'2
'1 TRT
TPVTPVnnn
21
12221121 TRT
TVPTVPnnn
Compare these two equations.
1221
122211
TVTV
TVPTVPP
Escape of gas from a container
Po = atmospheric pressureTo = external temperature
n
PVT
If we open the tap for a while, how much gas would escape?
Escape of gas from a container
Po = atmospheric pressureTo = external temperature
PV=nRT
There are n moles of gas.n
PVT
Escape of gas from a container
Po = atmospheric pressureTo = external temperature
The chamber is kept at constant temperature T.
Po
VT
Escape of gas from a container
Po = atmospheric pressureTo = external temperature
There remains n’ moles of gas in the container.
PoV=n’RTPo
VT
Po
VT
n’
Escape of gas from a container
Po = atmospheric pressureTo = external temperature
Po
VT
Po
VT
n’
The amount of gas escaped is
)(,oPP
RT
Vnnn
Escape of gas from a container
Po
VT
Po
VT
n’
The volume occupied by the escaped gas V’=
o
oo
o
o
P
RT
RT
PPV
P
TRn.
)(..
Kinetic theory model
• A microscopic point of view.
• How does the behaviour of gas molecules relate to its pressure, temperature, volume and mass?
http://www.phy.ntnu.edu.tw/java/idealGas/idealGas.html
Kinetic theory model• Ideal gas
– 1. There is not any intermolecular force between two gas molecules.
– 2. The volume of each molecule is zero.– 3. The collisions between molecules and the
container are perfectly elastic.– 4. The gas molecules are in random motion.
Pressure from kinetic theory
• Suppose that there are N gas molecules each of mass m in a cubical box of side L.
L
L
L
Resolve a vector• Revision: Resolve a vector c into 2
components (2-dimensional case).y
x
c
What is the relation between c and its 2 components?
Pressure from kinetic theory
x
y
z
Take one molecule, the ith molecule which has speedci.
ci
Pressure from kinetic theory
x
y
z
Resolve its velocity ci intothe three components ui, vi and wi.
ciui
vi
wi
2222iiii wvuc
Pressure from kinetic theory
• Consider its motion along the x-axis.
ui
x
L
LL
Pressure from kinetic theory
• Consider its motion along the x-axis.
ui
x
L
LL
The molecule hitsthe right wall withspeed ui after travellinga distance L.
Pressure from kinetic theory
• Consider its motion along the x-axis.
uix
L
LL
The moleculerebounds at thesame speed ui
because the collisionis perfectly elastic.
Pressure from kinetic theory
• Consider its motion along the x-axis.
ui
x
L
LL
The molecule hitsthe right wall againafter travellinga distance 2L.
Pressure from kinetic theory
ui
x
L
LL
-ui
What is the change of momentum of the ith molecule along x-axis?
(mui) = (-mui)-(mui)= -2mui
Pressure from kinetic theory
What is the time elapsed between two successive impacts on the right wall?
uix
L
LL
Pressure from kinetic theory
What is the time elapsed between two successive impacts on the right wall?
uix
L
LL
Pressure from kinetic theoryWhat is the time elapsed between two successive impacts on the right wall?
ui
x
L
LL
iu
Lt
2
Pressure from kinetic theory
ui
x
L
LL
-ui
What is the average force fi’ on the ith molecule by the right wall?
(mui) = -2mui
iu
Lt
2
L
mu
t
muf iii
2, )(
fi’
Pressure from kinetic theory
ui
x
L
LL
-ui
What is the average force fi on the right wall by the ith molecule?
By Newton’s 3rd lawfi = -fi’
fi
L
mui2
Pressure from kinetic theory
• As there are N molecules, the total force on the right wall is Fx = f1 + f2 + … + fN
L
L
LF
N
iiuL
m
1
2.
Pressure from kinetic theory
L
L
LF
N
iix u
L
mF
1
2.Define the mean square speed as
N
uu
N
ii
1
2
2
Pressure from kinetic theory
L
L
LFx
Define the mean square speed as
N
uu
N
ii
1
2
2
L
uNmFx
2
Pressure from kinetic theory
L
L
LFx
L
uNmFx
2
Pressure on the right wallis
V
uNm
L
uNm
L
F
A
FP x
2
3
2
2
Pressure from kinetic theory
L
L
LFx
Pressure on the right wall isV
uNmP
2
Pressure from kinetic theory
x
y
z
ciui
vi
wi
Pressure on the right wall isV
uNmP
2
How does relate to c?2u
Pressure from kinetic theory
x
y
z
ciui
vi
wi
2222iiii wvuc
2222
2222
wvuc
N
w
N
v
N
u
N
c iiii
Pressure from kinetic theory
x
y
z
ciui
vi
wi
2222 wvuc
As the gas molecules aremoving randomly,
222 wvu
22 3uc
Pressure from kinetic theory
L
L
L
Pressure isV
cNm
V
uNmP
3
22
2
3
1cNmPV
Pressure from kinetic theory
L
L
L
Pressure is
22
3
1
3
1cc
V
NmP
where is the density of gas
Root mean square speed cr
• Definition of r.m.s. speed of gas
2ccr
22rcc
Root mean square speed cr
2
3
1rNmcPV
or
2
3
1rcP
r.m.s.speed at s.t.p.
• Given: Molar mass of O2 gas = 0.032 kg
• Find the r.m.s. speed of oxygen at s.t.p.
• Use 2
3
1rAmcnNnRTPV
Average r.m.s. speed of air at room temperature
• Given: The average density of air at room temperature is 1.2 kg m-3.
• The atmospheric pressure = 1.01 105 Pa.
• Use2
3
1rcP
P
cr3
Distribution of molecular speed
• The r.m.s. speed of ideal gas molecules is given by
• The r.m.s speed increases with temperature and decreases with molecular mass.
2
3
1rmcnMnRTPV
Some r.m.s. speeds:http://www.physics.mun.ca/~gquirion/P2053/html21/slide9.html
Distribution of molecular speed
• The actual speed of each gas molecules varies.• If temperature is kept constant, the speeds of gas molecules follow the Max
wellian distribution.
No. of molecules with speeds of 0.5 ms-1 of speed v
speed v
dv
dN
low temperature
high temperature
Distribution of molecular speed
No. of molecules with speeds of 0.5 ms-1 of speed v
speed v
dv
dN
low temperature
high temperature
• The area under the graph represents the total number of gas molecules.
Distribution of molecular speed
No. of molecules with speeds of 0.5 ms-1 of speed v
speed v
dv
dN
low temperature
high temperature
• As the temperature increases, the r.m.s. speed increases.
• The peak shifts to the right (higher speed).
Distribution of molecular speed
No. of molecules with speeds of 0.5 ms-1 of speed v
speed v
dv
dN
low temperature
high temperature
• As the temperature increases, the r.m.s. speed increases.
• The range of speeds spreads out.
Distribution of molecular speed
No. of molecules with speeds of 0.5 ms-1 of speed v
speed v
dv
dN
low temperature
high temperature
• As the temperature increases, the r.m.s. speed increases.
• The amplitude is lower because the area, representing the total number of molecules, under the curve must be the same.
http://cyniska.ubishops.ca/0002933/3dspeed.html
Example 7
• Double the gas temperature.
• What would happen to its r.m.s. speed?
• Use
mr M
RTc
3
Escape speed and atmosphere
No. of molecules with speeds of 0.5 ms-1 of speed v
speed v
dv
dN
escapespeed
These moleculesare able to escape.
Escape speed and atmosphere
• On earth– The escape speed is 11.2 103 ms-1
– The r.m.s. speed of air molecules is 502 ms-1
• Do the air molecules on earth escape easily?
Escape speed and atmosphere
• On the moon– The escape speed is 2.38 103 ms-1
– The r.m.s. speed of air molecules is 558 ms-1
• Do the air molecules on the moon escape easily?
Translational kinetic energy
• The average translational kinetic energy is
221
2
2
1
2
121
r
N
ii
k mccmN
mcE
Translational kinetic energy
• From the ideal gas equation,
TN
Rmc
RTN
NNmc
nRTPV
Ar
Ar
2
3
2
1
3
1
2
2
Translational kinetic energy
• So the average translational kinetic energy is
kTTN
RmcE
Ark 2
3
2
3
2
1 2
whereAN
Rk is the Boltzmann constant
k=1.38 10-23 JK-1
Translational kinetic energy• So the average translational kinetic energy is di
rectly proportional to the ideal gas temperature.
kTEk 2
3
kE
T/K0
Translational kinetic energy• At the same temperature, all gas molecules hav
e the same average translational kinetic energy.
kTEk 2
3
kE
T/K0
Translational kinetic energy• At absolute zero, all molecules are at rest.
kTEk 2
3
kE
T/K0
Example 8
• Find the total translational kinetic energy of 1 mole of O2 gas at room temperature 27oC.
• NA = 6.02 1023
• k = 1.38 10-23 JK-1
Internal energy of an ideal gas
• Internal energy of an object
= total kinetic energy of the molecules
+ total potential energy of the molecules
http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/inteng.html
Internal energy of an ideal gas
• There is not any intermolecular force among ideal gas molecules
Potential energy of ideal gas = 0
Internal energy of ideal gas
= Total kinetic energy of ideal gas molecules
Internal energy of an ideal gas• Consider an ideal gas of monatomic molecu
les only. There is not any rotational kinetic energy.
The ideal gas has translational kinetic energy only.
Internal energy U of a monatomic ideal gas
= total translational kinetic energy of molecules.
Internal energy of an ideal gas
TN
RnNkTNENU
AAk .
2
3).(
2
3..
nRTU2
3
where n is the number of moles
Internal energy of an ideal gas
nRTU2
3
Apply PV=nRT
PVU2
3 for a monatomic ideal gas
Example 9
• He gas with volume 1 litre (0.001 m3) and at 2 atmospheric pressure.
• Assume that it is an ideal gas.
• Find its internal energy.
Real Gases
• Ideal gas• Molecules do not have
size.
• Real gases• There is a finite size
for each molecule.
Real Gases
• Ideal gas• There is not any
intermolecular force among ideal gas molecules
• Real gases• When they are far
apart, there is attraction among the molecules.
• When they collide, there is repulsion.
Real Gases
• Ideal gas• May apply the equatio
n
P.V = nRT
• Real gases• Need to correct the
equation
Gas equation for real gases
• Effect of attractive intermolecular force
Gas pressure P is due to the collision of moleculeson the wall of the container.
P piston
Gas equation for real gases
• Effect of attractive intermolecular force
For real gas, the collision of a molecule on the wallis retarded by the attraction of all other gas molecules.
piston
Gas equation for real gases
• Effect of attractive intermolecular force
As a result, the measured real gas pressure is less thanthe pressure of an ideal gas under the same condition.
piston
Gas equation for real gases
• Effect of attractive intermolecular force
Add a positive correction k to the measured pressureP. (P + k) is the pressure due to an ideal gas underthe same condition.
pistonP+k
Gas equation for real gases
• Effect of finite size of gas molecules
P+k piston
Some portion of volume is occupied by the molecules. The measured volume is greater thanthe volume of an ideal gas.
Gas equation for real gases
• Effect of finite size of gas molecules
P+k piston
Add a negative correction –nb to the measured volume V. (V - nb) is volume of the ideal gasunder the same condition. n = number of moles.
V - nb
van der Waals equation
• (P + k).(V – nb) = n.R.T
• k =
• a and b are constants. Their values depend on the kind of real gas.
2
2
V
an
van der Waals equation
• (P + k).(V – nb) = n.R.T
P
V
P.V = nRT
Critical temperature
• An ideal gas has no forces among molecules.
• So it cannot become a liquid.
• At infinite high pressure or at the absolute zero, the volume of an ideal gas become zero.
Critical temperature
• Real gas has forces among molecules.
• It can become liquid.
• If the temperature is below its freezing point, it becomes liquid.
Critical temperature
• Experiments show that when a gas is below a certain temperature (which is above the boiling point), it can become liquid by applying high pressure without further decreasing the temperature.
• The temperature is called critical temperature.
Critical temperature
• Critical temperature
It is the temperature above which a gas cannot become liquid by applying high pressure only.
http://www.chem.purdue.edu/gchelp/liquids/critical.html
Isothermals for CO2
Critical temperature of CO2 is 31.2oC
Isothermals for CO2
At high temperature above the critical temperatureCO2 is gas.
Isothermals for CO2
In this region, CO2 could be compressed into liquid. It is called vapour.
Isothermals for CO2In this region, it is difficult to change its volume by applying pressure. It is in either liquid or solid state.
Isothermals for CO2In this region, both liquid and vapour exist in equilibrium.The vapour is called the saturated vapour.
Average separation of gas molecules
• When liquid changes into gas under one atmospheric pressure, its volume increases by 750 times.
• What is the average separation D of gas molecules under one atmospheric pressure in terms of the diameter ro of each molecule?
D = 9.1 ro
Thermodynamics
• To study the laws governing– conversion of energy– direction of heat flow– work
Heat and Internal Energy
• Heat flow– Heat is the energy transfer from a body at high temperature to
another body at low temperature.
hightemperature
lowtemperature
heat flow
Heat and Internal Energy
• Internal energy– Sum of kinetic energy and potential energy of the molecules of a
substance.
hightemperature
lowtemperature
heat flow
Lose internalenergy
Gain internalenergy
Potential energy of gas molecules
• For a real gas, its potential energy is a maximum because they are separated by an infinite distance. The potential energy is zero.
• For other states, its potential energy is negative.
Internal energy of Ideal gas
• Potential energy of an ideal gas is zero.
• The internal energy of an ideal gas is the sum of kinetic energy of its molecules.
• When the internal energy of an ideal gas increases, its temperature rises up.
• When the internal energy of an ideal gas decreases, its temperature drops down.
Internal energy of gas
• Heating the gas Increasing the internal energy (kinetic energy) of the gas Rise in gas temperature
• Cooling the gas Decreasing the internal energy (kinetic energy) of the gas A drop in gas temperature
Internal energy of gas
• Doing work on the gas (or the gas does a negative work e.g. compression of a gas) Increasing the internal energy (kinetic energy) of the gas Rise in temperature
• The gas does a positive work (e.g. expansion of a gas) Decreasing the internal energy (kinetic energy) of a gas A drop in temperatrue.
Work done by a gas
• Expansion of a gas
The gas does a positive work
The gas loses internal energy.
gasF
F = P.A
Work done by a gas
• Expansion of a gas
The gas does a positive work
The gas loses internal energy.
gas
x
F
F = P.A
Work done by a gas
• compression of a gas
The gas does a negative work
The gas gains internal energy.
gasF
F = P.A
Work done by a gas
• compression of a gas
The gas does a negative work
The gas gains internal energy.
gas
x
F
F = P.A
Work done by a gas
• Expansion of a gas
Work done by the gas W = F. x = P.V
gas
x
F
F = P.A
Work done by a gas
• Expansion of a gas
Work done by the gas W =
gas
x
F
F = P.A
2
1
.V
VdVPdW
Work done by a gas
gas
x
F
F = P.A
2
1
.V
VdVPdW• W =
• To evaluate the integration, it is necessary to express P in terms of V.
Example 10
• Assume that the force is a constant. Evaluate the integration.
• As the work done by the gas is positive, it loses internal energy.
The indicator diagram• The state of a fixed mass of gas is described
by its pressure, volume and temperature.
• The state can be represented by a point in the P-V graph. (P1, V1, T1)
P
VT1
P1
V1
0
The indicator diagram• In the following diagram, it shows that the
gas expands at a constant temperature.
P
VTi
Pi
Vi
0Pf
Vf
Initial state: Pi , Vi , Ti
Final state: Pf , Vf , Tf
Tf = Ti
The indicator diagram• The area indicates the work done by the
gas.
• It is also the loss of internal energy of the gas.
Initial state: Pi , Vi , Ti
Final state: Pf , Vf , Tf
Tf = Ti
P
VTi
Pi
Vi
0Pf
Vf
The indicator diagram• In the following diagram, it shows that the
gas is compressed at a constant temperature.
P
VTi
Pf
Vf
0Pi
Vi
Initial state: Pi , Vi , Ti
Final state: Pf , Vf , Tf
Tf = Ti
The indicator diagram• The area indicates the negative work done
by the gas.
• It is also the gain of internal energy of the gas.
P
VTi
Pf
Vf
0Pi
Vi
Initial state: Pi , Vi , Ti
Final state: Pf , Vf , Tf
Tf = Ti
Work done by the gas
• W > 0 if its volume expands Loss of internal energy
Vf > Vi
• W < 0 if its volume is compressed Gain of internal energy
Vf < Vi
Work done by a gas
• Example 11
• Find the gas pressure first
• Calculate the work done by the gas.
Changes of states of a gas
gas
heat reservoir
weight A gas is trapped inside acylinder with a movable piston.The gas temperature canbe regulated by a heatreservoir.
Movable piston
Isobaric process
• The pressure P is kept constant.
P
V
Po
Vi
Ti
Vf
Tf
Pi = Pf = Po
ExpansionW > 0
Changes of states of a gas
gas
heat reservoir
weight
Movable piston
Change the temperature of the heat reservoir fromTi to Tf. The gas expands with constant pressure.Pf = Pi = Po
P
V
Po
Vi
Ti
Vf
Tf
Changes of states of a gas
gas
heat reservoir
weight
Movable piston
There is heat flow to the gas while the pressure is kept at a constant.
P
VVi
Ti
Vf
Tf
Po
Changes of states of a gas
gas
heat reservoir
weight
Movable piston
The area represents the net work done by the gas.Heat is absorbed.The internal energy of the gas increases.
P
VVi
Ti
Vf
Tf
Po
Changes of states of a gas
gas
heat reservoir
weight
Movable piston
W1 = Po.(Vf – Vi) is the work done by the gas.Q1 is the heat flow to the gas.U is the increase in the internal energy of the gas.
Q1
P
VVi
Ti
Vf
Tf
Po
Changes of states of a gas
gas
heat reservoir
weight
Movable piston
Q1 = U + W1 or U = Q1 - W1 > 0
The heat flow increases the internal energy.The work done by the gas decreases the internal energy.
Q1
P
V
Po
Vi
Ti
Vf
Tf
Changes of states of a gas
gas
heat reservoir
weight
Movable piston
There may be another way to reach the same final state.
P
V
Po
Vi
Ti
Vf
Tf
P’T’
Changes of states of a gas
gas
heat reservoir
weight
Movable piston
Step 1: Decrease both the pressure and temperature while keeping the volume constant.
P
V
Po
Vi
Ti
Vf
Tf
P’
1
T’
Changes of states of a gas
gas
heat reservoir
weight
Movable piston
Step 2: Increasing both the volume and temperature while keeping the pressure constant.
P
V
Po
Vi
Ti
Vf
Tf
P’ 2T’
Changes of states of a gas
gas
heat reservoir
weight
Movable piston
P
Step 3: Increasing both the pressure and temperature while keeping the volume constant.
V
Po
Vi
Ti
Vf
Tf
P’
3
T’
Changes of states of a gas
gas
heat reservoir
weight
Movable piston
The area represents the work done by the gas.Heat is absorbed by the gas.The internal energy of the gas increases.
P
V
Po
Vi
Ti
Vf
Tf
P’T’
Changes of states of a gas
gas
heat reservoir
weight
Movable piston
W2 = P’.(Vf – Vi) is the work done by the gas.Q2 is the heat flow to the gas.U is the increase in the internal energy of the gas.
Q2
P
V
Po
Vi
Ti
Vf
Tf
P’T’
Changes of states of a gas
gas
heat reservoir
weight
Movable piston
Q2 = U + W2
Q2
P
V
Po
Vi
Ti
Vf
Tf
P’T’
Changes of states of a gas• Comparison of the t
wo processes:• Both U are equal because they have the s
ame initial and final states.
• W1 = Po (Vf – Vi) > W2 = P’ (Vf – Vi) Q1 > Q2
P
V
Po
Vi
Ti
Vf
Tf
P’T’
process 1: process 2:
Q = U + W
First Law of Thermodynamics
Q = U + W
The heat supplied to the gas, Q, enables it to do work W and to increase its internal energy U .
Example 12
• Assume that the gas volume decreases in the process.
Example 13
• On expansion, there is loss of internal energy.
• By heating, the internal energy is supplemented.
Adiabatic process• In this process, there is not any heat transfer.
• Q = 0 U + W = 0
U = - W
• On adiabatic expansion
W > 0 U < 0
• On adiabatic compression
W < 0 U > 0
Adiabatic Expansion• Q = 0• W > 0 U < 0
P
VLow T
High T
Pi
Vi
Pf
Vf
The shaded area representsthe positive work done W.
Adiabatic Compression• Q = 0• W < 0 U > 0
P
VLow T
High T
Pf
Vf
Pi
Vi
The shaded area representsthe negative work done W.
Adiabatic process
• Using a container with good heat insulation adiabatic process Q = 0.
• Change the gas volume quickly adiabatic process Q = 0. (I) Quick compression of gas A rise in
temperature. (II) Quick expansion of gas A drop in
temperature.
Adiabatic process: Experiment
• Change the gas volume quickly
adiabatic process Q = 0.
(I) Quick compression of gas A rise in temperature.
(II) Quick expansion of gas A drop in temperature.
Example 14
• Expansion of gas quickly.
Constant volume process• The gas volume does not change W = 0.• U = Q.• Heat is supplied to the gas and
increases its internal energy (kinetic energy) Temperature rises up.
• Heat is extracted from the gas and decreases its internal energy (kinetic energy) Temperature drops down.
Heating under constant volume
P
VLow T
High T
Pf
Vo
Pi
•Vi = Vf = Vo
•W = 0 U = Q
Heating under constant volume
P
VLow Ti
High Tf
Pf
Vo
Pi
•For monatomic ideal gas, U = nRT2
3
)(2
3if TTnRUQ
Vi = Vf = Vo
Cooling under constant volume
P
VLow Tf
High Ti
Pi
Vo
Pf
•For monatomic ideal gas, U = nRT2
3
)(2
3if TTnRUQ
Vi = Vf = Vo
Example 15
• Monatomic ideal gas.
Isothermal process
• Temperature is kept constant. U =0.
• Q = W.
• Q > 0 W > 0 Expansion of gas
• Q < 0 W < 0 Contraction of gas
Isothermal process
• Temperature is kept constant. U =0.• Q = W > 0 Expansion
P
VTo
Pi
Vi
0Pf
Vf
The shaded area represeentsthe work done.
Ti = Tf = To
Isothermal process• Temperature is kept constant. U =0.• Q = W > 0 Expansion• If , find W.
nRTPV
2
1
2
1
)ln(1
2V
V
V
V V
VnRTdV
V
nRTPdVW
P
VTo
Pi
Vi
0Pf
Vf
Ti = Tf = To
Isothermal process• Temperature is kept constant. U =0.• Q = W < 0 Contraction• If PV = nRT , find W.
P
VTo
Pf
Vf
0Pi
Vi
2
1
2
1
)ln(1
2V
V
V
V V
VnRTdV
V
nRTPdVW
Ti = Tf = To
Example 16
• Compression of a gas without temperature changes.
Isobaric process
• The pressure is kept constant.
P
V
Po
Vi
Ti
Vf
Tf
Isobaric expansion
• The pressure is kept constant on expansion.
P
V
Po
Vi
Ti
Vf
Tf
W > 0 Q > 0 and U > 0
Isobaric expansion
• The pressure is kept constant on expansion.
P
V
Po
Vi
Ti
Vf
Tf
If it is a monatomic ideal gas,find W, U and Q.
Isobaric expansion
• The pressure is kept constant on expansion.
P
V
Po
Vi
Ti
Vf
Tf
W = Po(Vf – Vi)
)(2
3ifo VVPU
)(2
5ifo VVPQ
Example 17
• Expansion of a monatomic ideal gas without change in pressure.
Cycle• A gas changes from one state to another
state and back to the initial state through another path.
U = 0.
P
V
P1
V1
0P2
V2
CycleU = 0.
• In the following process,
W > 0 Q > 0.
P
V
P1
V1
0P2
V2
CycleU = 0.
• In the following process
W < 0 Q < 0.
P
V
P1
V1
0P2
V2
Example 18
• Discussion of Figure 24.
Typical exampleA cylinder with a movable piston is immersed inice-water.
gas ice-waterOoC
Typical example1. A quick compression (adiabetic process)
gasice-water
P
VV1V2
1
0oC
T2>0oC
Typical example1. A quick compression (adiabetic process)2. Allow the gas to cool down at constant vol
ume.
gasice-water
P
VV1V2
1
0oC
T2>0oC2
Typical example1. A quick compression (adiabetic process)2. Allow the gas to cool down at constant volume.3. Expands the gas along an isothermal.
gasice-water
P
VV1V2
1
0oC
T2>0oC2
3
Example 19
• Discuss the energy change.