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Thermodynamics and Statistical Mechanics
First Law of Thermodynamics
Thermo & Stat Mech - Spring 2006 Class 3
2
Review of van der Waals Critical Values
227
278
3
ba
P
Rba
T
bv
C
C
C
Thermo & Stat Mech - Spring 2006 Class 3
3
van der Waals Results
bRa
Tba
Pbv ccc 278
271
3 2
bRa
R
bba
RTvP
c
cc
278
3271
2 375.0
83
Thermo & Stat Mech - Spring 2006 Class 3
4
van der Waals Results
Substance Pcvc/RTc
He 0.327
H2 0.306
O2 0.292
CO2 0.277
H2O 0.233
Hg 0.909
Thermo & Stat Mech - Spring 2006 Class 3
5
Configuration Work
đW = PdV
Gas, Liquid, Solid:
Thermo & Stat Mech - Spring 2006 Class 3
6
Kinds of Processes
Often, something is held constant. Examples:
dV = 0 isochoric or isovolumic processdQ = 0 adiabatic processdP = 0 isobaric processdT = 0 isothermal process
Thermo & Stat Mech - Spring 2006 Class 3
7
Work done by a gas
f
i
V
VPdVW
For isochoric process dV = 0, so W = 0For isobaric process dP = 0, so W = PV
đW = PdV
Thermo & Stat Mech - Spring 2006 Class 3
8
Work done by a gas
Thermo & Stat Mech - Spring 2006 Class 3
9
Work done by an ideal gas
f
i
V
VPdVW
For isothermal process dT = 0, so T = constant.
VRT
P
Thermo & Stat Mech - Spring 2006 Class 3
10
Isothermal Process
i
f
VV
V
V
V
VRTW
VRTW
VdV
RTW
f
i
f
i
ln
ln
Thermo & Stat Mech - Spring 2006 Class 3
11
Heat Capacity
Heat capacity measures the amount of heat that must be added to a system to increase its temperature by a given amount. Its definition:
where y is a property of the system that is kept constant as heat is added.
yy dT
QdC
Thermo & Stat Mech - Spring 2006 Class 3
12
Heat Capacity
Properties that are usually kept constant for a hydrostatic system are volume or pressure. Then,
PP
VV dT
QdC
dTQd
C
or ,
Thermo & Stat Mech - Spring 2006 Class 3
13
Heat Capacity
Clearly, the heat capacity depends on the size of the system under consideration. To get rid of that effect, and have a heat capacity that depends only on the properties of the substance being studied, two other quantities are defined: specific heat capacity, and molar heat capacity.
Thermo & Stat Mech - Spring 2006 Class 3
14
Specific Heat Capacity
Specific heat capacity is the heat capacity per mass of the system. A lower case letter is used to represent the specific heat capacity. Then, if m is the mass of the system,
P
PP
V
VV dT
Qdmm
Cc
dTQd
mmC
c
1
or ,1
Thermo & Stat Mech - Spring 2006 Class 3
15
Molar Heat Capacity
Molar heat capacity is the heat capacity per mole of the system. A lower case letter is used to represent the molar heat capacity. Then, if there are n moles in the system,
P
PP
V
VV dT
Qdnn
Cc
dTQd
nnC
c
1
or ,1
Thermo & Stat Mech - Spring 2006 Class 3
16
Shorter Version
PP
VV dT
qdc
dTqd
c
or ,
Use heat per mass.
Thermo & Stat Mech - Spring 2006 Class 3
17
cP – cV
đq = du + Pdv where u(T,v)
dvPvu
dTTu
qd
dvvu
dTTu
du
Tv
Tv
Thermo & Stat Mech - Spring 2006 Class 3
18
Constant Volume
dTdu
Tu
dTqd
cvv
v often ,
Thermo & Stat Mech - Spring 2006 Class 3
19
Constant Pressure
vPvu
cc
vPvu
cc
dTdv
Pvu
Tu
dTqd
Tvp
Tvp
PTvp
Thermo & Stat Mech - Spring 2006 Class 3
20
Ideal Gas
u is not a function of v.
Rcc vP
Thermo & Stat Mech - Spring 2006 Class 3
21
Adiabatic Process
For an ideal gas, and most real gasses,đQ = dU + PdV đQ = CVdT + PdV,.
Then, when đQ = 0, VC
PdVdT
Thermo & Stat Mech - Spring 2006 Class 3
22
Adiabatic Process
nRVdPPdV
CPdV
nRVdPPdV
dTnRPV
T
V
Then,
and ,
For an ideal gas, PV=nRT, so
Thermo & Stat Mech - Spring 2006 Class 3
23
Adiabatic Process
nRVdP
nRCCnR
PdV
nRVdP
nRCPdV
nRVdPPdV
CPdV
nRVdPPdV
dTnRpV
T
V
V
VV
0
110
Then,
and ,
Thermo & Stat Mech - Spring 2006 Class 3
24
Adiabatic Process
V
p
V
P
PV
V
V
C
C
VdPPdVVdPCC
PdV
CCnR
VdPC
CnRPdV
where,
0
0
Thermo & Stat Mech - Spring 2006 Class 3
25
Adiabatic Process
constant
constantlnlnln
constantlnln
,integrated becan which ,0
0
PV
PVPV
PVP
dPVdV
VdPPdV
Thermo & Stat Mech - Spring 2006 Class 3
26
Adiabatic Process
constant
constant
as, expressed be alsocan this
of help With the
constant
1
1
PT
TV
nRTPV
PV
Thermo & Stat Mech - Spring 2006 Class 3
27
for “Ideal Gasses”
33.162
1 :polyatomic
40.152
1 :diatomic
67.132
1 :monatomic
21