chapter 2: internal energy (u), work (w), heat (q...
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Winter 2013 Chem 254: Introductory Thermodynamics
Chapter 2: Internal Energy, Work, Heat and Enthalpy 13
Chapter 2: Internal Energy (U), Work (w), Heat (q), Enthalpy (H) ................................................ 13
Heat Capacities ......................................................................................................................... 16
Calculating ΔU, ΔH, w, q in Ideal Gas ........................................................................................ 18
Isothermal Compression ........................................................................................................... 21
Reversible Process (limiting process) ....................................................................................... 22
Isothermal Expansion ............................................................................................................... 22
Chapter 2: Internal Energy (U), Work (w), Heat (q), Enthalpy (H)
Internal Energy (excludes motion and rotation of vessel)
Look at isolated part of universe
system EnvironmentU U U
Total = isolated
First law of thermodynamics:
- Total U for isolated system is constant
- Energy can be exchanged between various components
- Energy forms can be interconverted
Eg. Chemical En Heat Work
0total system environementU U U
Winter 2013 Chem 254: Introductory Thermodynamics
Chapter 2: Internal Energy, Work, Heat and Enthalpy 14
Work
In classical mechanics, move object a distance d with force F in direction of
displacement is work N m = J
F mg d h
w mgh (kg m s-2 m = N m = J)
cosw mgd cosh
d
h
w mgd mghd
General formula
w F dL Line integral
PV work (constant external pressure)
m applies constant force F
PA
1 2( ) ( )ext
Fw mgh Fh Ah P V V
A
( )ext final initialw P V V Joules, or L Bar (1 L Bar = 100 J)
Winter 2013 Chem 254: Introductory Thermodynamics
Chapter 2: Internal Energy, Work, Heat and Enthalpy 15
More general formula for PV work, P does not need to be constant
f
i
V
extV
w P dV
Sign Convention : Work done on the system raises internal energy of system ( 0w )
Work done by the system lowers the internal energy ( 0w )
Other forms of work:
- electrical work
w Q Q is charge in coulombs
difference in potential (in Volts or J/C)
Run a current over
Q I t I is current (in Amps or C/s)
w I t
Important: Work is associated with a process, with change. Work is transitory. You
cannot say that a system contains that amount of energy or heat
Heat: associated with a process going from State 1 State 2
systemU q w q is heat; w is the work
Heat is exchanged between system and environment
0q : system loses energy
0q : system gains energy
system environmentq q
note: system environmentT T for heat to flow
Isolated system
Winter 2013 Chem 254: Introductory Thermodynamics
Chapter 2: Internal Energy, Work, Heat and Enthalpy 16
outer innerT T (regulate)
So there is no flow of heat
0system environmentU U
0innerU
Beaker + Lab +… = environment (isolated)
0IU ; 0IIU
0I IIU U
Chemical Energy 2 2 2Butane + O CO H
Note : 0IIU even if temperature increases!
Why? Chemical energy of butane is converted to heat.
Heat Capacities
The amount of energy (heat) required to raise the temperature of 1 gram of substance
by 1 oC. Heat capacity of water is 4.18 J/g K = 1 calorie
1) Heat capacity is dependent on heat
Eg. 10 oC 11 oC and 80 oC 81 oC, require slightly different energies
2) At least 2 types of heat capacity
a) Keep volume constant VC
b) Keep pressure constant PC
3) Heat capacity is proportional to amount of substance
Molar heat capacities : ,P mC , ,V mC
n moles : ,V V mC nC , ,P P mC nC
4) General formula
Winter 2013 Chem 254: Introductory Thermodynamics
Chapter 2: Internal Energy, Work, Heat and Enthalpy 17
f
i
T
V VT
q C dT
If VC is constant over temperature range:
f f
ii
T T
V V V V f iTTq C dT C T C T T
( )V Vq C T
And ( )P Pq C T
Which is larger PC or
VC ?
Relation for PC and
VC for ideal gas?
2 1V V ; 2 1T T
2 1( )P P extU q w q P V V PV nRT
2 1( )PU q nR T T
P Pq C T ; V VU q C T
P VC T C T nR T
P VC C nR or , ,P m V mC C R
Therefore PC is larger than
VC . At constant P , the system also does PV work when
raising T . (analysis for ideal gas)
No work because V is constant
VU q w Vq
VU C T
Bomb calorimetry
system surrounding Calorimeter
V V V measureq q C T
reactionU
Winter 2013 Chem 254: Introductory Thermodynamics
Chapter 2: Internal Energy, Work, Heat and Enthalpy 18
system surroundings
P Vq q
system Calorimeter
P V measureq C T
system
P reactionq H
True definition of Enthalpy
( )H U PV 2 2 1 1PV PV PV ; for 1 1 2 2PV PV ;
H U PV
At constant Pressure
2 2 1 1H U PV PV
H U P V
( ) ( )P PH q w P V q P V P V
P PH q C T
Completely general : ,U H are function of state
specify , ,T V P
2 2 2 1 1 1( , , ) ( , , )U U T P V U T P V
2 2 2 1 1 1( , , ) ( , , )H H T P V H T P V
Change in ,U H are the same for both paths
Change in ,q w are different for different
paths
Calculating ΔU, ΔH, w, q in Ideal Gas
1) Calculating ,U H is easy if T is known
( ) ]f
f
ii
T T
V V T V f iT
U U T U C dT C T C T T
VU C T for any process
( ) .....H H T
Winter 2013 Chem 254: Introductory Thermodynamics
Chapter 2: Internal Energy, Work, Heat and Enthalpy 19
PH C T for any process (if PC is constant)
We know P VC C nR
Special cases:
Isothermal Process T is constant 0T ; 0U H
2) Work: extw P dV PV work only
- Constant V 0i fV V w ; Vq q U
- Constant Pext ( )f
i
V
ext ext f iV
w P dV P V V ; Pq q H
Isothermal reversible process: (Reversible process: delicate, see later)
1
extP nRTV
nRT is constant
lnf f
ii
V V
VV
dVw nRT nRT V
V
ln ln lnf
f i
i
VnRT V V nRT
V
lnf
i
Vw nRT
V
3) Heat
Adiabatic process : 0q by definition
U q w ; U w
Adiabatic Reversible Process
0q , U w , ext
nRTP
V
f
i
V
V
nRTU w dV
V
,V m
nRTnC dT dV
V
,V m
dT nRnC dV
T V
Winter 2013 Chem 254: Introductory Thermodynamics
Chapter 2: Internal Energy, Work, Heat and Enthalpy 20
,
f f
V mi i
dT dVnC nR
T V
, ln lnf f
v m
i i
T VnC nR
T V
,
ln lnf fv m
i i
T VC
R T V
For adiabatic reversible process:
,
ln lnf fv m
i i
T VC
R T V
OR
,ln ln
fP m i
i f
TC P
R T P
OR ,
,
ln lnf P m i
i V m f
V C P
V C P
1) ,
ln lnf fv m
i i
T VC
R T V
,
,
ln ln lnV m
R
Cf f f
i V m i i
T V VR
T C V V
,V m
R
Cf f
i i
T V
T V
2) ,
ln lnf fv m
i i
T VC
R T V
,
ln lnf fV m i
i f i
T nRTC P
R T P nRT
lnf i
i f
T P
T P
ln lnf i
i f
T P
T P
,
1 ln lnfV m i
i f
TC P
R T P
,
ln lnfV m i
i f
TC R P
R T P
,ln ln
fP m i
i f
TC P
R T P
Winter 2013 Chem 254: Introductory Thermodynamics
Chapter 2: Internal Energy, Work, Heat and Enthalpy 21
Adiabatic Isobaric Process
Constant external pressure AND 0q
Isothermal Compression
Constant external pressure 0f f iw P V V
0q w (because 0U because isothermal)
What is work in 2-step process?
2 int int inti f fw P V V P V V
2 1w w ; 2 1q q
Winter 2013 Chem 254: Introductory Thermodynamics
Chapter 2: Internal Energy, Work, Heat and Enthalpy 22
Conclusion: w and q depend on details of process, not only on initial and final state.
Repeat for 3 step, 4….
5 4 3 2 1w w w w w ; 5 4 3 2 1q q q q q
The more steps, the less w and less heat
Reversible Process (limiting process)
ext gasP P at each step
ext
nRTP
V
Isothermal Reversible Process
f f
i i
V V
extV V
dVw P dV nRT
V
ln | lnf
i
V f
V
i
VnRT nRT
V
work, q is minimal
Isothermal Expansion
Winter 2013 Chem 254: Introductory Thermodynamics
Chapter 2: Internal Energy, Work, Heat and Enthalpy 23
1 0f f iw P V V ;
1 1 0q w
2 1w w ; 2 1q q
3 2 1w w w ; 3 2 1q q q
More processes more work ( w ), more heat ( q )
5 4 3 2 1w w w w w ; 5 4 3 2 1q q q q q
Limiting Expansion Work exp
l imit l imit
compression ansionw w
limit lnf f
i i
V V f
extV V
i
VdVw P dV nRT nRT
V V
Winter 2013 Chem 254: Introductory Thermodynamics
Chapter 2: Internal Energy, Work, Heat and Enthalpy 24
Grains of sand : I can run process either way
The thermodynamic work is the same both ways for reversible process
Irreversible Process (Big chunks of mass)
Follows arrows in reverse: add mass, piston rises? ; removes mass, piston lowers?
This is absurd, hence:
Why do irreversible processes run in one way and not another?
What is special about irreversible?