Thermodynamics
Heat Content and
the First
Law Gases)
(Mostly
of Ideal
Dh
Mechanical Energy: Work and Heat
Vacuum Pump
w   p  Dh
Dh
Ideal Gas: Diluted ensemble of N structure-less,
independent particles with macroscopic, observable
properties described by an Equation of State.
(Sole interactions: Elastic scattering)
When enclosed in a container of volume V, IG exerts a
measurable pressure p on the container walls.
p V  N  R T
extensive variable
 Compression and decompression of a gas volume
(N=const.) change its (kinetic-) energy content U.
U = capacity to perform work w. Charged particles with
structure (atoms,...): additional work types (wel , rxns).
U of gas can be changed by heating or cooling = transfer
of disorganized energy = “heat” q (motion of particles in
container walls).
1. Law of Thermodynamics
(Conservation of Energy in isolated system):
p  Force Area (  pext )
incomplete differentials
dU  dq  dw  dq  p  dV  ...
Dw   F  Dh   p  DV |
DV  0 : Expansion
DV  0 : Compression
real
gases
Sign Convention: Work and heat are counted positive (w, q > 0), when they increase
the internal energy U of the gas, and negative when done or emitted by the gas.
Random (IG) Velocity/Kinetic-Energy Spectra
dP  u 
dP(u)/du (a.u.)
4
dP   
d
Velocity u (km/s)
dP()/d (a.u.)


u dP  u 
2
kBT
 4



e
3
3 /2
m d u
 kBT 

Maxwell 
Boltzmann
Mean thermal energy
 
3
 kBT
2
Plausibility of EOS (Gas Law) p  V  N  kB  T :
N =# of particles colliding with a wall
<u>
 momentum transfer to wall per particle
<u>
2
 pressure on wall p <u>
T (see EOS)
Kinetic Energy (10-20 J)
Transition
Lattice
Real Gas
Ideal Gas
W. Udo Schröder, 2015
Bound Lattice low T
Unbound Gas, T=300K
LatticeFluid/Real Gas
i (t)/kB (i=1,..,1000)
i (t)/kB
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 m 
 mu2 
2
IG particles (mass m) :
 4 
u
exp



du
2

k
T

B

 2kBT 
 dP  u  

8kBT
Mean thermal speed  u   du u 


0
du 
m

Heat Flow
Heat conduction, flux=current density through area A
jq   dQ dt  / A
5
 T
T
T 
Fourier ' s Law : jq     T  r      
i 
j
k

x

y

z


Thermal conductivity  (W mK )
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Heat convection: Heat transfer via mass flow
dQ
 h  A  T  Tambient 
dt
Heat transfer coefficient h (W m2K )
Newton ' s Law of cooling
Heat radiation: Heat transfer via elm. photons
Stefan  Boltzmann Law

4
Radiated thermal flux jQ     SB  T 4  Tambient

Emissivity  (often  1)
Stefan  Boltzmann constant  SB  5.6703  108 W m2K 4
W. Udo Schröder, 2015
The Ideal-Gas Equation of State
p·V = n·R·T;
T U
n=# moles; equivalent: p·V = N·kB·T (N=# particles)
Interacting only via elastic scattering, no bonding  Only gas phase!
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p(V,T)= n R T/V
{p, V, T}
Ideal Gas Constant R
R = 0.0821 liter·atm/mol·K
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A
R = 8.3145 J/mol·K
R = 8.2057 m3·atm/mol·K
R = 62.3637 L·Torr/mol·K or
L·mmHg/mol·
Boltzmann Constant kB
kB= 1.381·10-23 J/K
State “functions:” {p, V, T} (N=const.). Molar {p, V, T} hyper-plane (monotonic)
contains all possible gas states A. Set of “independent coordinates.” There are no other
states of the gas.  Other state functions can be expressed in {p, V, T}.
W. Udo Schröder, 2015
Reversible Processes
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7
T
Of interest for cyclic
machines.
Slow, equilibrium
processes A  B,
subject to boundary
conditions of:
1. Dp = 0
(isobaric)
2. DV = 0
(isochoric)
3. DT = 0 (isothermal)
q=0
4.
q=0
(adiabatic)
follow well-defined,
constrained routes in the
{p, V, T} hyper-plane of
states. Can easily be
inverted reversible
processes.
Reversibility is not guaranteed for all processes involving an ideal gas.
W. Udo Schröder, 2015
Transitions Between States
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Process 1
A B
along Path 1
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A
Two different states
B
(A and B)
of the same gas.
Process 2
A B
along Path 2
State functions p, V, T,… describe the system states but not the processes connecting
states. Two states A, B can be reached by different processes representing different
1
2
 B and A 
B
pathways on the {p,V,T} hyperplane. The two processes A 
AB differ by different relative amounts of energy transfer via absorption of heat and work.
W. Udo Schröder, 2015
Reversible Isothermal Expansion/Compression
w = - area under curve p(V)
Total work (1 2):
Use p  V  R  T for expanding 1 mole
2
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w    p (V ) dV   R  T 
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1
2
1
dV

V
V 
 R  T  ln  1   0
 V2 
w  0 implies system does work
on surroundings
But DU  DT  0  q  0 (absorbs heat )
Intersection of {p,V,T} hyperplane with plane T=const.
W. Udo Schröder, 2015
1. Law of Thermodynamics :
V 
 q  DU  w   w   R  T  ln  1   0
0
 V2 
Reversible Isobaric Compression
Compress 1 mole at p=const.
Work done on system :
2
2
w    p (V )dV   p   dV  0
1
1
10
  p  DV   R  DT > 0 Shaded Area
EOS
Heat transfer
DT  0: system also cools by emitting
5
p  DV
5
R
 w
2
R
2
Total
heat(transfer
(1 2)
Enthalpy
change
for p  const
.) :
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q  C p  DT 
DH  C p  DT  C p  [T2  T1 ]  q  0
DV
 emitted heat (internal energy )
DU  q  w  (C p  R)  DT ( DH )
Internal energy change
 CV  [T2  T1 ]  0
Inverse process: heating at constant p,
e.g., p=patm , leads to
expansion, V2  V1>V2  drives piston out of its cylinder.
W. Udo Schröder, 2015
Reversible Decompression
Isochoric (V = const.) decompression
 of 1 mole w =-pDV=0
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q  CV  DT  0
Work done on system
w0
Heat transfer
But DU  0,  system emits heat
q  CV  DT  CV  [T2  T1 ]
Total energy change (1 2)
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1. Law of Thermodynamics :
DU  q  w  q  CV  [T2  T1 ]  0
Enthalpy change
DH  DU  D ( pV )  (CV  R)  DT
 C p  [T2  T1 ] (always DH  C p  DT )
BUT : DH  q ( since p  const )
Inverse process: heating at constant V, leads to increased temperature
and pressure.
W. Udo Schröder, 2015
Expansion-Compression Cycles
Observation: IG systems absorbing external (random) heat can produce mechanical work
on surroundings (engine). Continuous operation requires cyclic process (in p-V-T space).
1
work dw=-p·dV
2) Isochoric decompression at V2=const.,
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1) Isothermal expansion at T1=const.
3) Isothermal compression at T2 =const.
4) Isochoric compression V1=const.,
T1
4
2
T2
V2
In one cycle the gas absorbs net heat
energy and does net work,
w = w1 + w2 = -q = CV∙[T2-T1]
Not all absorbed heat is converted,
some has to be dumped as waste heat.
W. Udo Schröder, 2015
1) gas does work
2) gas emits heat
3
V1
Energy balance:
w1 = - q1; DU = 0
q < 0;
DU < 0
3) gas receives work
w 2 = - q2 ;
DU = 0
4) gas absorbs heat
q > 0;
DU > 0
Total energy change:
Total work done:
Total heat absorbed:
DU = 0 (cyclic)
w = w1 + w2 < 0
q = q1+ q2=-w > 0
Thermal Engines: Principle of Operation
p
Net work
done by gas
T1
13
q1
Make an arbitrary cyclic process out of
elementary isothermal and isochoric
processes 
Heat energy q1 is absorbed at a high
temperature(s) T1, and partially
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dumped, |q2| < |q1|, at a lower
q2
T2
V
Horizontal paths traveled in both
directions do not contribute net work
 Area within closed p-V paths =
total work done in cyclic process.
temperature(s) T2.
The difference (q1 + q2)= q1-|q2| is
converted into useful work w < 0 done
on surroundings by the gas.
Random heat energy is converted into orderly collective energy
(work, pushing a piston, turning a wheel) !!!!!!!  Practical use
W. Udo Schröder, 2015
Work/Heat in Reversible vs. Irreversible Processes
A
rev
14
B
irrev
pext
System interacts with environment, is not isolated
(DT=0).
pext
In process A  B, carried out so that system is
always at equilibrium (e.g, pext dV= pgasdV+q),
system produces maximum work.
(balance by including the –sign, sign convention!):
pext
A’
X
wrev < wirrev  |wrev |> |wirrev|
pgas
A
A’
irrev
decompresion
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Opening valve t0  Expansion
Where did the difference Dw go ? Nowhere!
Also less/no heat absorbed on irreversible path.
1. Law TD, and since U is a state function,
If DUAB = DUA’B  (q+w)rev = (q+w)irrev
B’
pext
B
V

wrev < wirrev 
Wrev is largest amount deliverable (negative) reversibly 
qrev is largest amount of heat system can absorb reversibly
and convert into work.
Irreversible, spontaneous processes:
Less efficient conversion of absorbed heat into useful work.
W. Udo Schröder, 2015
qrev > qirrev
Entropy: The Arrow of Time
A small volume Vin containing randomly moving (thermal) particles “expands,” or a cluster
of many particles disintegrates, particles diffuse into and occupy all available states W.
Vin

T
j, i 
time 
S ≈ Smax 
Spontaneously from simple
to ever more complex.
Essentially irreversible.
Vfin
T
 S ≈ Smax
2. Law of TD : # of occupied states (W) never decreases
 S : k B  n W   k B   pij n pij increases (" Entropy ")
i, j
W
In principle, all transitions are individually reversible.
But: In the aggregate, it is unlikely that all particles move back to
exactly the same positions (states) from which they came. Probability
for complete reversal is non-zero but very small.
Poincaré Recurrence Time: trec ~ W  
Number (W ) of occupied
Only equilibrium (random) states are equivalent,
states increases in time.
connected by reversible processes.
Carnot Cycles
p
Adiabatic (q  0) EoS
(p1, V1)
T V
16
q=0
 1
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(p2, V2)
work
q=0
T2
DS1 
(p3, V3)
q1
q
  2  DS2
T1
T2
Entropy is conserved in reversible
cyclic processes : DS1  DS2  0.
 S  state function (descriptor )
q1  T1 DS1
q2  T2 DS2
q AB
T
 sign for reversible A  B only.
For any process : DS AB 
W. Udo Schröder, 2015
  c p cV
T1
(p4, V4)
" Entropy "
 const;
V
Similar to previous examples:
Isothermal expansion at Th=T1
Adiabatic expansion Th Tc=T2
Isothermal compression at Tc <Th
Adiabatic compression Tc Th,
Adiabatic works cancel.
Adiab. expansion/compr. 
V4/V1= V3/V2  V4/V3= V1/V2
Energy balance: w = q1 + q2 > 0
on isothermal portions:
q1   w1  
V2
V1
V 
p dV  R  T1  ln  2   0
 V1 
1 V
V4
V 
q2   w2   pdV  R  T2  ln  4   0
V3
 V3 
w0
Reversible adiabatic exp./compr.: DS = q/T= 0
since q= 0.
Irreversible adiabatic exp./compr.: DS > 0.
Efficiency of Carnot Engines
Theoretical Carnot efficiency
Th
C 
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qh= DS·Th
DS = const
C  1 
-w = qh+qc=DS·(Th- Tc)
qc= -DS·Tc
Tc
 w qh  qc

qh
qh
qc
T
 1  c 
1
Th 
qh
Th
U  const
Process at p, T  const. 

 q  DH 
T T 
w   Th  Tc   DS    h c  Th  DS
 Th 
T T 
   h c  DH Th   D  H  T  S 
 Th 
DG
In practice, Th depends on fuel heating value (max temperature Tad).
Transfer from fuel to hot reservoir:  F  Tad  Th  Tad  Tc 
 Effective Carnot efficiency:
W. Udo Schröder, 2015

   C   F  1 

Tc
Th
  Tad  Th 


T

T
  ad c 
Entropy Flow in Carnot Engines
p
(p1, V1)
Th
18
q=0
qh= DS·Th
Th
(p2, V2)
work
(p4, V4)
q=0
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Tc
(p3, V3)
Hydrodynamic Power Plant
Inlet
Reservoir
Turbine
-w = qh+qc=
= DS·(Th -Tc)
V
DM∙g∙h1
DM∙g∙h2
qc= -DS·Tc
Tc
Entropy DS from the hot reservoir
enters the engine with a heat energy
of DS·Th,
produces work and leaves it again
with a heat energy of DS·Tc,
which is dumped into the cold sink.
Outlet
Analog: Stream of water DM from a reservoir carries energy DM∙g∙h1 , enters a hydroturbine, produces work, and leaves with an energy DM∙g∙h2 , which is dumped into the
river.
Mass flow j  dM dt . Entropy flow j  dS dt
M
W. Udo Schröder, 2015
S
Steady-Flow Processes
1. Law of Thermodynamics
E U 
19
(Conservation of total energy in isolated system):
M= mass, = velocity, Vpot =potential energy (often ≈0)
Mass density r = rm (kg/m3), homogeneous
Internal energy density u (J/m3)
Enthalpy density h = u + p
Internal energy
Kinetic energy
dUi  ui  (A i  dx i )
dK i  1 2  ri  i2  (A i  dx i )
1
M  2  V pot
2
uo , ro , o
dVo
dVi  A i  dx i
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Mechanical work dWi  pi  (A i  dx i )
Carried by mass flow through system
incoming
outgoing
1
rii2  (A i dxi )
2
1
dUo  uo  (A o dxo )  po  (A o dxo )  r0o2  (A o dxo )
2
dVi
ui , ri , i
dUi  ui  (A i dx i )  pi  (A i dx i ) 
Mass conservation :
dMi   r A  i   r A  o  dMo
Additional mechanical work output dW , heat output dQ
dE d Uin  Uout   dW dQ 
dE



Steady state :
0

dt
dt
dt 
dt
 dt
dW dQ dW 
1
1





  hi  rii2  A i i   ho  roo2  A o o
dt
dt
dt
2
2




well insulated
W. Udo Schröder, 2015
Power
Fuel Flow
2
dW  hi i2   ho o   dM
  



 
dt
2   ro
2   dt
 ri
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20
Spontaneous Reactions Require Free Energy
G
W. Udo Schröder, 2015
W. Udo Schröder, 2015
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