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Introduction First Law of Thermodynamics
Thermodynamic Property Measurable quantity characterizing the state
of a system
•
•
•
•
•
•
Temperature
Volume
Pressure
Voltage
Magnetic field strength
Energy, etc.
h
Notable exceptions to this are Work and Heat – It doesn’t make any
sense to say that some object has a certain amount of work or heat. These quantities
can’t always easily be distinguished!
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Introduction First Law of Thermodynamics
Thermodynamic equilibrium
• mechanical equilibrium
• thermal equilibrium
• chemical equilibrium
The state of a system refers to the condition of the system as
described by the thermodynamic parameters characterizing
the system.
Equation of state
A functional relationship among the thermodynamic parameters characterizing
the system.
This reduces the number of independent
f ( p,V , T )  0 parameters characterizing the system from
three to two.
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Mechanical Work Terms
General form
W = pdV; hydrostatic work
= -dZ; electrical work
W = Intensive x d (Extensive)
= -HdM; magnetic work
= -gdA; surface work
= -Fds
Intensive variable - mass independent
Extensive variable- mass dependent
 W = Si Xidxi = pdV - dZ – HdM +…
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Work
If F and s are in the same direction, W > 0.
If F and s are in the opposite direction, W < 0.
Massless, frictionless piston cylinder arrangements
vacuum
F=0
W=0
Free expansion
F = mg
vacuum
W = -Fdz
expansion against mg
Pext
dW = -Fdz
F = pexA
W = - pex(Vf-Vi )
expansion against a constant
atmospheric pressure, Pext
Hydrostatic Work
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Heat
Historically, the concept of heat was difficult to quantify because
it is an energy flow based on microscopic actions rather than
Correlated macroscopic action.
If energy “enters a system” by some correlated motion, i.e., the
movement of a piston, we consider that as work.
If energy enters a system owing to some microscopic transport,
i.e., hot piston uncorrelated motion of molecules, we consider that
as heat.
If, E + *W ≠ 0, the remainder of the energy change
is due to heat. Sign convention
E + W = Q
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Heat Transport
Quite generally all transport processes can be described by an
equation of the form, v = m x F, where v is a generalized velocity
or flux of transport and F is a generalized force.
This works for thermal transport, mass transport, electrical
transport, etc. All transport equations take the following form,
j = -k grad .
In the case of heat transport the heat flux is given by,
jheat = -k grad T or j  T
The minus sign is there because the flux is always in a direction
of decreasing temperature.
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First Law of Thermodynamics
First Law - Conservation of Energy
dE   Q   W
dE
dU
Q
W
d(KE) + d(PE) + dU
change in internal energy
heat added to the system
work done by the system
For example - In a conservative
field the quantity
 
W   F  dr
is path independent, i.e.,
 
W ( B)  W ( A)   dW   F  dr
B
B
A
A
dU is an exact differential whereas
Q and W are not. *What’s and exact
F  W and  F  0
 
Note that   F  0
leads
differential? Is it possible to somehow turn a differential
quantity that isn’t exact in to an exact differential?
directly to Maxwell the
reciprocal relations.
* Q is positive for heat flow in to the system
W is positive for work done by the system.
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Adiabatic Work
p
Adiabatic walls
P1, V1
P2, V2
Quasistatic compression
of an ideal gas.
Vf
U   W     W
p = constant/Vg
Vi
What’s the physical manifestation
of the increase in internal energy?
Vf
Vi
V
Sign convention: The work is taken negative if it increases the energy in the system. If the volume
of the system is decreased work is done on the system, increasing its energy; hence the positive sign
in the equation W  pdV . The unusual convention was established to fit the behavior of heat
engines whose normal operation involves the input of heat and the output of work. So according to
this unfortunate convention work is positive if the engine is doing its job. Note that some textbooks
e.g., Equilibrium Thermodynamics by Adkins will write the first law as dU   Q  W which is
more in line with results in the opposite sign convention for work. In this convention positive work is
work going in to a system raising its energy. So, the manner in which you write the first law
establishes the sign convention that you use.
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Adiabatic Work
Composite system
2
p
If a system is caused to change to an initial
state to a final state by adiabatic processes, the
work done is the same for all adiabatic paths
connecting the two states.
P1, V1
A
C
P2, V2
B
1
V
Different adiabatic paths between two states of a fluid.
• 1A2. An adiabatic compression followed by electrical work at
constant volume performed via a “heater” of negligible thermal
capacity immersed in the fluid.
• 1B2. The same process but in reverse order
• 1C2. A complex route requiring simultaneous electrical and
mechanical work.
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Non adiabatic Work
Heat reservoir at fixed T
Diathermal walls
P1, V1
Heat reservoir at fixed T
P2, V2
Q
Quasistatic isothermal compression
of an ideal gas.
p
T
U  0, W  Q
Q
p = nRT/V
Vf

Vi
pdV
Vf
Vi
V
Heat
The heat flux to a system in any process is
simply the difference in internal energy
between the final and initial states plus
the work done in that process.
Q  U  W
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Simple thermodynamic system
A system that is macroscopically homogeneous, isotropic, and uncharged, that is
large enough so that surface effects can
be neglected, and that is not acted on by
external electric, magnetic or gravitational
fields.
All thermodynamic systems have an enormous number of atomic coordinates. Only
a few of of these survive the statistical
averaging associated with a transition to a
macroscopic description. Certain of these
coordinates are “mechanical” in nature.
Thermodynamics is concerned with
consequences of atomic coordinates that,
by virtue of the coarseness of observation,
do not appear in a macroscopic description
of the system.
Thermodynamic Equilibrium
In all systems there is a tendency to
evolve toward states in which the
properties are determined by intrinsic
factors and not by previously applied
external influences. Such simple
terminal states are time independent
and are called equilibrium states.
Equilibrium states of simple systems
are characterized completely by the
internal energy U, the volume V, and
the mole numbers N1, N2, … of the
chemical components.
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Thermodynamic Parameters
Ideal Gas Temperature Scale
pV/nk
Measurable macroscopic quantities
associated with the system, e.g., p, V,
T, H, etc.
pV = nkT
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divisions
Equation of State
0K
Functional relationship among the
thermodynamic parameters for a system
in equilibrium. If p,V and T are the
parameters,
T
f (p, V, T) = 0,
T
Fp H20
Bp H20
Surface representing the
equation of state.
Equilibrium point
This reduces the number of independent
parameters. Any point lying on this
surface represents a state on equilibrium.
V
p
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Mathematical digression
Suppose that three variables are related as in an equation of state:
F ( x, y, z )  0
This expression can be rearranged to yield any one variable in
Terms of the other two independent variables,
x  x( y, z ).
Differentiating this expression,
 x 
 x 
dx    dy    dz.
 z  y
 y  z
(A)
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Mathematical digression
The terms in brackets are partial derivatives which are defined
In a manor analogous to normal derivatives
x  y  dy, z   x  y, z 
 x 
.
   lim
y

0
dy
 y  z
Also since, z =z (x, y)
 z 
 z 
dz    dx    dy.
 x  y
 y  x
Substituting in equation (A) for dz,
 x   x   z  
 x   z 
dx      dx          dy
 z  y  x  y
 y  z  z  y  y  x 
(B)
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Mathematical digression
This result is true whatever pair of variables we choose as independent.
In expression (B) suppose, dy = 0, then
1
 x   z 
 x 

1
or

   
 
 z  y  x  y
 z  y  z 
 
 x  y
This is the reciprocal theorem which allows us to replace any partial
derivative by the reciprocal of the inverted derivative with the same
variable(s) held constant.
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Mathematical digression
In expression (B) suppose, dx = 0, and dy ≠ 0, then,
 x 
 x   z   y 
 x   z 


or
   
 
       -1
 z  y  y  x
 y  z
 y  z  x  y  z  x
by repeated application of the reciprocity theorem.
Taylor series expansion in 1 variable.
1  d2 f 
2
 df 
f ( x A )  f ( xo )   

x


x
   2     ...
2!  dx  x  x
 dx  x  xo
o
f(xo)
xo
x = xA - xo
x
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Mathematical digression
Taylor series expansion in 2 variables, x = x(y,z).
z
B
2
1
A
z
y
y
Given x at y1, z1, we want to evaluate the value
of the function x at point 2. Proceeding from 1
to A,
 x 
1  2 x 
2
x A  x1    dy   2   dy   ...
2  y  x  x
 y  x  x1
1
And then from point A to 2,
1  2 x 
2
 x 
x2  x A   
dz   2 
dz
   ...
2  z  x  x
 z  x  xA
A
Substitution for xA from the 1st relation,
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Mathematical digression
 x 
1  2 x 
2
 x 
x2  x1   
dy   
dz   2 
 dy 
2  y  z , x  x
 z  y , x  xA
 y  z , x  xA
A
1  2 x 
  x 
2
+  2
dz

dydz  ...
 
 
2  z  y , x  x
z  y  z , x  xA
A
If we had proceed from 1 to B to 2,
 x 
1  2 x 
2
 x 
x2  x1   
dy   
dz   2 
dy
 

y

z
2

y
  y , x  xA
  z , x  xA

 z , x  xA
1  2 x 
  x 
2
+  2
dz

dzdy  ...
 
 
2  z  y , x  x
y  z  y , x  xA
A
Since the result must be the same, the last 2 terms in each of the
Expressions must be equal,
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Mathematical digression
  x 
  x 
2 x
2 x

.
     , or
z  y  z y  z  y
zy yz
(C)
Exact differentials
If x is a function of y and z, we can write an expression for an
infinitesimal change in x owing to infinitesimal changes in y and z,
dx  Ydy  Zdz.
Here Y and Z correspond to the partial derivatives. In principle, dx
can always be integrated since all we need are the initial and final
states.
If a quantity is not an exact differential, W, then in order to integrate
To get the work, we need to know the path.
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Mathematical digression
Since dx is an exact differential,
 Y   Z 

 
 ,
 z  y  y  z
Which just expresses the condition (C) on the previous page.
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Internal Energy
Macroscopic systems have a definite and precise energy energy subject to
a conservation principle. The energy of the universe is the same
today as it was eons ago.
Thermodynamics is concerned with measuring energy differences of a
system owing to external influences. We can easily determine the energy
difference between two equilibrium states of a system by enclosing the
system with adiabatic walls so that energy change can only occur by
doing some form of mechanical work on the system.
For a simple hydrostatic system (pdV work term) the change in internal energy
is describable in terms of any two of the variables p, V and T since the third is
by an equation of state (for an ideal gas, pV = nRT):
 U 
 U 
U  U ( p, T ); dU  
 dp  
 dT
 T  p
 p T
 U 
 U 
U  U (V , T ); dU  
 dV  
 dT

V

T

T

V
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Heat
The heat flux to a system in any process is simply the difference in
internal energy between the final and initial states plus the work done in
that process,
.
Q  U  W
Heat is what is adsorbed by a system if its temperature changes while no
work is done. If Q is the small amount of heat adsorbed in a system causing
a temperature change of dT the ratio Q /dT is called the heat capacity C, of
that system. The specific heat c, is the heat capacity per unit mass. For
example the heat capacity per mole of substance or the molar heat capacity
is defined as;
C 1 Q
.
Specific heat c  
n
n dT
In the case of a hydrostatic system the heat capacity is unique when constraints
of either constant pressure or volume are imposed so that
Q 
 H 
Q 
 U 
Cp  

and
C


V






 .
 dT  p
 T  p
 dT V  T V
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Here the symbol H is a thermodynamic potential or “energy” called the enthalpy
which I will define shortly.
The second of the set of equalities is easily demonstrated just by writing out the
first law,
 Q  dU  pdV ,
and substituting for dU
 U 
 U 
U  U (V , T ); dU  
dV



 dT
 V T
 T V
obtaining

 U 
 U  
Q  
 dT   p  
  dV ,

T

V

V

T 

At constant volume,
Q 
 U 




  CV
 dT V  T V
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If we try to do the same thing for Cp we run in to a problem because of
the form of the internal energy function,
 Q  dU  pdV ,
 U 
 U 
U ( p, T ); dU  
 dp  
 dT
 T  p
 p T
At constant pressure
Q 
 U 
 V 
Cp  

dT

p





 dT  p  T  p
 T  p
You can see that this is messy compared to the simple equation for CV. It is
convenient to define a new function H, such that dH = d(U+pV). Then
dH  dU  pdV  Vdp   Q  Vdp
dU   Q  pdV
Q 
 H 
Introduction of the enthalpy function
Cp  




allows for a simple relation for Cp.
 dT  p
 T  p
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