PY2P/T10: Thermodynamics Prof. Graham Cross

PY2P/T10: Thermodynamics
Prof. Graham Cross
Graham.Cross@tcd.ie
Web page:
8.10.2014
www.tcd.ie/physics/people/Graham.Cross
PY2P10 Thermodynamics Lect. 2
2
PY2P/T10: Thermodynamics – Lecture 1
1
Temperature. Zeroeth Law of Thermodynamics
Wed. Oct. 1
2
Reversible and irreversible processes. Types of work.
Wed. Oct. 8
3
Internal Energy, heat. First law of thermodynamics.
Wed. Oct. 15
4
Specific heat.
Wed. Oct. 22
5
Heat engines, Carnot cycles, Joule-Kelvin effect.
Wed. Oct. 29
6
Second law of thermodynamics. Thermodynamic (absolute)
temperature and entropy.
7
More discussion of the 2nd Law of Thermodynamics. Combined
first and second laws: Central equation.
8
Thermodynamic potentials U, H, F, G and Maxwell's relations.
9
Energy equation and applications of Maxwell relations.
10
Application of thermodynamic potentials.
11
Phase changes.
12
Magnetic systems and the Third Law of Thermodynamics
13
Tutorial
14
Exam
8.10.2014
End of academic year
PY2P10 Thermodynamics Lect. 2
3
Review
• Systems, surroundings, boundaries
• Equilibrium state
• Thermal equilibrium, mechanical equilibrium, thermodynamic equilibrium
• State variables: Eg. gas in piston: (P, V, T), stretched elastic band (F, L, T)
• Zeroth Law of Thermodynamics
• Temperature
Surroundings
Gas system
Moving piston
Wall
8.10.2014
PY2P10 Thermodynamics Lect. 2
4
PY2P/T10: Thermodynamics – Lecture 2
1
Temperature. Zeroeth Law of Thermodynamics
Wed. Oct. 1
2
Reversible and irreversible processes. Types of work.
Wed. Oct. 8
3
Internal Energy, heat. First law of thermodynamics.
Wed. Oct. 15
4
Specific heat.
Wed. Oct. 22
5
Heat engines, Carnot cycles, Joule-Kelvin effect.
Wed. Oct. 29
6
Second law of thermodynamics. Thermodynamic (absolute)
temperature and entropy.
7
More discussion of the 2nd Law of Thermodynamics. Combined
first and second laws: Central equation.
8
Thermodynamic potentials U, H, F, G and Maxwell's relations.
9
Energy equation and applications of Maxwell relations.
10
Application of thermodynamic potentials.
11
Phase changes.
12
Magnetic systems and the Third Law of Thermodynamics
13
Tutorial
14
Exam
8.10.2014
End of academic year
PY2P10 Thermodynamics Lect. 2
5
Processes
Some more important definitions:
Thermodynamics is concerned with the changes in the state functions of a
system as it changes from one equilibrium state to another.
A process is the means or mechanism of making this change occur, in our
course this is includes various types of mechanical and thermal interactions
The end points of the process are the initial and final equilibrium states.
(P1,V1)
(P2,V2)
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PY2P10 Thermodynamics Lect. 2
6
Reversible and Quasistatic Processes
A reversible process, eg. from state (P1,V1) to (P2,V2), must have two features:
1. It is possible to return the system to the original state (P1,V1).
2. When returning to the original state, the surroundings are left unchanged as
well.
How do we realize a reversible process?
Requires a quasistatic process, which is one such that the system only departs
from thermodynamic equilibrium by an infinitesimal amount at any instant.
• The thermodynamic state of the system is changed by an interaction of
infinitesimal magnitude at each step, such that the system is never far from
equilibrium.
Then, reversible processes are quasistatic processes where, in addition, no
dissipative forces such as friction are present.
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PY2P10 Thermodynamics Lect. 2
7
Reversible Process in an Ideal Gas
Example: A mechanical interaction with an ideal gas system with frictionless piston
(P1,V1)
F1  P1 A
Initial system state (P1,V1)
A is area of piston face
F is force
Let the walls of the piston now be diathermal and the
surroundings be a thermal reservoir
Here, the surrounding temperature never changes
P
Now increase the force F
infinitesimally, allow equilibrium to
be re-established.
(P2,V2)
F  F1  d F  ( P1  dP) A
(P2,V2)
Continue this until a new final
state (P2,V2)
(P1,V1)
PV=nRT always
8.10.2014
PY2P10 Thermodynamics Lect. 2
V
8
Irreversible Process in a Gas
(P1,V1)
P
(P2,V2)
Sudden compression…
Finite temperature and
pressure gradients
(P2,V2)
After some time
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(P1,V1)
V
Not in equilibrium: not uniform or steady!
PV=nRT equation of state is not valid
during this process, only at end points.
PY2P10 Thermodynamics Lect. 2
9
Multivariate Calculus: Partial Derivatives
Derivative for a single variable function:
y  f ( x) dy  lim f ( x  x)  f ( x)
dx x0
x
z  f ( x, y)
Partial derivative for a multi-variable function:
z  f ( x, y)
z
f ( x  x, y )  f ( x, y)
 lim
x x0
x
z  f 
 
x  x  y
8.10.2014
z
Holding y constant!
PY2P10 Thermodynamics Lect. 2
y
x
10
Multivariate Calculus: Implicit Function
An example of a multivariate function with an implicit definition:
Only 2 of x, y, z are
independent, eg. :
F ( x, y, z )  0
x  x( y, z ) z  z ( x , y )
z  z ( x, y)
Lets write the total differentials:
 x 
 x 
dx    dy    dz
 z  y
 y  z
 y 
 y 
dy    dx    dz
 x  z
 z  x
 x   y   x  
 x   y 
dx      dx          dz
 y  z  z  x  z  y 
 y  z  x  z
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PY2P10 Thermodynamics Lect. 2
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Multivariate Calculus: Reciprocal Relation
Let x and z be the independent variables in
F ( x, y, z )  0
 x   y   x  
 x   y 
dx      dx          dz
 y  z  z  x  z  y 
 y  z  x  z
Independence of x and z means that even for dz = 0, dx remains arbitrary
 x   y 
dx      dx
 y  z  x  z
 x   y 
1    
 y  z  x  z
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1
 x   y 
   
 y  z  x  z
PY2P10 Thermodynamics Lect. 2
Reciprocal Relation
12
Multivariate Calculus: Cyclical Relation
Let x and z be the independent variables in
F ( x, y, z )  0
 x   y   x  
 x   y 
dx      dx          dz
 y  z  z  x  z  y 
 y  z  x  z
Independence of x and z means that for dx = 0, dz remains arbitrary
 x   y   x  
0          dz
 y  z  z  x  z  y 
 x   y   x 
0       
 y  z  z  x  z  y
 x   x   y 
      
 z  y  y  z  z  x
 x   y   z 
1       
 y  z  z  x  x  y
 x   z   y 
1       
 y  z  x  y  z  x
Cyclical Relation
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PY2P10 Thermodynamics Lect. 2
13
Multivariate Calculus: Exact Differential
Consider a function f of x and y:
f  f ( x, y)
 f 
 f 
df    dx    dy
 x  y
 y  x
f  f ( x2 , y2 )  f ( x1 , y1 )  
x2 , y2
x1 , y1
This is an exact differential
df
For a finite change in f, f
depends only on initial points
(x1,y1) and final points (x2,y2)
If you can vary the (x,y) path arbitrarily, the integral is path independent.
For the integral to be path independent, the integrand must be an exact differential
8.10.2014
PY2P10 Thermodynamics Lect. 2
15
Multivariate Calculus: Exact Differential
For any differential of the form:
df  X ( x, y)dx  Y ( x, y)dy
If
 X ( x, y )   Y ( x, y ) 

 


y

x
y

x 
Then df is an exact differential.
df  2 xy 4 dx  4 x 2 y3dy
For example, consider
 
4
3
2
xy

8
xy



y

x
and
 
2 3
3
4
x
y

8
xy


 x
y
Thermodynamic state variables and state functions of a system are exact
See Appendix B of Finn
8.10.2014
PY2P10 Thermodynamics Lect. 2
16
Bulk Modulus and Expansivity
We define some new state variables here, which are other parameters
of a thermodynamic system that can be directly measured in experiment
Spatial thermal parameters:

1  V 


V  T  P
1  L 
  
L  T F
Volume thermal expansivity
(Expansivity)
  3
One dimensional analogue
Linear expansion coefficient
Elastic moduli:
1
 P 
K  V 
 
 V T 
Y
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L  F 


Area  L T
Bulk modulus – 3D “breathing mode”
Inverse is , the compressivity
Young’s modulus
1D: Uniaxial tension or compression
PY2P10 Thermodynamics Lect. 2
17
The Thermodynamic Method
Tension F of wire rigidly clamped between supports at temperature T
F1
T1
Assume some equation of state of the form:
F1
g (F , L, T )  0
F = F ( L, T )
Q: What is the increase in tension F with cooling?
F2
T2
L
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Consider a Reversible Process:
Go from equilibrium state (F1,T1) to (F2, T2),
while holding L constant
F2
In general, at any point in the reversible process:
 F
dF  
 T
 F
dF  
 T

 F 
 dT  
 dL
L
 L T

Because of constant
 dT
length L
L
PY2P10 Thermodynamics Lect. 2
18
The Thermodynamic Method
Change of tension of wire
clamped between supports:
F1
T1
F1
F2  F1  F =
1  L 
  
L  T F
F ( x, y, z )  0
F2
T2
L
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F2

T2
T1
 F 

 dT
 T  L
L  F 
Y


Area  L T
 x   z   y 
1       
 y  z  x  y  z  x
 F   L   T 
1  
 
 

 T  L  F T  L F
 F 

  L
 T  L 

F
PY2P10 Thermodynamics Lect. 2 
1
  T 
 


L
T 
F
19
The Thermodynamic Method
Tension of wire clamped between supports
F1
T1
F2
T2
L
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F1
F2
1
 x   y 
   
 y  z  x  z
1
 F 

  L
  T 
 T  L 

 

 F T  L F
 F 
 F   L 





 


T

L

T

L

T 
F
Area  F 
 L 
Y

L   

L
 L T
 T F
 F 

  Y Area 
 T  L
PY2P10 Thermodynamics Lect. 2
20
The Thermodynamic Method
Tension of wire clamped between supports
F1
T1
F1
 F 

  Y Area 
 T  L
Change in tension:
F =
F2
T2
F2

T2
T1
Y Area  dT
If we assume Y, Area, and  are
approximately constant over the
temperature change T1 to T2:
T2
 Y Area   dT
T1
L
 Y Area  (T2  T1 )
(reversible)
Tension increases with cooling...
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PY2P10 Thermodynamics Lect. 2
21
Work
We will first introduce work in the case of a
specific reversible process: The quasistatic
expansion of our gas-piston system.
Let the process proceed by infinitesimal steps
of reduction of the force F, such that our gas
system never departs appreciably from
equilibrium: “Reversible expansion”
dW  F dx  P Area dx  PdV
V2
W   PdV
F
P Area
dx
F  P Area
(reversible)
(reversible)
V1
• This is also the maximum work that can be obtained from a gas expansion.
• More rapid expansion will lead to faster drops in F for a given x...
Energy goes elsewhere than just work on the piston!!
8.10.2014
PY2P10 Thermodynamics Lect. 2
23
Work
Use indicator diagrams to discuss gas
expansion, with an infinite number of
paths connecting (P1,V1) to (P2,V2).
Consider isothermal, reversible
expansion of an ideal gas, when the
gas-piston system is in thermal contact
with a thermal reservoir
nRT
P
V2
W   PdV  nRT V
V2
V1
1
Ideal gas
(P1,V1)
(P2,V2)
V
1
dV
V
 V2 
 nrT ln  
 V1 
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P
V
Work of isothermal reversible process
is not equivalent to reversible
isochoric, isobaric process.
PY2P10 Thermodynamics Lect. 2
24
Some Conclusions about Work
Work is a path dependent quantity, and is not given by the difference in end
point values of a system state function*. This fact is unaffected by reversibility.
This is unlike volume or (as we saw) tension F in a wire, which are state
variables of a state function. V = V2-V1 etc. for going from state 1 to 2 in any
manner.
Work is NOT a thermodynamic state variable!!
For infinitesimal values of a quantity like work, we write đW, not dW, to indicate
that the differential is in general not exact.
Our sign convention: When system does work on surroundings, work is NEGATIVE
Eg. For reversible gas expansion:
đW = -PdV
(Engineers usually do the opposite: what can the system do…?)
*except for adiabatic processes.
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PY2P10 Thermodynamics Lect. 2
25
Example: General Reversible
Process Work Calculation
Consider reversible compression of a non-ideal gas,
from initial state (P1,T1) to (P2,T2).
Procedure: Write the equation of state in a form that gives the state function
whose change we wish to find, in terms of the other two state functions whose
changes are given.
Won’t use specific (eg. ideal) gas law here, we will use
V  V ( P, T )
an abstract form for the state equation
 V 
 V 
dV  
dP



 dT
 P T
 T  P
V
dV   dP  VdT
K
Using:
Since state variable V is exact
 P    1  V 
K  V 

V  T  P

V

T
Bulk Modulus
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PY2P10 Thermodynamics Lect. 2
Expansivity
26
Another Reversible Process Calculation
V
đW  P dP  PVdT
K
P2
T2
V
Total work done going from state 1 to 2: W 
P1 P K dP  T1 PVdT
đW = -PdV
For some simple processes, we can integrate. Eg. Isothermal
W 
P2
P1
( P22  P12 )
V
P dP  V
K
2K
dT  0 :
Provided V and K stay reasonably
constant during the process of
going from P1 to P2.
For more rigorous justification , see second mean value theorem of calculus:

x2
x1
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x2
f ( x) ( x)dx   ( )  f ( x)dx
True for some z where x1    x2
x1
PY2P10 Thermodynamics Lect. 2
27