The fundamental theorem of calculus: f = f(x)
x2
x2
 df 
x  dx  dx  x df  f  x2   f  x1 
1
1
In 1-D only one way to integrate this.
In 2-D infinite paths possible
y
z = z(x,y)
A
C
a
b
x
For an exact differential
b
a
b
b
a
b
a
a
 dz   dz   dz   dz   dz  0
C
Green’s theorem
y
A
C
a
b
x
 N M 


Mdx

Ndy


dxdy
C
A  x y 
In general, the differential dz = Mdx + Ndy is ‘exact’ if:
M N
1.

y
x
2. dz  0
C
b
3. dz
Is path independent
a
e.g. state variables, P = P(V,T)
Inexact differentials do not satisfy these conditions
e.g. dz = ydx - xdy
Sum of two inexact differentials can be exact.
Heat: Spontaneous flow of energy caused by
the temperature difference between two
objects
Work: Any other kind of energy transfer
Internal Energy: Total energy in a system
First law of thermodynamics (in words)
Increase in internal energy = Heat added + work
done on the system
U = Q + W
(W is the work done ON the system)
First law of thermodynamics (in words)
Increase in internal energy = Heat added + work
done on the system
U = Q + W
(W is the work done ON the system)
Phlogiston?
First law of thermodynamics
dQ = dU - dW
Show PdV
Configuration Work
This is the work done in a reversible process given by the product
of some intensive variable (y) and the change in some extensive
variable (X). The most general case would be:
đW 
 y dX ,
i
i
i  1,2,....n.
i
•đW is called the configuration work; it is an inexact differential,
i.e. work is not a state variable.
•The amount of work done changing the configuration of a system
from one state to another depends on how the work is done, i.e.
on the path taken between the final and initial states. The path
must be specified in order to calculate work via integration.
What do we mean by reversible?
Why is it an inexact differential?
Configuration Work
This is the work done in a reversible process given by the product
of some intensive variable (y) and the change in some extensive
variable (X). The most general case would be:
đW 
 y dX ,
i
i
i
i  1,2,....n.
Dissipative Work
• This is the work done in an irreversible process; it is always done
‘on the system’.
• Total work is the algebraic sum any configuration work and any
dissipative work.
• If a process is reversible, then dissipation is necessarily zero.
Examples: Stirring
Resistive electrical heating
Frictional work
Plastic deformation
Many chemical reactions
Microwave oven
Do not confuse between exact
differentials and reversible
processes.
Reversible processes need not be
exact differentials.
Why is it called reversible then?