2
Thermodynamics : Summary
•
Extensive and intensive variables
: The equilibrium state of a thermodynamic system is char-
acterized by specifying a number of
state variables
which can be either
extensive
(scaling lin-
early with system size), or
intensive
(scaling as the zeroth power of system size). Extensive
quantities include: energy
E
, entropy
S
, particle number
N
, magnetization
M
,
etc.
Inten-
sive quantities include temperature
T
, pressure
p
, number density
n
, magnetic field
H
,
etc.
The ratio of two extensive quantities is intensive,
e.g.
n
=
N/V
. In the
thermodynamic
limit
, all extensive state variables tend to infinity (in whatever units are appropriate), while
their various ratios are all finite. A full description of the state of any thermodynamic sys-
tem must involve at least one extensive variable (but may or may not include intensive
variables).
•
Work
: The internal energy of a thermodynamic system can change as a result of a
gener-
alized displacement
dX
i
, as a result of work
W
done
by
the system. We write the differential
form of
W
as
¯
dW
=
−
summationdisplay
i
y
i
dX
i
−
summationdisplay
a
µ
a
dN
a
,
where
−
y
i
is the
generalized force
conjugate to the generalized displacement
X
i
, and
µ
a
is
the
chemical potential
of species
a
, which is conjugate to the number of particles of that
species,
N
a
.
Think of chemical work as the work required to assemble particles out of
infinitely remote constituents. The slash through the differential symbol indicates that
¯
dW
is an
inexact differential
,
i.e.
there is no function
W
(
T,p,V,...
)
.
•
Heat
: Aside from work done by or on the system, there is another way of changing
the system’s internal energy, which is by transferring
heat
,
Q
. Heat is a form of energy
contained in the random microscopic motions of the constituent particles. Like
¯
dW
, the
differential
¯
dQ
is also inexact, and there is no heat function
Q
(
T,p,V,...
)
. Transfer of heat
under conditions of constant volume or pressure and constant particle number results in a
change of the the thermodynamic state via a change in temperature:
dT
= ¯
dQ/C
, where
C
is the
heat capacity
of the system at fixed volume/pressure and particle number.
•
First Law
: The First Law of Thermodynamics is a statement of energy conservation which
accounts for both types of energies:
Δ
E
=
Q
−
W
, or in differential form
dE
= ¯
dQ
−
¯
dW
.
•
Single component systems
: A single component system is completely specified by three
state variables, which can be taken to be
E
,
V
, and
N
, and writing
¯
dW
=
pdV
−
µdN
, we
have
¯
dQ
=
dE
+
pdV
−
µdN.
If, for example, we want to use variables
(
T,V,N
)
, we write
dE
=
parenleftbigg
∂E
∂T
parenrightbigg
V,N
dT
+
parenleftbigg
∂E
∂V
parenrightbigg
T,N
dV
+
parenleftbigg
∂E
∂N
parenrightbigg
T,V
dN.