50
Heat engines and the second law
C D: The gas is expanded to a volume VD , at a constant temperature T1 . The
heat absorbed from the hot reservoir is Q1 = nRT1 ln(VD /VC ).
D A: The gas is expanded adiabatically to the larger volume VA . During this
tra
4.3
Reversible changes of temperature
57
states A and B. The rst, , is reversible, while the second, , is irreversible. For
the (irreversible) cyclic transformation + (), the Clausius theorem implies
0
dQ
=
T
dQ
+
T
d Qrev
,
T
B
A
dQ
,
T
(4.5)
where the
6.7
Statistical mechanics of paramagnetism
85
We shall now put this on a more quantitative basis. Imagine that a paramagnetic sample undergoes adiabatic demagnetization, during which its total magnetic
moment M goes from MA to zero. We want to calculate t
64
A
C
B
Entropy and the third law
value, and it then stays constant. Now, because the free expansion of a gas is an
irreversible transformation, we know that the systems entropy also increases as the
gas evolves toward equilibrium, and that it also stays
5.2
Enthalpy and the free energies
71
This is an example of a Legendre transformation, by which the original function
is shifted by a quantity that mixes the fundamental variables (x and y) with the
derived quantities (a and b). On what variables does the
78
Thermodynamic potentials
Chapter 6
Thermodynamics of
magnetic systems
6.1
Thermodynamic variables and equation of
state
In this last section of the course we will turn our attention to the thermal properties
of magnetic systems, and show how the genera
36
Transformations and the rst law
Chapter 3
Heat engines and the
second law
3.1
Conversion of heat into work
The eld of thermodynamics emerged in the nineteenth century as an eort to
perfect the design of machines that could convert the heat generated by
3.4
The refrigerator
43
P B
Q
T
hot
C
Q abs
Q rej
Tcold
Figure 3.3: The reversed Stirling cycle.
A
Q
D
V
44
Heat engines and the second law
D A: The gas temperature is raised at constant volume. The gas regains the
quantity Q of heat that was lost in the
Chapter 1
Thermodynamic systems and
the zeroth law
Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work,
died similarly in 1933. Now it is our turn to study statis
8
Thermodynamic systems and the zeroth law
1.7
Equation of state
The all-important relation
g(P, V ) =
P
3
2
1
V
is called the equation of state of the thermodynamic system. It states that at
equilibrium, the systems pressure, volume, and empirical tempe
1.11
Problems
15
a) Express a and b in terms of Vc and Tc .
b) Derive the relation
3
Pc V c
= .
nRTc
8
Test this prediction against the critical constants of carbon monoxide,
ethylene, and water. Is the prediction accurate?
c) Rewrite the van der Waals eq
2.7
Some formal manipulations
29
2.7.2 Heat capacity at constant pressure
A similar statement can be made about CP , the heat capacity at constant pressure.
Suppose that we now express U as a function of P and T , and that we use the
equation of state to
22
Transformations and the rst law
2.3.6 Integration of dierentials
This conclusion ceases to be true if the quantity being integrated is dG, a dierential.
In this case, the following theorem holds:
y
Q
Q
P
P
x
dG = G(Q) G(P ),
along any curve joining P a