Manual Solution Fermi Thermodynamics

Additional Notes and Solution Manual For:
Thermodynamics
by Enrico Fermi
John L. Weatherwax∗
August 1, 2014
Introduction
Here you’ll find some notes that I wrote up as I worked through this excellent book. I’ve
worked hard to make these notes as good as I can, but I have no illusions that they are perfect.
If you feel that that there is a better way to accomplish or explain an exercise or derivation
presented in these notes; or that one or more of the explanations is unclear, incomplete,
or misleading, please tell me. If you find an error of any kind – technical, grammatical,
typographical, whatever – please tell me that, too. I’ll gladly add to the acknowledgments
in later printings the name of the first person to bring each problem to my attention.
Acknowledgments
Special thanks to (most recent comments are listed first): Neill Warrington, Eduardo Becerra,
Giacomo Caria, Rosario Oliva, Juanico Loran, Glenn Musano, and Don Rintala for helping
improve these notes and solutions.
All comments (no matter how small) are much appreciated. In fact, if you find these notes
useful I would appreciate a contribution in the form of a solution to a problem that is not yet
worked in these notes. Sort of a “take a penny, leave a penny” type of approach. Remember:
pay it forward.
∗
wax@alum.mit.edu
1

Addendums/Clarifications/Derivations
Constraints on Motion in the p-V plane (Page 6)
Note that in general, there is no conditions or requirements on the motion of a piston in
connection with the gas to which it is attached. What this means is that without any other
information the path in (V, p) can be taken arbitrary and simply represents the projection
of points in (V, p, T) space onto the (V, p) plane. The only requirement is that at every
(V, p) point the gas is assumed to be in equilibrium and must satisfy its equation of state
f(p, V, T) = 0. In this way the temperature T maybe computed if desired.
If we are told more information about the specific type of path in the (V, p) plane (like in fact
that it is along an isentrope or an isotherm) then the path in the (V, p) plane is completely
specified by its two endpoints.
Equivalence of the Pressure and Volume Formulation of the Work
done during an Isothermal Expansion (Page 9)
Now p1 and p2 are the initial and final pressures for m grams of an ideal gas under an
isothermal expansion. As such evaluating the ideal gas law at the two end points gives
p1V1 =
m
M
RT (1)
p2V2 =
m
M
RT (2)
where T the common temperature. Dividing these two expressions we have
p1V1
p2V2
= 1 (3)
or separating volume and pressure to alternate sides of the equation give
V1
V2
=
p2
p1
. (4)
It is because of this relationship that we have the equivalence of the two work expressions
given in Fermi Eq. 10 from the book.
The Derivation of the change in Heat Q in terms of the Variables
T and p (Page 20)
The first law of thermodynamic in terms of its “canonical” variables U and V is given by
dU + pdV = dQ .

To express this in terms of the independent variables T and p we first express the differential
of internal energy U in terms of these variables (using standard calculus) as
dU =
∂U
∂p T
dp +
∂U
∂T p
dT . (5)
In the same way, we next express the differential of V in terms of the variables T and p as
dV =
∂V
∂p T
dp +
∂V
∂T p
dT . (6)
These two expressions are then inserted into the left hand side of the first law of thermody-
namics (replacing dU and dV ) giving
∂U
∂p T
dp +
∂U
∂T p
dT + p
∂V
∂p T
dp +
∂V
∂T p
dT = dQ . (7)
Now grouping dp and dT terms we obtain
∂U
∂p T
+ p
∂V
∂p T
dp +
∂U
∂T p
+ p
∂V
∂T p
dT = dQ . (8)
Which is Fermi Eq. 23.
Expressions Relating the Change in Q to the Specific Heats Cp and
CV (Page 24)
Now Fermi Eq. 30 is
CV dT + pdV = dQ , (9)
and taking the differential of the ideal gas law gives (Fermi Eq. 31) or
pdV + V dp = RdT . (10)
When Eq. 9 is subtracted from Eq. 10 we obtain
CV dT − V dp = dQ − RdT , (11)
or grouping all coefficients of dT we obtain
(CV + R)dT − V dp = dQ . (12)
This is Fermi Eq. 32. Now a transformation at constant pressure has dp = 0 (by definition)
and remembering the definition of Cp
Cp ≡
dQ
dT p
,

we obtain from Eq. 12
Cp =
dQ
dT p
= CV + R . (13)
Which is Fermi Eq. 33. As described in the text this same expression can be obtained for a
ideal gas in the following way. Since for an ideal gas the internal energy U is only a function
of only T the partial derivative holding p constant becomes a total derivative
∂U
∂T p
=
dU
dT
= CV . (14)
Because we know that the general relationship between Cp and CV is given by
Cp =
dQ
dT p
=
dU
dT p
+ p
dV
dT p
. (15)
Which for an ideal gas and Eq. 14 becomes
Cp = CV + p
∂
∂T
RT
p p
= CV + p
R
p
= CV + R . (16)
Which again is Fermi Eq. 33. For a monotonic gas with CV = 3
2
R the above gives
Cp =
3
2
R + R =
5
2
R (17)
For a diatomic gas (with CV = 5
2
R) the above calculation becomes
Cp =
5
2
R + R =
7
2
R . (18)
From the deﬁnition of K = Cp
CV
and the fact that Cp = CV + R we see that
K = 1 +
R
CV
(19)
For a monotonic gas with CV = 3
2
R we have that
K = 1 +
R
3
2
R
= 1 +
2
3
=
5
3
(20)
In the exact same way a diatomic gas has
K = 1 +
R
5
2
R
= 1 +
2
5
=
7
5
, (21)
for its ratio of speciﬁc heats.

Constraints on p, V , and T under an Adiabatic Transformations of
an Ideal Gas (Page 25-26)
For an adiabatic transformation dQ = 0 and from Eq. 9 we have
CV dT + pdV = 0 . (22)
Inserting the ideal gas law pV = RT in the above for p gives a differential expression in terms
of only T and V . From this, the following sequence of manipulations (assuming constant
CV ) integrates this obtaining the following
CV dT +
RT
V
dV = 0 (23)
dT
T
+
R
CV
dV
V
= 0 (24)
log(T) +
R
CV
log(V ) = constant . (25)
TV
R
CV = constant . (26)
Defining K = 1 + R
CV
the above becomes
TV K−1
= constant . (27)
For an initial temperature, volume configuration denoted by (T0, V0) the above expression
becomes
TV K−1
= T0V K−1
0 . (28)
For a diatomic gas we have K = 7
5
, so K − 1 = 2
5
= 0.4. For an expansion that has its final
volume V twice its initial volume size 2V0 we have
T =
T0
2K−1
=
T0
20.4
≈ 0.75 T0 . (29)
To derive the expression of an adiabatic expansion in terms of the variables p and V or p and
T we substitute the ideal gas law in Eq. 27 as follows. From ideal gas law we have pV = RT
or T = pV
R
when substitute into Eq. 27 gives
pV
R
V K−1
= constant , (30)
or
pV K
= constant . (31)
This is Fermi Eq. 39 and is the equation between the variables p and V that must be true for
an adiabatic transformation of an ideal gas. In terms of p and T (replacing V with V = RT
p
in Eq. 27) it looks like
T
(RT)K−1
pK−1
= constant . (32)
or
TK
pK−1
= const. (33)

which is equivalent to Fermi Eq. 40. Another way of obtaining the same expression can be
obtained by solving for V in Eq. 27 giving
V =
C
T
1
K−1
.
Here I will denote constants that don’t need to be further speciﬁed by C. Putting this
expression for V into the ideal gas law (pV = RT) gives
p C
T
1
K−1
= RT , (34)
or solving for p we obtain
p = C T1+ 1
K−1 = C T
K
K−1 , (35)
which we recognize as being equivalent to Fermi Eq. 40 again.
A Comparison of the p-V Representations of Isothermal and Adia-
batic Transformations (Page 26-27)
In the (V, p) diagram p = RT
V
= C
V
for an isothermal transformation. From the previous
section we see that an adiabatic transformation in the (V, p) diagram is governed by p = C
V K ,
with K the ratio of speciﬁc heats. To explicitly determine the slope for each of these
transformations in the (V, p) plane we must evaluate the following derivatives
dp
dV isotherm
=
−C1
V 2
(36)
and
dp
dV adiabatic
=
−C2
V K+1
(37)
We see that the adiabatic transformation will have a steeper slope when
−C2
V K+1
>
−C1
V 2
. (38)
In terms of V (assuming C1 ≈ C2 > 0 for simplicity) the above equation is equivalent to the
following sequence of algebraic transformations
−
1
V K+1
> −
1
V 2
(39)
1
V K+1
<
1
V 2
(40)
V K+1
> V 2
(41)
K + 1 > 2 (42)
K > 1 . (43)
Since this last equation is true for a general thermodynamical system, all previous manipu-
lations are valid and we conclude that the slope of isothermal curve is steeper in general.

An Example of Adiabatic Expansion in the Atmosphere (Page 27)
The weight of a constant density object in a gravitational field is given by the product of
the gravitational constant, the objects density, and the objects volume. For a fluid in a an
infinitesimal cylindrical slab of height dh and base area A this weight is given by
dW = gρAdh .
To balance this weight, a pressure differential must exist between the top and bottom of the
fluid (which is maintained by the internal pressure) and is given by
dp =
dW
A
= −
gρAdh
A
= −ρgdh . (44)
We have explicitly introduced a minus sign in the pressure differential under the expectation
that the pressure should decrease dp < 0 as we move up in the atmosphere dh > 0. From
the ideal gas law in terms of density ρ = Mp
RT
Eq. 44 can be written
dp = −
gMp
RT
dh (45)
To determine the molecular weight M of air we remember that since air is about 78%
Nitrogen, and 21% Oxygen so we can compute the molecular mass of air from its constitute
parts
MN2 = 2(14) = 28 g/mol (46)
MO2 = 2(16) = 32 g/mol . (47)
So the average molecular weight of air is given by an appropriately weighted linear combi-
nation of that for Nitrogen and Oxygen
Mair = 0.78(28) + 0.21(32) ≈ 28.56 g/mol . (48)
A more accurate calculation gives Mair ≈ 28.97 g/mol. Fermi Eq. 40 can be written as
Tp− K−1
K = C , (49)
assuming the gas is expanding adiabatically. Taking the logarithm of both sides of this
equation we obtain
log(T) −
K − 1
K
log(p) = log(C) . (50)
Further taking the differential of both sides of the above equation gives
dT
T
−
K − 1
K
dp
p
= 0 , (51)
or solving for dT
T
gives
dT
T
=
K − 1
K
dp
p
. (52)

Replacing the dp term with that from Eq. 45 we obtain
dT
T
= −
K − 1
K
gM
RT
dh . (53)
This gives for the change in temperate with respect to height the following expression
dT
dh
= −
K − 1
K
gM
R
, (54)
which is Fermi Eq. 42 in the book. Since air is mostly a diatomic gas we have that K = 7
5
.
Taking the remaining constants to be their standard CGS values of
g = 980 cm/s2
(55)
R = 8.314 107 dyne cm
mol K
, (56)
we obtain
dT
dh
= −9.8 10−5
deg/cm = −9.8 10−5
deg/kilometer . (57)
Postulates of Reversible Thermodynamic Engines (Page 37-39)
The expression
Ltotal = N′
L′
− NL ,
is represented in two terms, the first N′
L′
, is the work done by the first engine and the
second NL is the work done by the heat. The total heat absorbed from T2 is given by
Q2,total = N′
Q′
2 − NQ2 ,
while the total given up to temperature T1, is
Q1,total = N′
Q′
1 − NQ1 .
With the definitions given by Fermi Eq. 47 and Fermi Eq. 48 for (L′
and L) we have
Ltotal = N′
L′
− NL (58)
= N′
(Q′
2 − Q′
1) − N(Q2 − Q1) (59)
= N′
Q′
2 − NQ2 − N′
Q′
1 + NQ1 (60)
= Q2,total − Q1,total . (61)
If Q2,total = 0, from the above equation we have that Ltotal = −Q1,total which is Fermi Eq. 51.
Now as explained in the book Ltotal ≤ 0, and because of the equivalence between the mag-
nitude of Ltotal and Q1,total we therefore must have Q1,total ≥ 0. Remembering the definition

of Q1,total and Fermi Eq. 49 the following manipulations derive the fact that the efficiency of
a reversible cyclic engine must be greater than that from a non-reversible cyclic engine.
Q1,total ≥ 0 (62)
N′
Q′
1 ≥ NQ1 (63)
N′
N
Q′
1 ≥ Q1 (64)
Q2
Q′
2
Q′
1 ≥ Q1 (65)
Q2Q′
1 ≥ Q1Q′
2 (66)
Q2
Q′
1
≥
Q′
2
Q′
1
(67)
Q1
Q2
≤
Q′
1
Q′
2
(68)
1 −
Q1
Q2
≥ 1 −
Q′
1
Q′
2
(69)
ηrev. ≥ ηnon−rev. (70)
Carnot Cycles with an Ideal Gas (Pages 42-43)
We will step along each segment of the Carnot cycle, deriving expression that must be
satisfied due to the known nature of the transformation that occurs during the considered
segment. For instance, from the first law of thermodynamics we obtain that the transforma-
tion from A to B along an isothermal expansion must satisfy
UB − UA + LAB = QAB ≡ Q2 . (71)
For an isothermal expansion the work can be explicitly calculated and is found to be
LAB = RT2 log(
VB
VA
) (72)
Since the assumed substance is an ideal gas where the internal energy is a function of only
temperature we have that UB = UA and thus
Q2 = LAB = RT2 log(
VB
VA
) (73)
Along the symmetric isothermal contraction we have
UC − UD + LDC = QDC ≡ Q1 (74)
with (by the same reasoning UC = UD) and the above simplifies as before
Q1 = LDC = RT1 log(
VD
VC
) (75)

The two paths we have not considered are the two adiabatic ones CA and BD. On the CA
path the adiabatic constraint requires that
T1V K−1
C = T2V K−1
A , (76)
and the BD path requires
T2V K−1
B = T1V K−1
D . (77)
Dividing these two expressions gives
VC
VD
K−1
=
VA
VB
K−1
(78)
or taking the 1/(K − 1)th root of both sides we obtain
VC
VD
=
VA
VB
. (79)
Thus in terms of Q2 and Q1 we have
Q2
Q1
=
RT2 log(VB
VA
)
RT1 log(VD
VC
)
=
T2
T1
, (80)
which expresses the ratio of heat extracted and emitted in terms of the two operating reservoir
temperatures.
The Eﬃciency of the Carnot Cycle (Pages 43-44)
From Fermi Eq. 43 we have L = Q2 − Q1 which is equivalent to
1 =
Q2
Q1
−
L
Q1
(81)
Using the relationship above to express the heat ratio in terms of the reservoir temperatures
we obtain
1 =
T2
T1
−
L
Q1
(82)
Solving for Q1 gives
Q1 = L
T1
T2 − T1
, (83)
which is Fermi Eq. 60. Now solving the above for L gives
L =
T1 − T2
T1
Q1 = Q1
T2
T1
− 1 . (84)

Proof that the Total Entropy Must Increase under Heat Flow (Page
56)
The transfer of an amount Q between bodies A1 and A2 results in a decrease in entropy ∆S1
for body A1 and an increase in entropy of ∆S2 for body A2. In terms of the heat transferred
at the lower temperature T1 and the higher temperature T2 we have
∆S1 =
Q
T1
(85)
∆S2 = −
Q
T2
. (86)
The total entropy change in the entire system is given by
∆S1 + ∆S2 =
Q
T1
−
Q
T2
=
Q(T2 − T1)
T1T2
> 0 , (87)
which can be seen to be positive since T2 > T1.
A Derivation of the Functional form of the Entropy (Page 58)
Since entropy is additive while probabilities are multiplicative our entropy function f must
satisfy the relation f(xy) = f(x) + f(y). Replaying y with 1 + ǫ this becomes
f(x(1 + ǫ)) = f(x) + f(1 + ǫ) . (88)
Expanding side of the above in a Taylor series for small ǫ we obtain
f(x) + xǫf′
(x) +
x2
ǫ2
2
f′′
(x) + O(ǫ3
) = f(x) + f(1) + f′
(1)ǫ +
f′′
(1)
2
ǫ2
+ O(ǫ3
) (89)
or canceling the common f(x) we obtain
xǫf′
(x) + O(ǫ2
) = f(1) + f′
(1)ǫ + O(ǫ2
) . (90)
Matching powers of epsilon on both sides we have f(1) = 0 and xf′
(x) = f′
(1) ≡ k or
f′
(x) =
k
x
(91)
Integrating, we obtain a functional form for the entropy of f(x) = k log(x) + constant.
Explicit expressions for Entropy (Pages 59-61)
The heat dQ received during an inﬁnitesimal transformation is given by
dQ =
∂U
∂T V
dT +
∂U
∂V T
+ p dV , (92)

which is Fermi Eq. 79. Since for a reversible transformation dS = dQ
T
the above becomes
dS =
dQ
T
=
1
T
∂U
∂T V
dT +
1
T
∂U
∂V T
+ p dV (93)
For an ideal gas the heat received during an inﬁnitesimal transformation of an ideal gas
comes in two parts as dQ = CV dT + pdV where p = RT
V
, giving
dQ = CV dT +
RT
V
dV (94)
which is Fermi Eq. 84. A reversible transformation has dS = dQ
T
giving
dS =
CV
T
dT +
R
V
dV . (95)
Assuming that CV a constant (which is true for an ideal gas) we can integrate the above to
obtain
S = CV ln(T) + R ln(V ) + a . (96)
Which is Fermi Eq. 86. Since an idea gas has V = RT
p
we can determine the entropy in terms
of p and T obtaining
S(p, T) = CV ln(T) + R ln(
RT
p
) + a . (97)
When we expanding out the second ln term we get
S(p, T) = CV ln(T) + R ln(R) + R ln(T) − R ln(p) + a (98)
or
S(p, T) = (CV + R) ln(T) − R ln(p) + a + R ln(R)
= Cp ln(T) − R ln(p) + a + R ln(R) (99)
which is Fermi Eq. 87. In the general case where the internal energy U is a function of both T
and V (and Cv is not constant), then Fermi Eq. 80 requires (since S is an exact diﬀerential)
and we can equate mixed partials
∂
∂V
1
T
∂U
∂T V
=
∂
∂T
1
T
∂U
∂V T
+ p . (100)
evaluating the derivatives of both sides we obtain
1
T
∂2
U
∂V ∂T
=
∂U
∂V
+ p −
1
T2
+
1
T
∂2
U
∂T∂V
+
∂p
∂T
. (101)
Which upon canceling the second derivative term from both sides gives
∂U
∂V
+ p = T
∂p
∂T V
(102)
or
∂U
∂V T
= T
∂p
∂T T
− p (103)

which is Fermi Eq. 88 in the book. Note that all the derivations up to this point have been
independent of the equation of state. If in fact we further specify that our operating medium
is an ideal gas with an equation of state given by the ideal gas law
p =
RT
V
we can evaluate our general expression equation 103. For the ideal gas law this gives
∂U
∂V T
= T
R
V
−
RT
V
= 0 . (104)
From which we can conclude that U does not depend on T.
Using T and p as independent variables:
Choosing (T, p) as the independent state variables Fermi Eq. 23 gives for dQ
dQ =
∂U
∂T p
+ p
∂V
∂T p
dT +
∂U
∂p T
+ p
∂V
∂p T
dp (105)
For a reversible path we again have dS = dQ
T
(for a general path dS ≥ dQ
T
) and the above
becomes
dS =
1
T
∂U
∂T p
+ p
∂V
∂T p
dT +
1
T
∂U
∂p T
+ p
∂V
∂p T
dp (106)
Again since dS must be a perfect diﬀerential equating the cross derivatives we have
∂
∂p
1
T
∂U
∂T p
+ p
∂V
∂T p
=
∂
∂T
1
T
∂U
∂p T
+ p
∂V
∂p T
. (107)
when we expanding the derivatives above (remembering that in this formulation p is a
constant to T and vice versa) gives
1
T
∂2
U
∂p∂T
+
∂V
∂T
+ p
∂2
V
∂T∂p
=
∂U
∂p T
+ p
∂V
∂p T
−
1
T2
(108)
+
1
T
∂2
U
∂p∂T
+ p
∂2
V
∂T∂p
(109)
or when one cancels common terms, one obtains
∂U
∂p T
= −p
∂V
∂p T
− T
∂V
∂T p
(110)
which is Fermi Eq. 89 in the book.

Using p and V as independent variables:
With p and V as independent variables with Fermi Eq. 24 dQ can be expressed as
∂U
∂p V
dp +
∂U
∂V p
+ p dV = dQ . (111)
On a reversible path dS = dQ
T
and the above becomes
dS =
dQ
T
=
1
T
∂U
∂p V
dp +
1
T
∂U
∂V p
+ p dV . (112)
As in all previous derivations, using the knowledge that dS is a perfect differential we can
equate mixed partials of S obtaining
∂
∂V
1
T
∂U
∂p
=
∂
∂p
1
T
∂U
∂V
+ p (113)
which upon expanding the above gives us
∂
∂V
1
T
∂U
∂p
+
1
T
∂2
U
∂p∂V
=
∂
∂p
1
T
∂U
∂V
+ p +
1
T
∂2
U
∂p∂V
+ 1 (114)
or
−
1
T2
∂T
∂V p
∂U
∂p V
= −
1
T2
∂T
∂V p
∂U
∂V p
+ p +
1
T
. (115)
Solving for T (by multiplying by T2
) we have
T = −
∂T
∂V p
∂U
∂p V
+
∂T
∂p V
∂U
∂V p
+ p (116)
Which is Fermi Eq. 90. in the book.
The Derivation of the Clapeyron Equation (Page 65)
With the change in volume dV and dU given by
dV = (v2(T) − v1(T))dm (117)
dU = (u2(T) − u1(T))dm (118)
where vi and ui are the specific volume and specific internal energy of the substance. The
direct ratio with
u2 − u1 + p(v2 − v1) = λ
gives
dU
dV T
=
u2(T) − u1(T)
v2(T) − v1(T)
=
λ − p(v2(T) − v1(T))
v2(T) − v1(T)
=
λ
v2 − v1
− p (119)

Comparing this with equation 104 or
∂U
∂V T
= T
∂p
∂T T
− p (120)
we have by inspection that
T
dp
dT
=
λ
v2 − v1
. (121)
Which is Fermi Eq. 94.
An Example using the Clapeyron Equation (Page 66)
Now with the given value of λ
λ = 540cal/gm = 540(4.185 107
erg)/gm = 2259 107
erg/gm
and since water boils at 100 Centigrade, T = 100+273.15 = 373.15K, so that the Clapeyron
equation gives
dp
dT
=
(2259 107
erg/gm)
(373.15K)(1677 − 1.043)cm3/gm
= 3.61 104
erg/cm3
K
Remembering the conversion that
1
dyne
cm2
= 7.5 10−5
cm Hg
we have then
dp
dT
= 2.7
cm Hg
K
. (122)
Normally for gas-liquid or a gas-solid interface
v1 ≪ v2 (123)
Here v1 is the specific volume of the liquid/solid and v2 is the specific volume of gas. Fermi
Eq. 6 (the equation of state for an ideal gas) gives
p
V
m
=
1
M
RT . (124)
The expression V
m
is the specific volume or the volume per gram so, since index 2 corresponds
to the gas we have v2 = RT
Mp
, so that the Clapeyron equation becomes
dp
dT
=
λ
Tv2
=
λ
T RT
Mp
=
λMp
RT2
(125)
Which is Fermi Eq. 96 in the book.

For water vapor over its liquid at the boiling temperature we get (remembering that M =
18gm/mol)
dp
dT
=
(2260 107
erg/gm)(18gm/mol)(1atm)
(8.314 J/molK)(373.1K)2
. (126)
Remembering the unit conversion that 1erg = 10−7
J and 1atm = 1.01 105
Pa, the above
becomes
dp
dT
=
(2260J/gm)(18gm/mol)(1.01 105
Pa)
(8.314 J/molK)(373.1K)2
= 3.55 103
Pa/K = 3.55 104 dynes
cm2K
. (127)
Which agrees with the number given in the book.
In equation 125, if the heat of vaporization λ can be assumed independent of temperature
then this expression can be integrated giving
p = Ce
λM
RT (128)
As an example of a solid-liquid system consider melting of ice then we are told that
λ = 80cal/cm = 80 × 4.185 107
erg/gm = 334.8 107
erg/gm . (129)
so that the Clapeyron equation becomes
dp
dT
=
λ
T(v2 − v1)
=
335 107
erg/gm
(273.1K)(1.00013 − 1.0907)cm3/gm
= −1.35 108
erg/cm3
K (130)
Remembering that
1
dyne
cm2
= 9.86 10−7
atm
the above becomes
dp
dT
= −1.33 102
atm/K = −133 atm/K (131)
This increases in pressure by 134 atmospheres lowers the melting point by 1 Kelvin. The
more pressure ice is under the easier it is to melt it.
The Van der Waals Critical State in terms of the constants a and b
(Page 72-73)
A gas that satisﬁes the Van der Waal equation of state must satisfy
p +
a
V 2
(V − b) = RT . (132)
Given T and p, to obtain a cubic equation in V we multiplying both sides of the above by
V 2
and expanding the product on the left hand side producing
(V 2
p + a)(V − b) = V 2
RT (133)
V 3
p − bV 2
p + aV − ab = V 2
RT (134)
pV 3
+ (−bp − RT)V 2
+ aV − ab = 0 . (135)

Evaluating the above at the critical pressure p = pc and temperature T = Tc we have
pcV 3
− (bpc + RTc)V 2
+ aV − ab = 0 . (136)
Because the pressure and temperature are evaluated at the critical isotherm, this expression
must have Vc as a root of third order. Mathematically this means that the above should be
represented as pc(V − Vc)3
= 0 for some choice of Vc. Expanding this cube we have
pcV 3
− 3pcV 2
Vc + 3pcV V 2
c − pcV 3
c = 0 . (137)
Comparing these coeﬃcients to those in Eq. 136 we have
−3pcVc = −(bpc + RTc) (138)
3V 2
c pc = a (139)
−pcV 3
c = −ab (140)
This is a set of three equations involving three unknowns of (Vc, pc, Tc). Solving for Vc using
the second and third equation above by dividing the third equation by the second gives
Vc = 3b . (141)
Inserting this equation into the second equation from 138 we see that pc is given by
pc =
a
3V 2
c
=
a
27b2
. (142)
while the ﬁrst equation from 138 then gives for Tc
Tc =
3Vcpc − bpc
R
=
3(3b)( a
27b2 ) − b( a
27b2 )
R
=
a
b
1
3
− 1
27
R
=
8
27
a
Rb
(143)
which is Fermi Eq. 100. As suggested in the text introducing the non-dimensional variables
P =
p
pc
(144)
V =
V
Vc
(145)
T =
T
Tc
(146)
we get by substituting the following
p = Ppc (147)
V = V Vc (148)
T = T Tc , (149)
into the Van der Waals’ equation of state (Fermi Eq. 99)
p +
a
V 2
(V − b) = RT , (150)
the following
Ppc +
a
V 2
c V 2
(VcV − b) = RT Tc , (151)

or
P +
a
pcV 2
c V 2
V −
b
Vc
=
RTc
Vcpc
T . (152)
Now the factors involving our recently determined critical constants simplify as follows
a
pcVc
=
a
a
27b2 9b2
=
1
1
3
= 3
b
Vc
=
b
3b
=
1
3
RTc
Vcpc
=
R 8
27
a
Rb
3b a
27b2
=
8
3
and we get for equation 152
P +
3
V 2
V −
1
3
=
8
3
T (153)
which is Fermi Eq. 101.
Expressions for internal energy and entropy in a Van der Waals gas
(Page 73-75)
Using the Van der Waals equation of state expression (Fermi Eq. 99) by solving for the
pressure we obtain
p = −
a
V 2
+
RT
V − b
. (154)
Now for any thermodynamic system the internal energy change with respect to volume is
given by (holding temperature constant)
∂U
∂V T
= T
∂p
∂T V
− p . (155)
Evaluating this expression for the p = p(V ) relation given by the Van der Waals equation of
state we obtain
∂U
∂V T
= T
R
V − b
+
a
V 2
−
RT
V − b
=
a
V 2
. (156)
Integrating this expression (with respect to V ) gives
U = −
a
V
+ f(T) , (157)
where f(T) is an arbitrary function of temperature. This is Fermi Eq. 103. To derive the
entropy of a Van der Waals gas we ﬁrst evaluate the speciﬁc heat at constant volume given
by
CV =
∂Q
∂T V
=
∂U
∂T V
= f′
(T) , (158)

which if we assume that CV is constant, we can integrate with respect to T to obtain
f(T) = CV T + w .
This expression when put back into the expression for the internal energy U results in
U = −
a
V
+ CV T + w . (159)
Now to evaluate the entropy we can perform the following manipulations exactly as in the
book
dS =
dQ
T
=
1
T
(dU + pdV )
=
1
T
(CV dT +
a
V 2
dV ) +
1
T
−
a
V 2
+
RT
V − b
dV
= CV
dT
T
+
a
TV 2
dV −
a
TV 2
dV +
R
V − b
dV
= CV
dT
T
+
R
V − b
dT , (160)
which when integrated with respect to T gives
S = CV ln(T) + R ln(V − b) + constant . (161)
Which is Fermi Eq. 105. Compare this expression with Fermi Eq. 86 the similar expression
for an ideal gas which is
S = CV ln(T) + R ln(V ) + a
Now along an adiabatic transformation by definition dQ = 0, equivalently dS = 0, or
S = constant and the above expression can be manipulated as follows
CV ln(T) + R ln(V − b) = C1
ln(T) +
R
CV
ln(V − b) = C2
T(V − b)R/CV
= C3 (162)
where C1, C2, and C3 are all constants. This expression can be recognized as Fermi Eq. 106.
The derivation of the isochore of Van’t Hoff (Page 81)
The derivation given in the book proceeds smoothly until about the middle of the page
where the statement dF(A)/dT = −S(A) is made. This expression can be derived as
follows. Considering the definition of the free energy F(A) as F = U − TS we have that the
temperature derivative of F given by
dF(A)
dT
=
dU(A)
dT
− T
dS(A)
dT
− S(A)
=
1
dT
(dU(A) − TdS(A)) − S(A)
= −S(A)

where we have used the fact that from the first and second law of thermodynamics dU =
dQ − dW = TdS − dW or dU − TdS = −dW = 0 if no work is done.
Since S(A) = (U − F)/T from the definition of the free energy the temperature derivative
above can be written in terms of the free energy and internal energy as
dF(A)
dT
=
F(A)
T
−
U(A)
T
.
To relate this to the derivative of work with respect to temperature and derive the isochore
of Van’t Hoff recall Fermi Eq. 115 or
dL
dT
=
dF(A)
dT
−
dF(B)
dT
,
we can use the expression above to replace the derivatives of the free energy with respect to
temperature with expressions involving the free energy itself as
dL
dT
=
dF(A)
dT
−
dF(B)
dT
=
F(A) − U(A)
T
−
F(B) − U(B)
T
.
multiplying both sides by T we obtain
T
dL
dT
= F(A) − U(A) − (F(B) − U(B)) = F(A) − F(B) − (U(A) − U(B)) = L + ∆U
where we have used Fermi Eq. 114 of L = F(A) − F(B). Thus in summary we have derived
−∆U = L − T
dL
dT
(163)
which is Fermi Eq. 117 or the isochore of Van’t Hoff.
WWX: I have not finished this section ... start
WWX: page 2 of the second set of scanned notes
From Fermi Eq. 112 we have L ≤ −∆F
pdV = −
∂F
∂V T
(164)
so
−p =
∂F
∂V T
(165)
Since F = U − ST, for an ideal gas we get that
F = CvT + W − (CvT + R log(V ) + a)T (166)
= CvT + W − T(Cp log(T) − R log(p) + a + R log(R)) (167)

Derivation of the thermodynamic potential at constant pressure
(Page 82-83)
L ≤ −∆F gives pV (B) − pV (A) ≤ F(A) − F(B). Now defining
Φ = F + pV = U − TS + pV (168)
so that the above becomes
pV (B) + F(B) ≤ pV (A) + F(A) (169)
giving Φ(B) ≤ Φ(A). From the book: To find an equilibrium state for a system that has
pressure and temperature we look for the minimum of the Gibbs Free energy Φ. For a general
system, find what ... thermodynamic potential be it F − U − TS, Φ = U − TS + pV now
since Φ = U − TS + pV we have that
∂Φ
∂p T
=
∂U
∂p T
− T
∂S
∂p T
+ V + p
∂V
∂p T
(170)
since dQ = TdS = dU + pdV ,
T
∂S
∂p T
=
∂U
∂p T
(171)
putting this into the above we have
∂Φ
∂p T
=
∂U
∂p T
−
∂U
∂p T
− p
∂V
∂p T
+ V + p
∂V
∂p T
= V (172)
which is Fermi Eq. 123. In the same way
∂Φ
∂T p
=
∂U
∂T p
− S − T
∂S
∂T p
+ p
∂V
∂T p
(173)
TdS = dU + pdV (174)
so that
T
∂S
∂T p
=
∂U
∂T T
+ p
∂V
∂T p
(175)
when put in the above becomes
∂Φ
∂T p
=
∂U
∂T p
− S −
∂U
∂T p
− p
∂V
∂T p
+ p
∂V
∂T p
(176)
Φ = U − TS + pV . Now T is the same for both vapor and liquid phases and P is the same
for both vapor and the liquid at least at the surface or in zero gravity? Now Φ = Φ1 + Φ2,
i.e. on the space shuttle or with very little liquid. pV = RT and
Φ = m1φ1(T) + m2φ2(T) (177)

Let m1 → m1 + dm, then Φ becomes
Φ′
= (m1 + dm1)φ1(T) + (m2 − dm1)φ2(T) = Φ + dm1(φ1 − φ2) (178)
As Φ was at a minimum. In order that the value of Φ′
not change from that of Φ, we must
have φ1 = φ2, which is
U1 − TS1 + pV1 = U2 − TS2 + pV2 (179)
or
U2 − U1 − T(S2 − S1) + p(V2 − V1) = 0 (180)
taking the partial with respect to T we have
d
dT
(U2 − U1) − (S2 − S1) − T
d
dT
(S2 − S1) +
dp
dT
(V2 − V1) + p
d
dT
(V2 − V1) = 0 (181)
T
ds
dT
=
dU
dT
+ p
dv
dT
(182)
dU
dT
− T
dS
dT
+ p
dV
dT
= 0 (183)
−(S2 − S1) +
dp
dT
(V2 − V1) = 0 (184)
with S2 − S1 = λ
T
, we obtain
dp
dT
=
λ
T(V2 − V1)
(185)
Φ = U − TS + pV (186)
= cV T + W − T(Cp log(T) − R log(p) + a + R log(R)) + pV (187)
= cV T + W + pV − T(Cp log(T) − R log(p) + a + R log(R)) (188)
= cV T + W + RT − T(Cp log(T) − R log(p) + a + R log(R)) (189)
since CV + R = Cp the above becomes
Φ = CpT + W − T(Cp log(T) − R log(p) + a + R log(R)) (190)
dF(A)
dT
=
dU
dT
− T
dS
dT
− S(A) (191)
=
1
dT
(dU − TdS) − S(A) (192)
= −S(A) (193)
if no work is done? Since
dU = dQ − dW = TdS − dW (194)
so we have that dU − TdS = −dW = 0 if there is no work done.
dF(A)
dT
=
F(A)
T
−
U(A)
T
(195)

since F = U − ST, we have that −S = F −U
T
, now Fermi Eq. 114 is that L = F(A) − F(B)
and
dL
dT
=
dF(A)
dT
−
dF(B)
dT
(196)
then
T
dL
dT
= T
dF(A)
dT
− T
dF(B)
dT
(197)
each term in the above can be replaced as
T
dL
dT
= F(A) − U(A) − (F(B) − U(B)) = F(A) − F(B) − (U(A) − U(B)) = L + ∆U (198)
so we have
T
dL
dT
= L + ∆U (199)
giving
−∆U = L − T
dL
dT
(200)
which is Fermi Eq. 117. From Fermi Eq. 112 we have L ≤ −∆F
pdV = −
∂F
∂V T
(201)
so
−p =
∂F
∂V T
(202)
Since F = U − ST, for an ideal gas we get that
F = CvT + W − (CvT + R log(V ) + a)T (203)
= CvT + W − T(Cp log(T) − R log(p) + a + R log(R)) (204)
L ≤ −∆F gives pV (B) − pV (A) ≤ F(A) − F(B). Now defining
Φ = F + pV = U − TS + pV (205)
so that the above becomes
pV (B) + F(B) ≤ pV (A) + F(A) (206)
giving Φ(B) ≤ Φ(A). From the book: To find an equilibrium state for a system that has
pressure and temperature we look for the minimum of the Gibbs Free energy Φ. For a general
system, find what ... thermodynamic potential be it F − U − TS, Φ = U − TS + pV now
since Φ = U − TS + pV we have that
∂Φ
∂p T
=
∂U
∂p T
− T
∂S
∂p T
+ V + p
∂V
∂p T
(207)
since dQ = TdS = dU + pdV ,
T
∂S
∂p T
=
∂U
∂p T
(208)

putting this into the above we have
∂Φ
∂p T
=
∂U
∂p T
−
∂U
∂p T
− p
∂V
∂p T
+ V + p
∂V
∂p T
= V (209)
which is Fermi Eq. 123. In the same way
∂Φ
∂T p
=
∂U
∂T p
− S − T
∂S
∂T p
+ p
∂V
∂T p
(210)
TdS = dU + pdV (211)
so that
T
∂S
∂T p
=
∂U
∂T T
+ p
∂V
∂T p
(212)
when put in the above becomes
∂Φ
∂T p
=
∂U
∂T p
− S −
∂U
∂T p
− p
∂V
∂T p
+ p
∂V
∂T p
(213)
Φ = U − TS + pV . Now T is the same for both vapor and liquid phases and P is the same
for both vapor and the liquid at least at the surface or in zero gravity? Now Φ = Φ1 + Φ2,
i.e. on the space shuttle or with very little liquid. pV = RT and
Φ = m1φ1(T) + m2φ2(T) (214)
Let m1 → m1 + dm, then Φ becomes
Φ′
= (m1 + dm1)φ1(T) + (m2 − dm1)φ2(T) = Φ + dm1(φ1 − φ2) (215)
As Φ was at a minimum. In order that the value of Φ′
not change from that of Φ, we must
have φ1 = φ2, which is
U1 − TS1 + pV1 = U2 − TS2 + pV2 (216)
or
U2 − U1 − T(S2 − S1) + p(V2 − V1) = 0 (217)
taking the partial with respect to T we have
d
dT
(U2 − U1) − (S2 − S1) − T
d
dT
(S2 − S1) +
dp
dT
(V2 − V1) + p
d
dT
(V2 − V1) = 0 (218)
T
ds
dT
=
dU
dT
+ p
dv
dT
(219)
dU
dT
− T
dS
dT
+ p
dV
dT
= 0 (220)
−(S2 − S1) +
dp
dT
(V2 − V1) = 0 (221)
with S2 − S1 = λ
T
, we obtain
dp
dT
=
λ
T(V2 − V1)
(222)

Φ = U − TS + pV (223)
= cV T + W − T(Cp log(T) − R log(p) + a + R log(R)) + pV (224)
= cV T + W + pV − T(Cp log(T) − R log(p) + a + R log(R)) (225)
= cV T + W + RT − T(Cp log(T) − R log(p) + a + R log(R)) (226)
since CV + R = Cp the above becomes
Φ = CpT + W − T(Cp log(T) − R log(p) + a + R log(R)) (227)
Φi = Φi(T, p, mi,1, mi,2, mi,3, . . . , mi,n) (228)
Φi(T, p, kmi,1, kmi,2, kmi,3, . . . , kmi,n) = kΦi(T, p, mi1 , mi2 , . . . , min ) (229)
which is homogeneous of degree 1.
˙k = mi,k − δm (230)
mj,k′ = mj,k + δm (231)
δΦ = δΦi + δΦj (232)
=
∂Φi
∂mi,k
(−δm) +
∂Φj
δmj,k
δm (233)
=
∂Φj
∂mj,k
δm −
∂Φi
∂mi,k
δm = 0 (234)
∂Φj
∂mj,k
=
∂Φi
∂mi,k
(235)
How many equations like this do we have? f phases and n components, since each component
n of them in a given phase i can go to any of the other f − 1 phase (nothing happens if it
goes to itself. We have n(f − 1) equations for equilibrium. Each ∂Φ
∂·
depends only on ratios
of the mk,i. The number of ratios there are like this are n − 1. Then for all ∂Φi
∂·
is f(n − 1)
with T and p we have f(n − 1) + 2 variables. Let v be the number of unknowns minus the
number of equation. Then
v = (n − 1)f + 2 − n(f − 1) (236)
= 2 + n − f (237)
which is Fermi Eq. 131. Now example #1 we have
v = 2 + 1 − 1 = 2 (238)
Now example #2 we have
v = 2 + 2 − 1 = 3 (239)
Now example #3 is 2 phases solid and liquid has f = 2 and one component so n = 1 so we
have
v = 2 + n − f = 2 + 1 − 2 = 1 (240)

Now in example #4 we have n = 1 and f = 3 so we have
v = 2 + n − f = 2 + 1 − 3 = 0 (241)
we have L = ev. Power exerted by DC current is given by P = V I with V the voltage and
I the current. Now current is charge per unit time. Thus the work is voltage times charge.
U(T, e) = U(T) − eU(T) (242)
∆U = −eU(T) (243)
isochore of Vanft Hoﬀ
L − T
dL
dT
= −∆U (244)
eV − T
d(eV )
dT
= eU (245)
v − T
dv
dT
= u (246)
which is Fermi Eq. 134. e = Cv(T) and dL = 1
2
dCV 2
(T) energy of an isolated capacitor is
given by
1
2
e2
C
(247)
dL = −d(
1
2
e2
C
) =
1
2
e2
C2
dC (248)
But e = Cv and dL = 1
2
C2v2
C2 dC = 1
2
v2
dC
U(T, e) = U(T) − eu(T) = U(T) − Cv(T)u(T) (249)
plus energy now present in condenser, heat released/absorbed when
dQ = dU + dL = d(U(T) − CV (T)u(T) +
1
2
Cv2
(T)) +
1
2
dCv2
(250)
= dT
dU
dT
− C
dv
dT
u − Cv
dU
dT
+ Cv
dv
dT
(251)
+ dC −vu +
1
2
v2
+
1
2
v2
(252)
dS =
dQ
T
=
dT
T
dU
dT
− Cv
du
dT
− Cu
dv
dT
+ Cv
dv
dT
(253)
+
dC
T
v2
− vu (254)
∂
∂C
dU
dT
− CV
dU
dT
− Cu dv
dT
+ CV
dV
dT
T
=
∂
∂T
v2
− vu
T
(255)
1
T
−v
dU
dT
− u
dV
dT
+ v
dv
dT
(256)
= −
v2
− vu
T2
+
2vv′
− v′
u − vu′
T
(257)

which gives
−
v
T
du
dT
−
u
T
dv
dT
+
v
T
dv
dT
=
2v
T
dv
dT
−
u
T
dv
dT
−
v
T
du
dT
−
v2
T2
+
vu
T2
(258)
v2
T2
−
vu
T2
=
v
T
dv
dT
(259)
v
T
−
u
T
=
dv
dT
(260)
which gives
v − T
dv
dT
= u (261)
which is Fermi Eq. 134.
End page 10 of second set of scanned notes
begin page 13 of second set of scanned notes
In Fermi Eq. 136 if k(T) ≪ 1 then
[A1]n1
[A2]n2
· · · [Ar]nr
= k(T)[β1]m1
[β2]m2
· · · [βs]ms
(262)
Generally we have that [A1]n1
[A2]n2
· · · [Ar]nr
≪ 1, so the reaction is shifted to the right.
Now the deﬁnition of [A] is the number of moles of A divided by the volume. So [A]M is
the mass of A divided by the volume.
LI = −RT
n
i=1
ni (263)
p1V1 = n1RT (264)
since p ∝ n
V
RT we have that p ∝ [H]RT, so the pressure is proportional to the mole
concentration. To insure that we get m1 moles in the ﬁrst cylinder which we can extend to
+∞ we must require that
V1[β1] = m1 ⇒ V1 =
m1
[β1]
(265)
where [β1] concentration of elements β1.
L = LI + LII (266)
= RT
s
j=1
mj −
n
i=1
ni = −∆F (267)
as this process is isothermal and reversible. We have that
F1 = n1{Cv1 T + w1 − T(Cv1 log(T) + R log(
1
[A]
) + a1} (268)
= n1{Cv1 T + w1 − T(Cv1 log(T) − R log([A]) + a1} (269)
(270)

The Free energy initially is given by
FI =
r
i=1
ni{Cvi
T + wi − T(Cvi
log(T) − R log([Ai]) + ai} (271)
FF =
s
j=1
mj{C′
vj
T + w′
j − T(C′
vj
log(T) − R log([Bj]) + a′
j} (272)
FI − FF = L becomes
RT
s
j=1
mj −
n
i=1
ni =
n
i=1
ni{Cvi
T+wi−T(Cvi
log(T)−R log([Ai])+ai}−
s
j=1
mj{Cvi
T+wi−T(Cvi
log(T)
(273)
so we have then that
RT
s
j=1
mj −
r
i=1
ni =
r
i=1
niCvi
T+niwi−niTCvi
log(T)+niRT log([Ai])+niaiT−
s
j=1
mjC′
vj
T+mjw′
j−m
(274)
dividing by RT we have
s
j=1
mj −
r
i=1
ni =
r
i=1
niCvi
R
+
niwi
RT
−
niCvi
log(T)
R
+ ni log([Ai]) +
niai
R
(275)
+
s
j=1
mjCv′
j
R
+
mjw′
j
RT
−
mjC′
vj
log(T)
R
+ mj log([Bj]) +
mjaj
R
(276)
so we have that
r
i=1
ni log([Ai]) −
s
j=1
mj log([Bj]) = −
r
i=1
niCvi
RT
+
niwi
RT
−
niCvi
log(T)
R
+
niai
R
+ ni(277)
+
s
j=1
mjC′
vj
R
+
mjw′
j
RT
−
mjC′
vj
log(T)
R
+
mja′
j
R
+ mj(278)
which gives
log(
[A1]n1
[A2]n2
· · · [Ar]nr
[B1]m1 [B2]m2 · · · [Br]mr
=
r
i=1
niCvi
log(T)
R
−
s
j=1
mjC′
vj
log(T)
R
(279)
+ −
r
i=1
niwi
RT
+
s
j=1
mjw′
j
RT
(280)
+ −
r
i=1
niCvi
RT
+
niai
R
+ ni (281)
s
j=1
mjC′
vj
RT
+
mjaj
R
+ mj (282)

Therefore we have that
log(· · · ) = log(
T
n1Cv1
R T
n2Cv2
R T
n3Cv3
R · · · T
nrCvr
R
T
m1Cv1
R T
m2Cv2
R T
m3Cv3
R · · ·T
msCvs
R
) −
r
i=1
log(exp(
niwi
RT
)) +
s
j=1
log(exp(
mjw′
j
RT
))
(283)
−
r
i=1
log(exp(
niCvi
R
+
niai
R
+ ni)) +
s
j=1
log(exp(
mjC′
vj
R
+
mia′
j
R
+ mj)) (284)
or
log(· · · ) = log(T
1
R
r
i=1 niCvi − 1
R
s
j=1 mj C′
vj )+log(
exp( s
j=1
mj w′
j
RT
)
exp( r
i=1
niwi
RT
)
)+log(
exp( s
j=1
mj C′
vj
R
+
mj a′
j
R
+ mj)
exp( r
i=1
niCvi
RT
+ niai
R
+ ni)
)
(285)
Taking exponentials we have
[A1]n1
[A2]n2
· · · [Ar]nr
[B1]m1 [B2]m2 · · · [Bs]ms
= exp(
1
R
s
j=1
mj(R′
+ C′
vj
+ aj) −
r
i=1
ni(R + Cvi
+ ai) ) (286)
which is times
T
1
R
r
i=1 niCvi − s
j=1 mj C′
vj (287)
which is times the following
exp(−
1
RT
r
i=1
niwi −
s
j=1
mjw′
j (288)
which gives Fermi Eq. 139. This gives piV = niRT, so pi = niRT
V
with p = i pi = RT
V i ni
Since F = U − TS and Φ = U − TS + pV the Free energy is given by
Cv1 T+W1−T(Cv1 log(T)−R log([A1])+a1)+V [A1](Cv1 T+W1−T(Cv1 log(T)−R log([A1])+a1))
(289)
so
F = V
r
i=1
[Ai](Cvi
T+Wi−T(Cvi
log(T)−R log([Ai])+ai))+V
s
j=1
[Bj](C′
vj
T+W′
i −T(C′
vj
log(T)−R log([Bj])+
(290)
Which is Fermi Eq. 140. Thus the variation on F is given by
δF = −
∂F
∂[A1]
ǫn1−
∂F
∂[A2]
ǫn2−
∂F
∂[A3]
ǫn3−. . .−
∂F
∂[Ar]
ǫnr++
∂F
∂[B1]
ǫm1+
∂F
∂[B2]
ǫm2+
∂F
∂[B3]
ǫm3+. . .+
∂F
∂[Bs]
ǫms
(291)
which gives
−
∂F
∂[A1]
[A1]−
∂F
∂[A2]
[A2]−
∂F
∂[A3]
[A3]−. . .−
∂F
∂[Ar]
[Ar]++
∂F
∂[B1]
[B1]+
∂F
∂[B2]
[B2]+
∂F
∂[B3]
[B3]+. . .+
∂F
∂[Bs]
[Bs] = 0
(292)
From Fermi Eq. 140. we have
∂F
∂[Ai]
= V
r
i=1
(Cvi
+ Wi − T(Cvi
log(T) − R log([Ai]) + ai) + V
r
i=1
[Ai]
TR
[Ai]
= V TR (293)

This is incorrect. You are taking the derivative of the concentration [Ai] not the summation
variable [Ai]. The above is taking the derivative of the summation variable index [Ai] we get
∂F
∂[Ai]
= V {Cvi
T + wi − T(Cvi
log(T) − R log([Ai]) + ai}+V [Ai]
RT
[Ai]
= V {Cvi
T + wi − T(Cvi
log(T) − R lo
(294)
In the same way
∂F
∂[Bi]
(295)
Then
δF = −[A1]V (Cv1 T+w1−T(Cv1 log(T)−R log([Ai])+ai)+RT)−[A2]V (Cv2 T+w2−T(Cv2 log(T)−R log([Ai])
(296)
or
−
r
i=1
ni {Cvi
T + wi − T(Cvi
log(T) − R log([Ai]) + ai) + RT}+
s
j=1
mj C′
vj
T + w′
j − T(C′
vj
log(T) − R log(
(297)
∆U =
s
j=1
mj(C′
vj
T + w′
j) −
r
i=1
ni(Cvi
T + wi) (298)
H = −∆U =
r
i=1
ni(Cvi
T + wi) −
s
j=1
wj(C′
vj
T + w′
j) (299)
d log(k(T))
dT
=
1
TR
r
i=1
Cvi
ni −
s
j=1
C′
vj
mj +
1
RT2
(300)
To exclude the logrithm derivative of 141 write
k(T) = C1TC2
e−
C3
RT (301)
so that
log(k(T)) = log(C1) + C2 log(T) −
C3
RT
log(e) = log(C1) + C2 log(T) −
C3
RT
(302)
so that
d log(k(T)
dT
=
C2
T
+
C3
RT2
1
RT
r
i=1
Cvi
ni −
s
j=1
C′
vj
mj +
1
RT2
r
i=1
niwi −
s
j=1
mjw′
j
=
1
RT2
r
i=1
TCvi
ni + niwi −
s
j=1
TC′
vj
m
=
H
RT2

Note that H = H(T) thus we can see the temperature dependence. If n1 + n2 + . . . + nr <
m1 + m2 + . . . ms, then shifting the equation to the right increases the pressure.
[A1]n1
[A2]n2
· · · [Ar]nr
[B1]n1 [B2]n2 · · ·[Bs]ns
= k(T) (307)
Compress the system shrinking V causes the concentrations [A] and [B] to increase. Because
of the concentration inequality n1 + n2 + . . . + nr < m1 + m2 + . . . ms the left hand side
decreases to prevent this [A] increases by the [B] stays the same which implies that the
reaction shifts towards the reactants.
Problem 1
2A → A which gives
[A]2
[A2]
= k(T) (308)
is the equation of the law of mass action. Here [A] is in units of number of moles per the
volume and we are told that our reaction constant k is given by k(18C) = 1.7 10−4
. Dalton’s
law of partial pressure says that p = 1 atm = pA + pA2 with pA the partial pressure of the A
species and pA2 the partial pressure of the A2 species. Then we have that
pA =
nART
V
= [A]RT (309)
and
pA2 = [A2]RT (310)
so
p = ([A] + [A2])RT (311)
so [A]2
= k(T)[A2] which when put in above we have
p = ([A] +
[A]2
k(T)
)RT (312)
or
[A]2
k(T)
+ [A] −
p
RT
= 0 (313)
which is a quadratic equation for [A]. Since
p
RT
=
1.01 105
Pa
(8.314J/molK)(273.15 + 18K)
=
41.72J//m3
J/mol
= 41.72mol/m3
(314)
so we have for [A] the following
[A] =
−1 ± 1 − 4 1
k(T)
−P
RT
2 1
k(T)
=
−1 ± 1 + 4p
k(T)RT
2
k(T)
= 8.413 10−2
mol/V (315)

so the concentration of [A2]
[A2] =
RT
p
− [A] = 41.72 − 8.413 10−2
= 41.64 mol/m3
(316)
we can check this by considering
[A]2
[A2]
= 1.6997 10−4
(317)
since we are asked for the percentage of A we remember that nA = [A]V and nA2 = [A2]V
so the percentage of A is given by = nA
nA+nA2
= [A]V
[A]V +[A2]V
= [A]
[A]+[A2]
= 2.01 10−3
= 0.201
Problem 2
We have H = 50000cal/mol degree of dissociation a H > 0 which means that this is an
exerthermal reaction expect raising the temperature to shift the reaction towards the left
and we should have an increase in the concentration of A. If I write the equation for the
chemical reaction in the other order, I would flip the concentration ratio [A]2
[A2]
but would not
change the term e−H/RT
which would give a different equation. Where is the inconsistency.
Would be then than H should be given as the heat of reaction from the left to right. Which
if we switch the order of the chemical equation we switch the sign of H.
d log(k(T))
dT
=
H
RT2
(318)
so
log(k(T)) = C1 −
H
RT
(319)
so that
k(T) = C2e− H
RT (320)
therefore we have
[A]2
[A2]
= C2e− H
RT = k(T) (321)
What is C2? We know that k(T = 18 C = 291.15) = 1.7 10−4
so that
RT = (8.314 J/molK)(291.15K) = 2.42 103
J/mol
since we know that 1J = 0.2388cal we have that RT = 5.78 103
cal/mol, so that
H
RT
=
50000cal/mol
5.78 103cal/mol
= 8.64 101
(322)
C2 exp(−8.64 101
) = 1.7 10−4
(323)
which gives C2 = exp(8.64 101
) 1.7 10−4
= 5.66 1033
, which is huge. Therefore k(T = 18 C) =
5.66 1033
exp(− H
RT
) = 2.07 10−4
. Now to find the percentage of A we remember that p =
pA + pA2 = [A]RT + [A2]RT
[A] + [A2] =
P
RT
and [A]2
= k(T)[A2] (324)

[A] +
[A]2
k(T)
=
p
RT
⇒
[A]2
k(T)
+ [A] −
p
RT
= 0 (325)
solving for [A] we have
[A] =
−1 ± 1 − 4 1
k(T)
−P
RT
2 1
k(T)
= 9.26 10−2
> 8.41 10−2
(326)
so the concentration of species A2 is given by
[A2] =
[A]2
k(T)
⇒ [A2] = 4.14 101
< 4.16 101
(327)
so the percentage A is given by 2.22 10−3
or 0.22 percent which has increased the concentra-
tion of A as expected.
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Problem Solutions
Chapter 1 (Thermodynamic Systems)
Problem 1
Since the gas is expanding against a constant atmosphere (at pressure p) the total work L
done by the gas can be written as
L =
V2
V1
pdV = p(V2 − V1) (328)
Using the numbers given in the text we have
L = 2.34 atm(4.01 − 3.12) liters = 2.082 atm liters (329)
To convert this result into CGS units (centimeters, grams, and seconds) we remember that
1 atm = 1.01 105
Pa = 1.01 105 N
m2
(330)
1 liter = 10−3
m3
(331)
1 N = 105
dyne (332)
1 m = 100 cm (333)

Thus we have
1 atm liter = 1.01 105 N
m2
× 10−3
m3
(334)
= 1.01 102
N m (335)
= 1.01 102
× 105
dyne × 102
cm (336)
= 1.01 109
dyne cm . (337)
After unit conversion, our total work L is given by
L = 2.103 102
N m = 210.3 N m = 2.103 109
dyne cm (338)
Problem 2
Since we are told that the gas is hydrogen (H2), looking to the periodic table gives us that
hydrogen’s mass per unit mole (molecular weight) is given by M = 2g/mole. The rest of the
problem provides the gas’s mass m, volume V , and temperature T as
m = 30 g (339)
T = 18◦
C = 18 + 273.15 = 291.15 K (340)
V = 1 m3
(341)
Then one application of the ideal gas law
pV =
m
M
RT
gives
p =
m
M
R T
1
V
=
30g
2g/mole
× 8.314
J
mole K
× 291.15 K ×
1
1m3
= 3.63 104 J
m3
(342)
Since 1J = 1N m, the above simpliﬁes to
p = 3.64 104 N
m2
= 3.64 104
Pa . (343)
To convert to atmospheres, requires the conversion to atmospheres from Pascals given by
1 Pa =
1
1.01 105
atm = 9.9 10−6
atm .
With this, the pressure in atmospheres becomes
p = 3.59 10−1
atm . (344)

Problem 3 (calculate the density and specific volume of nitrogen)
Nitrogen exists as a diatomic molecule (N2) under standard conditions and thus has a molec-
ular weight of
M = 2 × 14 grams/mole = 28 grams/mole .
The idea gas law gives for the density ρ
ρ ≡
m
V
=
Mp
RT
. (345)
In the problem, we are assuming that the Nitrogen is at 0 Celsius or 273.15 K. As such, the
density will be completely determined once we have specified a pressure. Since none is given
in this problem we will assume that atmospheric conditions apply and that the pressure is
1 atm = 1.01 105
Pa. With this assumptions the density is given by
ρ =
(28 grams/mole) (1.01 105
Pa)
(8.314 J/mole K) (273.15 K)
= 1244
grams Pa
J
= 1244
(10−3
kg)(kg/ms2
)
kg m2/s2
= 1.244
kg
m3
= 1.244 10−3 g
cm3
.
Since the specific volume is defined to be the reciprocal of the density, we have numerically
that
v =
1
ρ
= 0.8038585
m3
kg
= 803.8585
cm3
g
. (346)
Problem 4
From Fermi Eq. 9 in the book we have the work for an isothermal expansion is given by
L =
m
M
RT log(
V2
V1
) =
m
M
RT log(
p1
p2
) . (347)
For this problem the provided inputs are
m = 10 g
M = 32.0 g/mol
R = 8.314J/molK
T = 20 C = 293.15 K
p1 = 1 atm
p2 = 0.3 atm
with these inputs the expression for work above becomes
L =
10g
32g/mol
8.314
J
molK
(293.15 K) log(
1
0.3
) (348)
which simplify to
L = 916.9J . (349)

Chapter 2 (The First Law of Thermodynamics)
Problem 1
From the first law of thermodynamics in canonical form (dU = dQ − dW) and the energy
unit conversion between cal’s and ergs of
1cal = 4.185 107
erg ,
we obtain
dW = 3.4 108
erg (350)
dQ = 32 cal = 32(4.185 107
erg) = 1.339 109
erg . (351)
From these we obtain for dU
dU = 1.339 109
− 3.4 108
= 9.9 108
erg = 23.8 cal (352)
Problem 2
The work done by an isothermal expansion is given by Fermi Eq. 9
L =
m
M
RT log
p1
p2
=
m
M
RT log
V2
V1
. (353)
Since m
M
is the number of moles of the gas under consideration for the specifics for this
problem we have that
L = (3mol)(8.314 J/molK)(0 + 273.15 K) log
5
3
= 3.48 103
J = 831.54 cal .
Since we are considering an ideal gas whos internal energy is a function of temperature only
the first law of thermodynamics dU = dQ − dW simplifies since the expansion considered is
isothermal (dU = 0) to give
dQ = dW .
Because of this relationship, we have that the number of calories absorbed, or Q, is given by
Q = W = L which is computed above.
Problem 3 (a straight line transformation in the (V, p) plane)
For this problem we desire to compute the amount of work performed by the gas when it
undergoes a straight line transformation between (V0, p0) and (V1, p1) in the (V, p) plane. In
the specific case given in this problem the initial and final volumes (and temperatures) are

specified while the initial and final pressures must be computed from the equation of state
for an ideal gas. Using the ideal gas law to derive the corresponding pressure p0 we have
p0 =
8.314 J/molK 291 K 1 mol
2.1 10−2 m3
= 1.15 105
Pa .
In the same way we find
p1 = 1.99 105
Pa .
Since we have not derived an expression for the work done under a straight line transforma-
tion in the (V, p) plane we do so now and then use its result to evaluate this problem. The
work a gas performs under any transformation in the (V, p) plane is given by
W = L = p dV = p(V ) dV .
Where we have expressed the path in the (V, p) plane as a function of V explicitly with the
notation p(V ). When the path in the (V, p) plane is a line connecting the state (V0, p0) to
(V1, p1) can derive and explicit formula for the path by equating the slope at any point on
the line to the slope between the two end points as
p(V ) − p0
V − V0
=
p1 − p0
V1 − V0
.
On solving for p = p(V ) we obtain
p = p0 +
p1 − p0
V1 − V0
(V − V0) .
Thus the work the gas must do as it traverses this path is given by the integral
L =
V1
V0
p0 +
p1 − p0
V1 − V0
(V − V0) dV .
The algebra needed to perform this integration is
L = p0(V1 − V0) +
p1 − p0
V1 − V0
(V − V0)2
2
V1
V0
= p0(V1 − V0) +
p1 − p0
V1 − V0
(V1 − V0)2
2
= p0(V1 − V0) +
1
2
(p1 − p0)(V1 − V0) = (V1 − V0) p0 +
p1
2
−
p0
2
=
1
2
(V1 − V0)(p1 + p0) .
Now V1 − V0 < 0 and thus the work is negative and the environment performs work on
the system. Note also that the absolute value of the above expression is the area of the
triangle connecting the three states (V0, p0), (V1, p1), and (V1, p0) and the rectangle beneath
this triangle. Evaluating the above with the given values of p and V gives
L =
1
2
(1.27 − 2.1) 10−2
m3
(1.99 + 1.15) 105
Pa = −1.3 103
m3
Pa .
To calculate the heat absorbed by the system we remember that for an ideal gas U(T) = CV T
and a diatomic gas specifically has CV = 5
2
R so we have the change in internal energy given
by ∆U = CV ∆T = CV (T1 − T0) or
∆U = CV (305 − 291) = CV (14 K) =
5
2
(8.314J/molK)(14K)(1mol) = 2.9 102
J .

From the first law of thermodynamics for small changes we have ∆Q = ∆U + ∆W and we
obtain for the heat absorbed by the system
∆Q = 2.9 102
J + (−1.3 103
J) = −10.1 102
J = −1.01 1010
erg .
Where we used the unit conversion between ergs and Joules of 1J = 107
erg.
Problem 4
In this problem we have one mole of a diatomic gas undergoing an adiabatic volume expan-
sion. If we define the initial volume to be V0 the final volume V1 is then 1.35V0. For an
adiabatic transformation we have
TV K−1
= constant . (354)
Restricting this expression to connect the two states (T0, V0) and (T1, V1) we obtain
T0V K−1
0 = T1V K−1
1 (355)
Solving for the final temperature T1 gives
T1 = T0
V0
V1
K−1
. (356)
We are given the initial temperature of T0 = 18C = 18 + 273.15K = 291.15K, the knowledge
that the gas is diatomic (so K = 7
5
), and the fact that ratio of volumes is given by V0
V1
= 1
1.35
.
From this information we can compute that
V0
V1
K−1
= 0.507 . (357)
With this we finally compute the final temperature as T1 = 291.15K (0.507) = 147.6 K.
Chapter 3 (The Second Law of Thermodynamics)
Problem 1
Referring to figure, XXX, we first characterize what we know along each path in the p-V
plane and then develop the necessary mathematics. First, paths BD and CA are adiabatic
therefore no head flows through them. Second, along AB and DC the temperature does not
change and since for an ideal gas the internal energy U is a function of temperature only
U = U(T). Therefore along AB and DC dU = 0 so from the first law we have that dQ = dW
which implies Q = W.

We will solve this problem for the total work performed by the gas by computing the work
along each path in a clockwise fashion beginning with the path AB. To perform this calcula-
tion we will require the work performed by an ideal gas along an adiabatic expansion. This
will be calculated ﬁrst and used in the calculations that follow.
We begin by deriving the work performed during an adiabatic transformation between two
general points in p − V space (p1, V1) and (p2, V2). Along an adiabatic transformation of an
ideal gas we have
pV K
= constant , (358)
or anchoring the “constant” in the above expression to the point (p1, V1) we have
pV k
= p1V k
1 (359)
or solving for p = p(V ) we obtain
p = p1
V1
V
K
= p1
V
V1
−K
(360)
Thus the work between to points can be calculated in a straight forward manner as
W12 =
V2
V1
pdV
=
p1
V −k
1
V2
V1
V −k
dV
=
p1
V −k
1
V −k+1
−k + 1
V2
V1
=
p1V k
1
1 − k
V −k+1
2 − V −k+1
1 . (361)
We now begin with the calculations required for this particular problem. First the work
performed by our ideal gas along the isothermal path AB is given by
WAB = RT2 log(
VB
VA
) , (362)
which when evaluated, using the numbers given in the problem, gives the following
WAB = (8.314J/molK)(400K) log(
5
1
) = 5352.3J . (363)
Note that this is also equal to the heat absorbed by our engine Q2, at this state of the cycle.
Now along the path BD, since it is adiabatic we have in terms of variables of this problem
T2V K−1
B = T1V K−1
D . (364)
Solving for VD we have
VD =
T2
T1
V K−1
B
1/K−1
= VB
T2
T1
1/K−1
. (365)

Using the numbers from the text, we get that
VD = 5liters
400
300
1/K−1
.
Now for a monotonic gas K = 5/3 giving the volume in the D state the value
VD = 7.69 liter .
Using equation 361 we have that for a monotonic gas
WBD =
(6.65 105
Pa)(5 10−3
m3
)5/3
1 − 5
3
(7.69 10−3
m3
)1− 5
3 − (5 10−3
m3
)1− 5
3 = 1.24 103
J .
For practice, the units in this expression work out as follows
Pa m5
(m3
)− 2
3 = Pa m5
m−2
= Pa m3
=
N
m2
m3
= N m = J
Along the isothermal path DC we have that pV = constant giving in terms of the variables
of this problem the expression
pDVD = pCVC , (366)
which when we solve for pC the expression
pC = pD
VD
VC
= 4.32 105
Pa
7.69
1.539
= 2.158 106
Pa . (367)
Since CA is another adiabatic curve we have again that
T2V K−1
A = T1V K−1
C ,
or solving for VC the expression
VC =
T2
T1
1/K−1
VA = 1 liter
4
3
1/K−1
= 1.539 liter . (368)
Then the work along this segment of the cycle is given by
WCA = RT1 log(
VD
VC
) = (8.314J/molK)(300K) log(
7.69
1.539
) = 4.01 103
J . (369)
Note that this equals the heat released by the process through this segment of the cycle or
Q1. Finally, along the segment CA we have (using the expression for the work performed by
an adiabatic transformation derived above) the following
LCA =
2.158 106
Pa (1.539 10−3
m3
)5/3
5
3
(10−3
m3
)−2/3
− (1.539 10−3
m3
)− 2
3 = −1.65 103
J
So the total work performed during the entire cycle is given by
Wtotal = WAB + WBD + WDC + WCA = 9.27 103
J (370)

Problem 2
First, convert the given temperatures into Kelvin as follows
T1 = 18 C = 291.15 K (371)
T2 = 400 C = 673.15 K , (372)
then using Fermi Eq. 59 we have a maximum possible eﬃciency between these two temper-
atures of
η = 1 −
T1
T2
= 1 −
291.15
673.15
= 0.567 . (373)
Problem 3
Converting the given temperatures into Kelvin we have
0 F = 255.92 K = T1 (374)
100 F = 310.92 K = T2 . (375)
Then the minimum amount of work will be achieved when the engines operating cycle is
reversible. As such, from Fermi Eq. 60. we have
L =
T2 − T1
T1
Q1 . (376)
With the heat extracted Q1 = 1 cal we obtain, in various units
L = 0.2149 Q1
= 0.2149 cal
= 0.2149 4.185 107
erg
= 8.99 106
erg
= 0.899 J .
Where in the above conversions we have used the conversion that 1 erg = 10−7
J.
Chapter 4 (The Entropy)
Problem 1
We begin by recognizing that 1kg H2O = 1000 gm H2O and converting the given temperatures
to Kelvin as follows
T1 = 0 + 273.15 K = 273.15 K
T2 = 100 + 273.15 K = 373.15 K

For liquid’s I’ll make the assumption that Cp ≈ CV and is constant. As given in the book
we will take its value to be 1cal/gm. A change in entropy along a reversible path is given by
∆S =
B
A
dQ
T
. (377)
From the first law we have dU = dQ − pdV and a constant volume transformation results
in dU = dQ = CV dT. Since along a constant volume transformation we have the heat ratio
defined by
dQ
dT V
= CV (378)
Putting this expression into equation 377 results in
∆S =
T2
T1
CV dT
T
= CV ln(
T2
T1
) . (379)
Which is an expression to be understood as per unit mass. Using the numbers provided for
this problem gives for the entropy change
∆S = (1000 gm)(1cal/gmK) ln(
373.15
273.15
)
= 311.9 cal/K = 311.9 (4.185 107
erg/K)
= 1.305 1010
erg/K
Problem 2
We are told that a body obeys the following equation of state
pV 1.2
= 109
T1.1
,
and asked to find the energy and entropy of such a system as a function of T and V . To
solve this problem in a more general setting consider a body that has an equation of state
given by
pV a
= dTb
(380)
With a = 1.2, d = 109
, b = 1.1, and CV = 0.1 cal/deg for this problem. Now the units of V
are liters. Now consider U = U(V, T) then one can show for a system performing p-V work
that
∂U
∂V T
= T
∂p
∂T V
− p ,
which is a nice expression because once an equation of state is specified empirically one can
compute/evaluate
∂U
∂V T
,
by explicitly inserting the given equation of state. In our case
p =
d Tb
V a
, (381)

so the required derivative is given by
∂p
∂T V
=
d b Tb−1
V a
, (382)
therefore
∂U
∂V T
=
dbTb−1
V a
−
dTb
V a
= (b − 1)
dTb
V a
= (b − 1)p . (383)
From this expression integrating U holding T fixed gives the following expression for U(V, T),
in terms of an arbitrary function of temperature C1
U(V, T) = (b − 1)dTb V −a+1
−a + 1
+ C1(T) .
To evaluate C1(T) consider the definition of CV (evaluated at V0 = 100 liters)
CV ≡
∂Q
∂T V
=
∂U
∂T V
= (b − 1)dbTb−1 V 1−b
1 − a V0
+ C′
1(T) .
which gives for C′
1(T) the following
C′
V (T) = CV − (b − 1)dbTb−1 V 1−b
0
1 − a
.
Integrating with respect to T we get
C1(T) = CV T − (b − 1)dTb V 1−b
0
1 − a
+ C2
with C2 a constant. Then U(V, T) is given by
U(V, T) = CV T + (b − 1) d Tb (V 1−a
− V 1−a
0 )
1 − a
+ C2 (384)
To compute the entropy remember that for an reversible transformation we have
dS =
dQ
T
=
dU + pdV
T
=
1
T
dU +
1
T
p dV . (385)
Using the expression for dU from differential calculus
dU =
∂U
∂T V
dT +
∂U
∂V T
dV ,
in the above expression for dU we obtain for dS
dS =
1
T
CV + (b − 1)
dbTb−1
1 − a
(V 1−a
− V 1−a
0 ) dT + (b − 1)dTb
V −a
dV +
1
T
dTb
V a
dV

or manipulating this expression we have
dS =
CV
T
dT +
(b1)dbTb−2
1 − a
(V 1−a
− V 1−a
0 )dT +
1
T
(b − 1)dTb
V −a
dV +
1
T
dTb
V a
dV
= CV
dT
T
+
bdTb
V −a
T
dV +
b(b − 1)Tb−2
dV 1−a
1 − a
dT −
(b − 1)dbV 1−a
0 Tb−2
1 − a
dT
= CV
dT
T
−
(b − 1)dbV 1−a
0 Tb−2
1 − a
dT + bdTb−1
V −a
dV +
d2
Tb
dT2
dV 1−a
1 − a
dT
= CV
dT
T
− b(b − 1)d
V 1−a
0
1 − a
Tb−2
dT +
d2
(Tb
)
dT2
d
V 1−a
1 − a
dT + d
d(Tb
)
dT
V −a
dV
We can verify our algebra by checking if this is an exact differential as it must be. As such
we must have
∂
∂V
CV
T
+ . . . +
d2
(Tb
)
dT2
d
V 1−a
1 − a
=
∂
∂T
d
d(Tb
)
dT
V −a
which becomes
d
d2
(Tb
)
dT2
V −a
= d
d2
(Tb
)
dT2
V −a
(386)
which is a true statement. Therefore one can integrate dS if one can find an integrating
factor. Since dS is an exact differential there exists a function S(T, V ) such that
∂S
∂T V
= d
d2
Tb
dT2
V 1−a
1 − a
+
CV
T
− b(b − 1)
dV 1−a
0
1 − a
Tb−2
(387)
and
∂S
∂V T
= d
d(Tb
)
dT
V −a
. (388)
The second expression gives that
S = d
d(Tb
)
dT
V 1−a
1 − a
+ F(T) , (389)
for an arbitrary function F(T). Taking the temperature derivative of the above gives
∂S
∂T V
= d
d2
(Tb
)
dT2
V −a
1 − a
+ F′
(T)
and setting this equal to equation 387 gives for F′
(T)
F′
(T) =
CV
T
− b(b − 1)Tb−2 V 1−a
0
1 − a
d
which can be integrated to give for F(T)
F(T) = CV ln(T) − bTb−1 V 1−a
0
1 − a
d + S0 .
In total we have for S(T, V ) the following
S(T, V ) = CV ln(T) +
dbTb−1
1 − a
(V 1−a
− V 1−a
0 ) + S0 (390)

Problem 3
From Clapeyron’s equation we have
dp
dT
=
λ
T(V2 − V1)
(391)
Here V2 is the specific volume of the gas phase and V1 is the specific volume of liquid phase.
In Kelvin, the boiling temperature for ethyl alcohol is given by T = 351.45K. Assuming
V1 ≪ V2 and that the value of V2 for ethyl alcohol is approximately the same as for H2 O of
1677cm3
gm
Clapeyron’s equation becomes
dp
dT
=
855 J
gm
(351.45K)(1677 10−6 m3
gm
)
= 1.45 103 J
m3 K
= 1.45 103 Pa
K
.
Chapter 5 (Thermodynamic Potentials)
Problem 1
WWX: I have not finished proofreading this section ... start this is page 11 from
the second set of scanned notes
The phase rule v = 2+n−f, saturated solution and a solid dissolved in the in the substitute
have two phase liquid solution and solid (know this is correct page 86 gives examples of salt
in H2 0 and two components n = 2 ( a solid and a liquid component ).
v = 2 + 2 − 2 = 2 (392)
therefore we can specify 2 variables T and p arbitrary. I would think that the correct answer
wold be T only. I know that increasing the temperature increases the solvability but am not
sure about the pressure)
Problem 2
I am told the amounts of H2 O and air. I’ll assume air contains N2, O2, and H2 only and
H2 O vapor, then f equals the number of phases (which is 2), and n equals the number of
components which is 4. We have that
v = 2 + 4 − 2 = 4 (393)
But we are told the amount of H2 0. and air so we are told

Problem 3
Now
v(t) = v0 + v1t + v2t2
(394)
v0 = 924 (395)
v1 = 0.0015 (396)
v0 = 0.0000061 (397)
The units of T are centigrade, the units of V are volts, and the units of e are Columnbs,
from Page 96 we have
dQ = dU + dL = −eU(T) + ev(T) (398)
but from the equation of Helmholtz u(T) = v − T dv
dT
, thus the equation of Helmholtz gives
the a functional form for the energy lost per unit charge.
u(T) = v0 + v1t + v2t2
− T (v1 + 2v2t) (399)
= (v0 + Tv1) + (v1 − 2v2T)t + v2t2
(400)
Then dQ = −e(v − T dv
dT
) + ev = eT dv
dT
therefore we need u(T) actually
∆Q = eT
dv
dT
= eT(v1 + 2v2t) (401)
Chapter 6 (Gaseous Reactions)
Chapter 7 (The Thermodynamics of Dilute Solutions)
Chapter 8 (The Entropy Constant)
WWX: I have not ﬁnished this section ... end

Manual Solution Fermi Thermodynamics

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