Math 7
Notes
Writing and Solving Inequalities
Review: Inequality symbols and the phrases/words they represent.
0 less than c greater than . less than or equal to o gr
Math 7
Homework
Write and solve an inequality to represent each situation.
1. DRIVING Louella is driving from Melbourne to
Pensacola, a distance of more than 500 miles. After
driving 240 miles, Louell
Eyewitness Video: Volcano
1. A volcano is the product of a _ planet.
2. _ are generated by some of the same forces that create
volcanoes.
3. In 79 A.D., Mt. _ erupted, buried Pompey.
4. How were the w
The Boltzmann Factor and Partition Functions 533
171 2. An approximate partition function for a gas of hard spheres can be obtained from the partition
function of a monatomic gas by replacing V in Equ
Partial Derivatives 523
Because these derivatives are equal, dx/T is an exact differential.
H12. Prove that
1 BY _1 a?
Y 3P 7 BP T
(BF) (BF)
_ =71
aY T aY T.n
where Y = Y(P, T, n) is an extensive var
The Boltzmann Factor and Partition Functions 53]
178. ll~ Nw is the number of protons aligned with a magnetic eld B2 and N0 is the number of protons
opposed to the eld, show that
finZ/kBT
N0
_ =3
NW
G
Partial Derivatives 5] 9
H4. Given that
U : kTZ (aln Q)
3T ~_v
1 2nkaT 2 N
where
and k3, m, and h are constants, determine U as a function of T.
We are given
U = kT2<81n Q)
8T AH,
and
27rkaT)3N/2
The Properties of Gases
We compare this with Equation 16.22,
B T B T _
Z=1+ ZVVU+ 3L(,)+0(V3)
V
Setting the coefcients of 1 /l7 and 1 /72 equal to one another gives
a
Bzv=
and
20113
B = 2
3" +RT
522 Mdlllclhlpkr ll
Thcn
(3210) _0 3 A
W V 4 V(V+B)r5/2
Substituting into the expression from Problem H8 gives
(acv) _T(82P> _ 3 A
av ,_ 8T2 v" 4 V(V+B)Tm
H10. Is
dV = yrrzdh + 27rrhdr
an exact or
The Boltzmann Factor and Partition Functions 529
Therefore, 2 cosh u = e + e. The partition function in Example 171 is
h B
Q(7 Bl) = ehyBZ/Z +ehyBZ/2 = 2COSh '8 y z
We use Equation 17.20 to writ
The Properties of Cases
Find the reduced parameters of each gas by dividing T by E/kB and BZV by 27ra3NA/3 (Table 16.7).
Below, we plot B;V(T) versus T for each gas.
0.5 I I I
U
0.0 In: DA G
A .u'
[:1
53 6 Chapter 17
and Equation 17.43 becomes
e ale 135E
g = 8_ = - =
( ) Z 1 1+ el 1+e
where we have dropped the subscript "1" on 8 in both of the above expressions for simplicity. We
can use Equation 1
5 3 4 Chapter 17
1715. Using the partition function given in Example 172, show that the pressure of an ideal diatomic
gas obeys P V = NkB T, just as it does for a monatomic ideal gas.
We can use the p
5] 8 MathChapler H
For an ideal gas, PV = nRT. Taking the partial derivative of both sides of this equation with
respect to T gives
P (a) = nR
8T P
3 V _ nR
(a), - 7
18V _nR
V BTP PV
OF
1
05:
T
H3.
508
Chapter 1 6
1644. Use the following data for argon at 300.0 K to determine the value of BZV. The accepted value
is 15.05 cm3-mol.
P/atm p/mol-L" P/atm p/mol-L"
0.01000 0.000406200 0.4000 0.0162535
Partial Derivatives 52]
H8. Given that the heat capacity at constant volume is dened by
C_8U
V_8TV
and given the expression in Problem H7, derive the equation
aCV _ aZP
(av >T T (WV
3U
Ha),
e) W) P 2
The Boltzmann Factor and Partition Functions 535
We can substitute this value into Equation 17.27 to obtain the expression
EV 5 2240 K 2 e-WK/T
W = E + T (1 _ e2240 K/T)2
We plot EV versus T below.
510
Chapter 16
The data points for all three gases fall on the same curve, consistent with the law of corresponding
states.
1648. In Section 164, we expressed the van der Waals equation in reduced u
532
Chapter 17
Substituting this last result into Equation 17.21 gives
SN 3
(E) =kBT2 = E(NkBT)
1710. A gas absorbed on a surface can sometimes be modelled as a twodimensional ideal gas. We
will lea
CHAPTER 1 7
The Boltzmann Factor and Partition Functions
PROBLEMS AND SOLUTIONS
171 . How would you describe an ensemble whose systems are oneliter containers of water at 25C?
An unlimited number of
The Boltzmann Factor and Partition Functions 53 7
This summation is easy to evaluate if you recognize it as the so-called geometric series
(MathChapter I)
>2
=0 l x
Show that
ehv/Z
qho(T) = 1 _ [W
514
Chapter 16
Expanding the fractions 1 /(1 B /V) and 1 /(1 + B N) (Equation 1.3) gives
RT B _
P=:[1+=+0(V 3]
B[15E0W
v v *2 ]
RT A 82 _._3
V TVZV V TVA2V V v
We then use the denition of Z to nd th
MATHCHAPTER I I
Partial Derivatives
PROBLEMS AND SOLUTIONS
H1 . The isothermal compressibility, KT, of a substance is dened as
1 8V
7:7 5?
T
Obtain an expression for the isothermal compressibility o
5 3 0 Chapter 17
AsT>0,>oo,so
h B h B h B
lim(E)=lim y Ztanh y 23: VI
T>0 >oo 2 2 2
AsT>oo,f3>0,so
h B h B
lim(E)=lim y Ztanh y 13
T>oo fl>0 2 2
_-_
177. Generalize the results of Example 17
520 MathChapter H
Then
R RT A2T/~ B _
VB 2T-/ v(V+B) (V~B) T1/2V(V+B)2
H6. Show explicitly that
< 82F >_( BZP >
War _ aTaV
for the Redlich-Kwong equation (Problem HS).
We found expresions for (B
The Properties of Cases 5] 3
For identical atoms or molecules, 11 = 12 = 1, a2 = 011 = a, and 1 = 122 = 12, and so the sum
becomes
21134 31012
2 2
( (1+ 3k T + 4 >
B
2(u) =
2 6
(47180) r
The coef
5 2 8 Chapter 1 7
Likewise, we differentiate with respect to y and nd
[alnf(x+y)] 20+ [81nf(y):|
M W
dlnf(x+y) [8(x+y):| _dlnf(y)
d(x+y) 3y I dy
dlnf(x+y)=dlnf(y) (2)
d(x+y) dy
Equations 1 and 2 are e