Page 48 - Physical Chemistry

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Section 1.8
where the subscripts on the differentials denote changes at constant T.Wewant to Integral Calculus
find V. Therefore, we need to integrate this equation. The two variables are V and
P, since T is constant. To integrate, we need to first separate the variables, putting
everything that depends on V on one side and everything that depends on P on the
other side. k is an intensive quantity that depends on T and P, and T is constant, so
k belongs on the P side, as is obvious from the equation for k given in the state-
ment of the problem. To separate the variables, we multiply (1.65) by dP to get
T
1
k dP dV
T T
V
Next, we integrate both sides from the initial state P , V to the final state P , V ,
1 1 2 2
where P , V , and P are known, and T is constant:
1 1 2
V 2 1 P 2 P 2 2
dV k dP 1a bP cP 2dP
V 1 V P 1 P 1
1
3
1
2
ln V0 V 2 1aP bP cP 2 0 P 2
V 1 2 3 P 1
1 2 2 1 3 3
1ln V ln V 2 ln1V >V 2 a1P P 2 b1P P 2 c1P P 2
2
1
1
2
2
1
1
2
2
3
1
2
3
6
1
ln311.002961 cm 2>V 4 45.259 10 bar 1400 bar2
2
1
2
8
2
2
11.1706 10 bar 21401 1 2bar 2
2
1
3
3
3
12.3214 10 12 bar 21401 1 2bar 3
3
3
ln311.002961 cm 2>V 4 0.0172123
2
3
11.002961 cm 2>V 1.017361
2
V 0.985846 cm 3
2
3
which agrees with the true value 0.985846 cm .
Exercise
A liquid with thermal expansivity a is initially at temperature and volume T 1
and V . If the liquid is heated from T to T at constant pressure, find an
1
1
2
expression for V using the approximation that a is independent of T.
2
[Answer: ln V ln V a1T T 2.4
2
2
1
1
Exercise
For liquid water at 1 atm, thermal-expansivity data in the range 25°C to 50°C
2
are well fitted by the equation a e f1t>°C2 g1t>°C2 , where t is the
5
5
1
1
Celsius temperature, e 1.00871 10 K , f 1.20561 10 K , and
1
8
g 5.4150 10 K . The volume of one gram of water at 30°C and 1 atm
3
is 1.004372 cm . Find the volume of one gram of water at 50°C and 1 atm.
3
3
Compare with the experimental value 1.012109 cm . (Answer: 1.012109 cm .)
Logarithms
Integration of 1/x gives the natural logarithm ln x. Because logarithms are used so often
in physical chemistry derivations and calculations, we now review their properties. If
s
s
x a , then the exponent s is said to be the logarithm (log) of x to the base a: if a
x, then log x s. The most important base is the irrational number e 2.71828...,
a
defined as the limit of (1 b) 1/b as b → 0. Logs to the base e are called natural