Page 35 - Essentials of physical chemistry

Introduction: Mathematics and Physics Review xxxiii

Here we introduce another silly but useful mnemonic. The number 4.184 is easy to remember but
which unit is larger? ‘‘A calorie is worth many jewels!’’ Using the rhyming word ‘‘jewels’’ in place
of ‘‘joules’’ conveys greater numerical value on the calorie unit so we can remember that in some
absolute amount of energy a single calorie is ‘‘worth’’ (larger than) many (4.184) joules. This silly
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process is important for a student to develop a feeling for a physical quantity, for instance that 1 cm
is about the size of a boullion cube of condensed soup and 18C is one hundredth of the temperature
difference between ice water and boiling water (at 1 atmosphere). It is important to reason out the
physical units in a calculation rather than to use numbers blindly on a calculator.
ð ð
dW ¼ F x dx þ F y dy þ F z dz so we can integrate this to dW ¼ F(x)dx

A related concept is ‘‘Power’’: Power (Work=time) but that unit is more useful for electrical
measurements than in physical chemistry.
Energy can be ‘‘kinetic’’ or ‘‘potential.’’ Note that work is energy:
v
ð x
2
ð ð ð ð
dv x dx v x
W ¼ F x dx ¼ ma x dx ¼ m dx ¼ m(dv x ) ¼ mv x dv x ¼ m T x
dt dt 2
0
so that work can be in the form of T, which is the usual symbol for kinetic energy (although you may
have used ‘‘K’’ in sophomore physics)
Another useful concept in much of physical chemistry is the conservation of energy. The main
associated concept is that a force is the negative derivative of some potential.() means ‘‘implies’’)
qV qV qV
.
qx qy qz
V(x, y, z) ) F x ¼ , F y ¼ , and F z ¼
For simplicity, let us just consider the x-component of the energy.
v

ð ð 2 2 v 2
mv x
W x ¼ mv x dv x ¼ m v x dv x ¼ ¼ T 2 T 1 :
2
v 1
v 1
ð ð
qV V 2
dV ¼ (V 2 V 1 ) ¼ (V 1 V 2 ).
But we also know that W x ¼ dx ¼
qx
V 1
Then since we have computed W x two different ways, we can equate the kinetic and potential
forms of work W x ¼ T 2 T 1 ¼ V 1 V 2 .
And so as long as there are potentials whose negative derivatives in space form forces (energy
goes down hill!) we have the equality: T 2 þ V 2 ¼ T 1 þ V 1 .
This important result tells us that energy can change form but the total value remains the same,
that is, conservation of energy!
Some important units we will need as well seem random but they are necessary:
1 hour ¼ 60 minutes ¼ 3600 seconds
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1 mile ¼ 5,280 feet ¼ 1.6093 kilometers ¼ 1.6093 10 cm
1 inch ¼ 2.54 cm
1 pound ¼ 453.6 grams (only on Earth, a pound is a force while a gram is a mass)
1 pound (Avoirdupois) ¼ 16 ounces
F (degrees) ¼ (9=5)C (degrees) þ 32 (degrees)
1 calorie ¼ 4.184 joules

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Another important point for fractions or units is that ¼ 3; that is the denominator of the
1=3
denominator can ‘‘flip up’’ to the numerator and that can be very useful.