Geology Reference
In-Depth Information
Equations A8 and A9 have the same form, in which
the separation distance
r
appears as 1/
r
2
. Both are
examples of an algebraic relationship which - for this
reason - is called an
inverse-square law.
2
in cm
3
mol
−1
would make it necessary to introduce a
correction factor of 10
6
.
Some physical parameters are 'dimensionless' and
therefore have no units. They are pure numbers.
Specific
gravity
(density of a substance ÷ density of pure water
at 4°C) is an example. The numerical values of such
numbers are independent of the units being used in
their computation. Thus in calculating specific gravity,
the units of density cancel out, provided that the two
densities are expressed
in the same units.
Graphs must show the units in which each of the
variables is expressed. Current good practice
3
followed
in this topic is to label each axis in the form 'quantity/
units', e.g.
T
/°C.
Dimensions and units in calculations
In writing the answer to any numerical problem, one
should give two items: the numerical answer
and
its
units. The number by itself is incomplete (unless it is a
dimensionless number
).
One must pay attention to units at every stage in a
calculation, checking that all the quantities are
expressed in compatible units. If one variable is entered
in millimetres when it should appear in metres, you
are immediately introducing an error of 1000 times.
The procedure for carrying units through a calcula-
tion can be illustrated by the Clapeyron equation
(Chapter 2). Suppose we wish to know the slope of the
phase boundary representing the reaction between
two (isochemical) minerals A and B:
Experimental verification of a theoretical
relationship
When a mathematical equation is proposed (com-
monly on theoretical grounds) to describe a phenom-
enon, one often wishes to test it against available
experimental observations. Does it describe the experi-
mental results accurately, or would some other form of
equation match the experimental results more closely?
The simplest way to answer this question is to plot the
experimental data and the theoretical equation together
in a suitable graph.
It is important that this be done in such a way that
the form of the graph is linear. To see why, consider the
verification of the Arrhenius equation relating rate
constant to absolute temperature:
Table A4
Molar entropy and volume.
Reaction
A
→
B
Units in published tables
of S and V
Molar entropies:
Molar volumes:
s
A
s
B
v
A
v
B
J K
−
1
mo1
−
1
m
3
mol
−
1
∆
∆
SS S
VVV
=−
=−
J K
−1
mo1
−1
m
3
mol
−1
B A
B A
E
RT
or kAe
ERT
−
/
kA
=
exp
−
a
=
(A10)
a
In the Clapeyron equation:
−
1
−
1
d
d
P
T
∆
∆
S
V
JK mol
mmol
If we were to plot experimental results for
k
against
T,
the results would define a curve as in Figure A3.
Unless we happened to know the constants
A
and
E
a
in advance - and in general we don't - it would be
difficult to determine whether the curve defined by
the experimental data has the shape predicted by
the equation. It would be necessary to use a compli-
cated curve-fitting calculation to establish the agree-
ment between experimental data and the theoretical
equation.
=
3
−
1
'mol
−1
' appears on top and bottom, and therefore can-
cels out, leaving the units for the gradient as (J K
− 1
) m
− 3
=
J m
− 3
K
− 1
= N m
− 2
K
− 1
= Pa K
− 1
. Using volumes expressed
2
Note that when we refer to an equation like A8 as a scientific
'law', it should not be taken to mean that this equation actually
governs
what Nature does: it is simply shorthand for saying that
the equation provides our best algebraic
approximation
to what is
seen to happen in the natural world. Any such 'law' is a human
construct.
3
See www.animations.physics.unsw.edu.au/jw/graphs.htm#Units.
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