Page 46 - Physical chemistry understanding our chemical world

THE PRACTICE OF THERMODYNAMIC MEASUREMENT 13

We should note, before proceeding, firstly how the units of on both top and
◦
bottom of this fraction cancel; and secondly, how C −1 is in the denominator of the
fraction. As a consequence of it being on the bottom of the fraction, it is inverted
◦
and so becomes C. In summary, we see how a simple analysis of the units in this
◦
sum automatically allows the eventual answer to be expressed in terms of C. We are
therefore delighted, because the answer we want is a temperature, and the units tell
us it is indeed a temperature.

Aside

This manipulation of units is sometimes called dimensional analysis. Strictly speaking,
though, dimensional analysis is independent of the units used. For example, the units
of speed may be in metres per second, miles per hour, etc., but the dimensions of speed
are always a length [L] divided by a time [T]:

[speed] = [L] ÷ [T]

Dimensional analysis is useful in two respects. (1) It can be used to determine the
units of a variable in an equation. (2) Using the normal rules of algebra, it can be used
to determine whether an equation is dimensionally correct, i.e. the units should balance
on either side of the equation. All equations in any science discipline are dimensionally
correct or they are wrong!
A related concept to dimensional analysis is quantity calculus, a method we find
particularly useful when it comes to setting out table header rows and graph axes.
Quantity calculus is the handling of physical quantities and their units using the normal
rules of algebra. A physical quantity is defined by a numerical value and a unit:

physical quantity = number × unit

e.g.

H = 40.7 kJ mol −1

which rearranges to
H/kJ mol −1 = 40.7

SAQ 1.1 A temperature is measured with the same platinum-resistance
thermometer used in Worked Example 1.1, and a resistance R = 11.4 ×
10 −4 determined. What is the temperature?