Page 30 - Essentials of physical chemistry

xxviii Introduction: Mathematics and Physics Review

taken such a course, so we have to introduce the basics here. It really is very simple in that partial
derivatives use the same rules as one-dimensional derivatives except that the other variables are
held constant!
Consider the ideal gas law, which should be familiar from freshman chemistry:
nRT nRT PV PV
PV ¼ nRT, P ¼ , V ¼ , T ¼ , and n ¼ :
V P nR RT
Although we know R is a constant, there are four variables that have an effect on the overall state of
an ideal gas.
There are actually 12 possible partial derivatives for the state of a sample of gas but we will only
show 3 here, the others are easy to understand considering each variable in turn. Let us treat the
possible partial derivatives of P.

qP nR qP nRT qP RT
, ¼ , and ¼
¼
qT V qV V 2 qn V
n,V n,T V,T
Thus we see the pressure of an ideal gas depends separately on three other variables and the
partial derivatives could be rendered into a numerical value by using the values of the other
variables. Strictly speaking, the lower right subscripts should be included to indicate all the
other variables are being held constant during the derivative evaluations, but if the moles are
understood to be constant in a given problem, n is oftenneglected.Evensoany lowersubscript
can be used to remind us what conditions are held constant. It is important to learn to verbalize

qP
the partial derivatives to give them physical meaning. For instance couldbereadas
qn V,T
‘‘the change in P with a change in moles while holding volume and temperature constant.’’
A student should find more meaning if the partial derivatives are verbalized but the strategy is to
use quantities that are measurable and then use mathematics to manipulate them in ways that
lead to new information.
There are two physical quantities that can be measured in the laboratory, which can be tabulated
in various handbooks and used in a number of ways to simplify other equations.

1 qV
. Note ‘‘isobaric’’ means P is constant.
Isobaric thermal expansion coefficient: a
V qT
P
Here we note this represents how the volume changes as the temperature changes while holding
the pressure constant. To make the measurement independent of the amount we divide by the total
volume, and this is a positive number. For solids and liquids, this is usually a small numerical value
(with units), but it can be large for gases. For an ideal gas we have

1 nR 1
:
V P T
a ¼ ¼

1 qV
. Note ‘‘isothermal’’ means constant T.
V qP
Isothermal compressibility coefficient: b
T
The b value needs some thought because most substances are compressed when the pressure is
increased so the amount by which the volume changes when the pressure increases is negative. Thus the
definition includes a minus sign so that the tabulated values will be positive. Nevertheless, the physical
phenomenon is that most materials will be compressed to smaller volume as the pressure increases.
Recall that in nature (P, V, T) variables are (seemingly) independent of each other for a given
quantity of gas n. Thus we could plot the three variables on an (x, y, z) grid. A useful cyclic rule can be
derived with some thought and a trick using the differential quantities. Although we
can plot independent variables on a grid, there may actually be some state function that relates