Page 31 - Essentials of physical chemistry

Introduction: Mathematics and Physics Review xxix

them in some dependent way. In the case of a given amount of ideal gas (P, V, T), the ideal gas equation
relates thevariables,butwecanbemore general using justverbal descriptionofthe(x,y,z)coordinates.

qz qz
dy:
qx qy
dz ¼ dx þ
y x
This is a very general differential that can be read as ‘‘the change in z equals how much z changes
when x changes, holding y constant, multiplied by the amount of change in x plus how much z
changes when y changes, holding x constant, multiplied by the amount of change in y.’’ A student
should try to read the meaning of the partial derivatives and the total differential before blindly
performing correct algebraic manipulations, although that manipulation may be desirable to lead to
a new result. Now consider that z is held constant so that dz ¼ 0. That leads to a new result that may
be useful later and is given here to illustrate how partial derivatives can be manipulated; after all, the
derivatives are really ratios of small numbers.

" #

qz qz
dy holding z constant:
qx qy
0 ¼ dx þ
y x
z

qx
Next we can use algebra to form from the individual differentials and noting the condition
qy
z

qz dx qz dy
that z is constant. Then ¼ by dividing through by dy. Note that at this
qx dy qy dy
y z x z

dx qx dy
point d becomes q so ¼ and of course ¼ 1.
dy qy dy
z z z

qy
Now multiply both sides by and rearranging into alphabetical order (x, y, z) (the order in
qz
x

qx qy qz
the product can be permuted) to find ¼ 1.
qy z qz x qx y
If we plot the independent variables (P, V, T ) on the (x, y, z) axis and assume there really is some
sort of state function that connects them somehow, then we can generalize the discussion earlier to a
similar cyclic relationship in terms of (P, V, T)as

qP qT qV
¼ 1:
qT qV qP
V P T
This expression can also be rearranged to use a and b. Thus we can rearrange the cyclic rule and

1=V
multiply by the factor of , which completes the definitions of a and b to obtain
1=V

qV 1
qP qT P V a
¼ ¼ :
qV
1
qT V qP T V b
This illustrates the sort of algebraic manipulations that are common in thermodynamics. Note that in
this case, we have assumed there is some overarching connection in the form of a state function that
relates P, V, and T (for a fixed value of moles), but formally it is not necessary to specify the state
function as long as the various partial derivatives exist.
While we are stretching your mind with the basics of mathematics that you will need for the
remainder of the text, these examples should rapidly introduce you to what we will need without
spending two or three more semesters in mathematics courses. All that can be said here is that