Page 36 - Physical Chemistry

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Section 1.6
weight of F is 38.0, and its molar mass is M 38.0 g/mol. The absolute temper- Differential Calculus
2
ature is T 20.0° 273.15° 293.2 K. Since we know a value of R involving
atmospheres, we convert P to atmospheres: P (188 torr) (1 atm/760 torr)
0.247 atm. Then
1
MP 138.0 g mol 210.247 atm2
4
r 3.90 10 g>cm 3
1
1
3
RT 182.06 cm atm mol K 21293.2 K2
Note that the units of temperature, pressure, and amount of substance
(moles) canceled. The fact that we ended up with units of grams per cubic cen-
timeter, which is a correct unit for density, provides a check on our work. It is
strongly recommended that the units of every physical quantity be written down
when doing calculations.
Exercise
Find the molar mass of a gas whose density is 1.80 g/L at 25.0°C and 880 torr.
(Answer: 38.0 g/mol.)

1.6 DIFFERENTIAL CALCULUS
Physical chemistry uses calculus extensively. We therefore review some ideas of dif-
ferential calculus. (In the novel Arrowsmith, Max Gottlieb asks Martin Arrowsmith,
“How can you know physical chemistry without much mathematics?”)

Functions and Limits
To say that the variable y is a function of the variable x means that for any given
value of x there is specified a value of y; we write y f(x). For example, the area of
a circle is a function of its radius r, since the area can be calculated from r by the
2
expression pr . The variable x is called the independent variable or the argument of
the function f, and y is the dependent variable. Since we can solve for x in terms of
y to get x g(y), it is a matter of convenience which variable is considered to be the
independent one. Instead of y f(x), one often writes y y(x).
To say that the limit of the function f(x) as x approaches the value a is equal to c
[which is written as lim f(x) c] means that for all values of x sufficiently close to
x→a
a (but not necessarily equal to a) the difference between f(x) and c can be made as
small as we please. For example, suppose we want the limit of (sin x)/x as x goes to
zero. Note that (sin x)/x is undefined at x 0, since 0/0 is undefined. However, this
fact is irrelevant to determining the limit. To find the limit, we calculate the following
values of (sin x)/x, where x is in radians: 0.99833 for x 0.1, 0.99958 for x
0.05, 0.99998 for x 0.01, etc. Therefore

sin x
lim 1
xS0 x
Of course, this isn’t a rigorous proof. Note the resemblance to taking the limit as P → 0
in Eq. (1.15); in this limit both V and V become infinite as P goes to zero, but the limit
tr
has a well-defined value even though q/q is undefined.
Slope
The slope of a straight-line graph, where y is plotted on the vertical axis and x on the
horizontal axis, is defined as (y y )/(x x ) y/ x, where (x , y ) and (x , y )
2 1 2 1 1 1 2 2
are the coordinates of any two points on the graph, and (capital delta) denotes the