Page 37 - Physical Chemistry

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Chapter 1 change in a variable. If we write the equation of the straight line in the form y mx
Thermodynamics
b, it follows from this definition that the line’s slope equals m. The intercept of the
line on the y axis equals b, since y b when x 0.
The slope of any curve at some point P is defined to be the slope of the straight
line tangent to the curve at P. For an example of finding a slope, see Fig. 9.3. Students
sometimes err in finding a slope by trying to evaluate y/ x by counting boxes on the
graph paper, forgetting that the scale of the y axis usually differs from that of the x axis
in physical applications.
In physical chemistry, one often wants to define new variables to convert an equa-
tion to the form of a straight line. One then plots the experimental data using the new
variables and uses the slope or intercept of the line to determine some quantity.

EXAMPLE 1.2 Converting an equation to linear form

According to the Arrhenius equation (16.66), the rate coefficient k of a chemical
reaction varies with absolute temperature according to the equation k Ae E a >RT ,
where A and E are constants and R is the gas constant. Suppose we have mea-
a
sured values of k at several temperatures. Transform the Arrhenius equation to
the form of a straight-line equation whose slope and intercept will enable A and
E to be found.
a
The variable T appears as part of an exponent. By taking the logs of both
sides, we eliminate the exponential. Taking the natural logarithm of each side of
k Ae E a >RT , we get ln k ln1Ae E a >RT 2 ln A ln1e E a >RT 2 ln A E /RT,
a
where Eq. (1.67) was used. To convert the equation ln k ln A E /RT to a
a
straight-line form, we define new variables in terms of the original variables k
and T as follows: y ln k and x 1/T. This gives y ( E /R)x ln A.
a
Comparison with y mx b shows that a plot of ln k on the y axis versus 1/T
on the x axis will have slope E /R and intercept ln A. From the slope and
a
intercept of such a graph, E and A can be calculated.
a
Exercise
The moles n of a gas adsorbed divided by the mass m of a solid adsorbent often
varies with gas pressure P according to n/m aP/(1 bP), where a and b are
constants. Convert this equation to a straight-line form, state what should be
plotted versus what, and state how the slope and intercept are related to a and b.
(Hint: Take the reciprocal of each side.)

Derivatives
Let y f(x). Let the independent variable change its value from x to x h; this will
change y from f(x) to f(x h). The average rate of change of y with x over this inter-
val equals the change in y divided by the change in x and is

¢y f1x h2 f1x2 f1x h2 f1x2

¢x 1x h2 x h
The instantaneous rate of change of y with x is the limit of this average rate of change
taken as the change in x goes to zero. The instantaneous rate of change is called the
derivative of the function f and is symbolized by f :
f1x h2 f1x2 ¢y
f¿1x2 lim lim (1.25)*
hS0 h ¢xS0 ¢x