It sccms rcasonablc ro bclicvc that if wc made rhc intcrvat At smaltcr
CHAPTER 9
DIFFERENTIAL CALCULUS
In Chaptcr 7, wc discusscd thc functional dcpendcncc of onc variablc,
anothcr variablc, .r. Many timcs, w6 arc intcresred in thc rate of
upon
,,
change of .;r with rcspect to r. Wc may, lor cxamplc, wish to know how the
conccntration o[ a spccics taking part in a rcaction changcs with timc.
Problcms of this rypc arc bcst trcatcd by a branch of mathcmatics known
as diffcrcntial calculus, r.'hich is conccrncd. with the cffcq that a small
change in onc variablc, r, has upon thc valuc of anothcr variablc, /. Wc
shall restrict our discrrssion of diffcrcnrial calculus to thosc asPccts of the
disciplinc which arc most rclevant to gcncral chcmistry.
9.I
THE MEANING OF THE DERIVATTVE
To illustratc hou' thc conccPt of a dcrivativc ariscs, lct us considcr a
problcm rypical o[thosc cncountered in chcmical kinetics. ln studying thc
ratc o[ the rcaction:
X.zO.(s)
*
zNOr(s)
wc might accumulatc the fotlowing data for thc conccntration of NOz as
a
funcrion of timei
conc. NOl
timc (hrs)
0.00 0.80 1.28 r.57 1.74
o
r
2
3
.4
1.81
5
How can wc usc this data to dctcrminc how rapidly the conccntrati l. d1/dr > Q'
9'2)'
Clcarly. thc point : = ! rcPrcscnts a minimum (scc Tablc
thc
at
that
wc
conctudc
rcasoning,
o[
fy pr..it.ty ih. ,"-. linc
Wc scc that whcn
point
r:
-1, wc havc a maximum'
,t - -1.10; fildx - 3'6'- I - 0'6; d1/dx > 0
5 = :-0.90; dj/dx = 2-4 - I - -0'6.; $ldx
DIFFERENTIAL CALCULUS
In Chaptcr 7, wc discusscd thc functional dcpendcncc of onc variablc,
anothcr variablc, .r. Many timcs, w6 arc intcresred in thc rate of
upon
,,
change of .;r with rcspect to r. Wc may, lor cxamplc, wish to know how the
conccntration o[ a spccics taking part in a rcaction changcs with timc.
Problcms of this rypc arc bcst trcatcd by a branch of mathcmatics known
as diffcrcntial calculus, r.'hich is conccrncd. with the cffcq that a small
change in onc variablc, r, has upon thc valuc of anothcr variablc, /. Wc
shall restrict our discrrssion of diffcrcnrial calculus to thosc asPccts of the
disciplinc which arc most rclevant to gcncral chcmistry.
9.I
THE MEANING OF THE DERIVATTVE
To illustratc hou' thc conccPt of a dcrivativc ariscs, lct us considcr a
problcm rypical o[thosc cncountered in chcmical kinetics. ln studying thc
ratc o[ the rcaction:
X.zO.(s)
*
zNOr(s)
wc might accumulatc the fotlowing data for thc conccntration of NOz as
a
funcrion of timei
conc. NOl
timc (hrs)
0.00 0.80 1.28 r.57 1.74
o
r
2
3
.4
1.81
5
How can wc usc this data to dctcrminc how rapidly the conccntrati l. d1/dr > Q'
9'2)'
Clcarly. thc point : = ! rcPrcscnts a minimum (scc Tablc
thc
at
that
wc
conctudc
rcasoning,
o[
fy pr..it.ty ih. ,"-. linc
Wc scc that whcn
point
r:
-1, wc havc a maximum'
,t - -1.10; fildx - 3'6'- I - 0'6; d1/dx > 0
5 = :-0.90; dj/dx = 2-4 - I - -0'6.; $ldx