94 Chapter 4Differentiation
If the pressurepof the gas is changed by an amount∆pat constantTandn,the
change in the volume is
Figure 4.2 shows that the graph ofVagainstp(at constant Tand n) is not a straight line.
The gradient at any point on the curve is defined as the gradient of the tangent to the
curve at that point. The gradient changes from point to point, and is not given by∆V 2 ∆p.
The branch of mathematics concerned with the determination of gradients and,
therefore, with rates of change is the differential calculus.
4.2 The process of differentiation
Let the value of a variable xchange continuously from pto q. The difference (q 1 − 1 p)
is called the changeor incrementin x. In the differential calculus this change is
denoted by*
∆x 1 = 1 q 1 − 1 p (4.1)
We note that∆x 1 > 10 ifq 1 > 1 p, and∆x 1 < 10 ifq 1 < 1 p.
Lety 1 = 1 f(x)be a function of xthat changes continuously and smoothly from point
P to point Q (Figure 4.3). The values of yat P and Q arey
P1 = 1 f(p)andy
Q1 = 1 f(q). The
change ∆yin ycorresponding to change ∆xin xis therefore
y
Q1 − 1 y
P1 = 1 ∆y 1 = 1 f(q) 1 − 1 f(p)
The quantity
(4.2)
is the gradient of the line PQ, and can be interpreted as the averagerate of change of y
with respect to xbetween P and Q.
yy
qp
y
x
QP−
−
=
∆
∆
∆
∆
∆
∆
V
nRT
pp
nRT
p
nRT
p
pp p
=
−=−
()+
P
Q
pq
y
P
y
Q
∆x
∆y
y
x
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Figure 4.3
*Sometimes, if the change in xis supposed to be ‘small’, δxis used instead of ∆x.