4 Differentiation
4.1 Concepts
In the physical sciences we are interested in the value of a physical quantity and how
it is related to other physical quantities. In addition, we are interested in how the value
of the physical quantity changes on going from one state of the system to another,
and in the rate of change with respect to time or with respect to some other physical
quantity.
Consider the equation of state of the ideal gas
pV 1 = 1 nRT
Because any one of the four variables p, V, T, and ncan be expressed as a function
of the other three, the state of the system is determined by three of the four quantities.
If the temperature Tis changed by an amount∆T(read as ‘delta t’), keeping the
pressure pand the amount of substance nfixed, the volume changes from
and the change in volume is
Figure 4.1 shows that the graph of Vagainst T(at constant pand n) is a straight line
with gradient, or slope,
The gradient is the change in Vper unit change in T, or the rate of changeof Vwith
respect to T.
∆
∆
V
T
nR
p
==constant
∆∆V
nR
p
= T
V
nRT
p
VV
nR T T
p
=+=
to ∆
()∆
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TT+∆T
T
V
V+∆V
V
∆V
∆T
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pp+∆P
p
V
V+∆V
V
∆V
∆P
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Figure 4.1 Figure 4.2