species change as a function of time. Consider the formation of silver
nanoparticles by the thermal decomposition of silver oxalate at 140
o
C:
Ag 2 C 2 O 4 s
ð Þ ! 2Ag s
ð Þ + 2CO 2 g
ð Þ
(3.1)
At the beginning of the reaction, the rate of disappearance of Ag 2 C 2 O 4 will
be large, and this rate will decrease as the concentration of the reactant
decreases, as seen in Figure 3.1. The rate of reaction is the slope of the
reactant concentration at any point in time. The rate at t = 0 is the initial
rate. The larger the initial concentration of Ag 2 C 2 O 4 , the larger the initial
rate will be. Once the rate of change of the concentration reaches zero, the
reaction is complete. In general, we can define the instantaneous rate of
the reaction, n(t), as
n t
ð Þ = −
d Ag 2 C 2 O 4
½
Š
dt
(3.2)
The negative sign in front of the derivative in Equation 3.2 tells us that the
concentration of Ag 2 C 2 O 4 is decreasing with time because Ag 2 C 2 O 4 is a
reactant. The units of reaction rate are mol dm
–3 s
–1 and its numerical
value changes with time. The rate of increase in product is proportional to
0.001
0.002
0.003
0.004
0.005
0.006
0
0
1 0
2 0
3 0
4 0
5 0
Time (min)
[Ag
2 C
2 O
4 ](mol dm –3
)
Half life, t ½
Initial rate = v 0 (t = 0) = –
d[Ag 2 C 2 O 4 ]
dt
0
Rate at t 2 = v(t = t 2 ) = –
d[Ag 2 C 2 O 4 ]
dt
t 2
t 2
Figure 3.1 A plot of [Ag 2 C 2 O 4 ] versus time. The product concentration decreases
exponentially. The instantaneous rate at any point in time is given by the derivative of
concentration with respect to time at t. Instantaneous rates are shown at two different time points: t = 0 (the initial rate) and t = t 2 . The half-life for the reaction is also
indicated.
CHAPTER 3: Kinetics and Transport in Nanoscience
64