11.8 Exercises 335
Use the method in Exercise 26 to find the general solution:
Section 11.4
29.Find the intervalτ
12 nin which the amount of reactant in a first-order decay process is
reduced by factor n.
30.Solve the initial value problem for the nth-order kinetic process A 1 → 1 products
x(0) 1 = 1 a (n 1 > 1 1)
31.The reversible reactionA 1 j 1 B, first order in both directions, has rate equation
Findx(t)for initial conditionx(0) 1 = 10.
32.A third-order process A 1 + 1 2B 1 → 1 products has rate equation
where aand bare the initial amounts of A and B, respectively. Show that the solution that
satisfies the initial conditionx(0) 1 = 1 0 is given by
Section 11.5
Find the general solution:
- (n 1 ≠ 1 −1) 40.
Section 11.6
41.The system of three consecutive first-order processes is
modelled by the set of equations
where(a 1 − 1 x),y, and zare the amounts of A, B, and C, respectively, at time t. Given the
initial conditionsx 1 = 1 y 1 = 1 z 1 = 10 att 1 = 10 , find the amount of C as a function of t. Assume
k
1
1 ≠ 1 k
2
,k
1
1 ≠ 1 k
3
,k
2
1 ≠ 1 k
3
.
42.The first-order process is followed by the parallel first-order processes
and , and the system is modelled by the equations
da x
dt
ka x
dy
dt
ka x k ky
()
() ()( )
−
=− − , = − − +
1123
,, = , =
dz
dt
ky
du
dt
ky
23
BD
k3BC→
k2→
AB
k1
→
da x
dt
ka x
dy
dt
ka x ky
()
( ) ( )
−
=− − , = − − ,
112
ddz
dt
=−ky kz
23
ABCD
kkk123→→→
dy
dx
a
y
x
x
n+=
dy
dx
ax y bx
nn+=,
dy
dx
+=(tan) sin2xy x
dy
dx
y
xx
−=
22
dy 4
dx
y
x
+= x
2
2cos
dy
dx
ye
x+=
−
3
3
dy
dx
−= 4 xy x
dy
dx
+= 24 y
kt
ab
ab x
ba x
x
babb x
=
−
−
−
−−
1
2
22
22
2
()
ln
()
()( )( )
dx
dt
=− −ka x b x()( ) 2
2
dx
dt
=−−ka x kx
11 −
()
dx
dt
kx
n=−
dy
dx
xy
xy
=
−
−+ 2
dy
dx
=++ 23 xy