The Chemistry Maths Book, Second Edition

4.13 Exercises 125

82.Confirm that the cubicy 1 = 1 x

3

1 − 17 x

2

1 + 116 x 1 − 110 , discussed in Example 2.23, has local

maximum and minimum values atx 1 = 12 andx 1 = 1823.

83.Find the maximum and minimum values and the points of inflection ofy 1 = 12 x

5

1 − 15 x

4

1 + 13.

Sketch a graph to show the positions of these points.

84.The Lennard-Jones potential for the interaction of two molecules separated by distance

Ris

where Aand Bare constants. The equilibrium separation R

e

is that value of Rat which

U(R)is a minimum and the binding energy isD

e

1 = 1 −U(R

e

). Express (i)Aand Bin terms

of R

e

and D

e

, (ii)U(R) in terms of R, R

e

and D

e

.

85.The probability that a molecule of mass min a gas at temperature Thas speed vis given

by the Maxwell–Boltzmann distribution

where kis Boltzmann’s constant. Find the most probable speed (for which f(v)is a

maximum).

86.The concentration of species B in the rate process ,consisting of

two consecutive irreversible first-order reactions, is given by (whenk

1

1 ≠ 1 k

2

)

(i)Find the time t, in terms of the rate constants k

1

and k

2

, at which B has its maximum

concentration, and (ii) show that the maximum concentration is

Section 4.11

87.A particle moving along a straight line travels the distances 1 = 12 t

2

1 − 13 tin time t. (i)Find the

velocity vand acceleration aat time t. (ii)Sketch graphs of sand vas functions of tin the

intervalt 1 = 101 → 12 , (iii)find the stationary values, and describe the motion of the particle.

88.A particle moving on the circumference of a circle of radiusr 1 = 12 travels distance

s 1 = 1 t

3

1 − 12 t

2

1 − 14 tin time t. (i)Express the distance travelled in terms of the angle θ

subtended at the centre of the circle, (ii)find the angular velocity ωand acceleration 7

around the centre of the circle, (iii)Sketch graphs of θ, ωand 7 as functions of tin the

intervalt 1 = 101 → 14 , (iv)find the stationary values, and describe the motion of the particle.

Section 4.12

Find the differential dy:

89.y 1 = 12 x 90.y 1 = 13 x

2

1 + 12 x 1 + 11 91.y 1 = 1 sin 1 x

92.The volume of a sphere of radius risV 1 = 14 πr

3

23. Derive the differential dVfrom first

principles. Give a geometric interpretation of the result.

[] []

max

()

BA=

















−

0

1

2

221

kkk

k

k

[] []= ( )BA

−

−

−−

0

1

21

12

k

kk

ee

kt kt

ABC

kk

12

→→

f

m

kT

e

mkT

()vv

v

=













−

4

2

32

22

2

π

π

UR

A

R

B

R

()=−

12 6