The Chemistry Maths Book, Second Edition

3.10 Exercises 91

28.Find the cartesian coordinates of the points whose polar coordinates are

(i)r 1 = 1 3, θ 1 = 12 π 2 3, (ii)r 1 = 1 3, θ 1 = 14 π 2 3.

29.Find the polar coordinates of the points whose cartesian coordinates are

(i)(3, 2), (ii)(3, −2).

30.Find the polar coordinates of the points whose cartesian coordinates are

(i)(−3, 2), (ii)(−3, −2).

31.A solution of the equation of motion for the harmonic oscillator is given in Example 3.8

asx(t) 1 = 1 A 1 cos 1 ωt. Show thatx(t)can be interpreted as the x-coordinate of a point

moving with constant angular speed ωin a circle in the xy-plane, with centre at the origin

and radius A.

Section 3.6

32.Simplify

(i)e

2

e

3

(ii)e

3

e

− 3

(iii)e

3

e

− 4

(iv) e

3

2 e

2

(v)e

5

2 e

− 4

33. (i)Write down the expansion ofe

−x 23

in powers of xto terms inx

5

.

(ii)Use the expansion to calculate an approximate value ofe

− 123

. Determine how many

significant figures of this value are correct, and quote your answer to this number of figures.

34. (i)Write down the expansion of in powers of xto terms inx

15

.

Use the expansion to calculate an approximate value of that is correct to 12

significant figures for the following values of x, in each case giving the smallest number of

terms required: (ii) 10

− 1

, (iii) 10

− 2

, (iv) 10

− 3

, (v) 10

− 4

, (vi) 10

− 5

.

35.Sketch the graphs ofe

2 x

ande

− 2 x

for values−1.5 1 ≤ 1 x 1 ≤ 1 1.5.

36.For a system composed of Nidentical molecules, the Boltzmann distribution

gives the average fraction of molecules in the molecular state iwith energy ε

i

.

(i)Show that the ration

i

2 n

j

of the populations of states iand jdepends only on the

difference in energy of the two states. (ii)What is the ratio for two states with the same

energy (degenerate states)?

Section 3.7

37.Simplify:

(i)log

10

100 (ii)log

2

16 (iii)ln 1 e

− 5

(iv) (v)

(vi)ln 1 e

−kt

38.Express the following as the log of a single number:

(i)ln 121 + 1 ln 13 (ii)ln 121 − 1 ln 13 (iii) 51 ln 12 (iv)ln 131 + 1 ln 141 − 1 ln 16

39.Simplify:

(i)ln 1 x

3

1 − 1 ln 1 x (ii)ln 1 (2x

3

1 − 13 x

2

) 1 + 1 ln 1 x

− 2

(iii)ln 1 (x

5

1 − 13 x

2

) 1 + 121 ln 1 x

− 1

1 − 1 ln 1 (x

3

1 − 1 3)

(iv)ln 1 e

x

(v)

40.The barometric formula

p 1 = 1 p

0

e

−Mgh 2 RT

gives the pressure of a gas of molar mass Mat altitude h, when p

0

is the pressure at sea

level. Express hin terms of the other variables.

41.The chemical potential of a gas at pressure pand temperature Tis

μμ=+

ο

ο

RT

f

p

ln

lneeln

x

2

+ 33

−

ln

()

e

−++ax bx c

2

lne

x

2

n

N

e

i

kT

i

=

−ε/

e

−x

3

e

−x

3