The Chemistry Maths Book, Second Edition

626 Chapter 21Probability and statistics

Section 21.7

23.The variable xcan have any value in the continuous range 01 ≤ 1 x 1 ≤ 11 with probability

density functionρ(x) 1 = 16 x(1 1 − 1 x).(i)Derive an expression for the probability,P(x 1 ≤ 1 a),

that the value of xis not greater than a.(ii)Confirm thatP(0 1 ≤ 1 x 1 ≤ 1 1) 1 = 11. (iii)Find the

mean 〈x〉and standard deviation σ. (iv)Find the probability that〈x〉 1 − 1 σ 1 ≤ 1 x 1 ≤ 1 〈x〉 1 + 1 σ.

24.The variable rcan have any value in the ranger 1 = 101 → 1 ∞with probability density

functionρ(r) 1 = 14 r

2

e

− 2 r

. Derive an expression for the probabilityP(0 1 ≤ 1 r 1 ≤ 1 R)that the

variable has value not greater than R.

25.Confirm thatp(r)in Exercise 24 is the radial density function (in atomic units) of the 1s

orbital of the hydrogen atom. Find (i) the mean value 〈r〉, (ii) the standard deviation σ,

(iii) the most probable value of r.

Section 21.10

26. (i)Find the linear straight-line fit for the following data points. (ii) If the errors in yare

all equal toσ 1 = 11 , find estimates of the errors in the slope and intercept of the line.

x 123456789101112

y 4.4 4.9 6.4 7.3 8.8 10.3 11.7 13.2 14.8 15.3 16.5 17.2

27.The results of measurements of the rate constant of the second-order decomposition of

an organic compound over a range of temperatures are:

T 2 K 282.3 291.4 304.1 313.6 320.2 331.3 343.8 354.9 363.8 371.7

k 210

− 3

1 dm

3

mol

− 1

s

− 1

.0249 .0691 0.319 0.921 1.95 5.98 19.4 57.8 114. 212.

The temperature dependence of the rate constant is given by the Arrhenius equation

, orln 1 k 1 = 1 −E

a

2 RT 1 + 1 ln 1 A, in which the activation energyE

a

and pre-

exponential factor Amay be assumed to be constant over the experimental range of

temperature. A plot ofln 1 kagainst 12 Tshould therefore be a straight line. (i) Construct

a table of values of 12 Tandln 1 k, and determine the linear least-squares fit to the data

assuming only kis in error. (ii)Calculate the best values ofE

a

and A. (iii) Assuming that

the errors inln 1 kare all equal toσ 1 = 1 0.1, use equations (21.55) to find estimates of the

errors inE

a

and A.

kAe

ERT

=

−

a