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MATHS 110: Math for Science

Midterm Test

2018

1. (10 marks)

(a) A foot is an old unit of measurement, and there are 3.28 feet in 1 metre. Convert a

speed of 35.0 km/hour into units of feet/second. Give your answer to 3 signiﬁcant ﬁgures.

(b) The rate, r, of a chemical reaction has units of mol L− 1 s − 1 . Suppose we know that r = k1 S2 + k2 S,

where S is the concentration (in units of mol/L) of the species that is reacting, and where k1 and k2 are constants. What are the units of k1 ? What are the units of k2 ?

2. (10 marks) Newton’s law of gravitation says that the force, F , between two objects of mass m1 and m2 , a distance d apart, is given by

Gm1m2

F =

2 ,

where G is the gravitational constant G = 6.674 x 10 − 11 N m2 /kg2 .

(a) Given that those are the units of G, and if F has units of Newtons, what units

must m1 , m2 and d be expressed in?

(b) If m1 = m2 = 1000 土 0.2 g and if d = 0.20 土 0.001 cm, what is the relative error

in the calculation of F?

3. (10 marks) The Weber-Fechner law says that the relationship between stimulus and perception is logarithmic. Thus, if p denotes perception and S denotes the stimulus,

then

p = k ln / 、 ,

for some constants k and S0 . (The units of p and S are unspeciﬁed for this question.)

(a) What is p(S0 )?

(b) The Weber-Fechner law is often written in the form p(S) = k ln(S) + C. Work out

an equation for C in terms of S0 and k .

(d) Set k = S0 = 1. Sketch (by hand) the graphs of p(S) and S(p).

4. (10 marks) The Arrhenius equation relates the reaction rate, k, to the temperature:

− E0

k = Ae RT ,

where E0 is called the activation energy, and A is called the pre-exponential factor. As usual, R is the gas constant, with the value of 8.314 J K − 1 mol− 1 .

(a) By taking the natural log (ln) of both sides, write a linear equation that relates

ln(k) to 1/T.

(b) Suppose a scientist measures a rate constant k at two diﬀerent temperatures, T.

When T = 300 K, k = 1 µM s − 1 . When T = 400 K, k = 5 µM s − 1 . Use this information to calculate A and E0 .

5. (10 marks) Sulphur dioxide (SO2 ) is a toxic gas and a major air pollutant. The molecule has a planar shape, and is sketched in the ﬁgure, which shows the relative positions of the atoms. The coordinates of each atom are given in units of angstroms, or A˚ .

(-0.6993,1.25)