Maths Questions
Fractions (useful for chemists when dealing with spin of electrons)
Simplify/evaluate the following
1.
2.
3.
4.
1
1
1
−5+2
3
1
2
1
+9−7
4
1
5
(3 × 2) − 6
1
+4
2 1
−
5 6
1
3
3
5. Convert 8 to a decimal.
6.
𝑥
2
+ 3𝑥 +
𝑥
3
7.
𝑦
6
− 2𝑦 +
𝑦
9
Percentages (very useful for working out percentage yields!)
8. Convert 0.25 to a percentage
2
9. Convert 6 to a percentage.
10. If the maximum theoretical yield of a product is 2.0 g and we collect a yield of 1.7 g,
what percentage yield do we have?
Indices or Powers (useful when dealing with units, e.g. can work out the conc = mass
/ vol)
Simplify/evaluate
11. y2 x y3
12. a6 x a-10
13. z4 ÷ z3
14. a0
15. (a4)6
16. 4.6 x 10-8 x 5.6 x 109
17. 7.9 x 1011 ÷ 2.8 x 108
Fractional powers (again useful for units and for using terms in equation, ie some
equations have things to ½ and if you don’t know what it means, you can’t use the
equation)
18. 41/2
19. (27)1/3
20. (27)2/3
Units (conversion from one unit to another, or to work out equations when you can’t
remember what they are)
The International System of Units (abbreviated SI from French: Système international
d'unités) formes a collection of units called SI units that are used around the world. These
include distance (m), mass (kg), time (s), and temperature (K). We can use these units to
describe any other unit. For example, the most commonly used unit of energy is the Joule, J.
This can be written in SI units as mass, time and distance: using the definition of kinetic
energy k.e. = ½ mv2, mass x velocity squared. Velocity is speed, so distance ÷ time. Units of
mass are kg, distance is m and time is s. If we put all that together, we get that J units = kg
(mass) (m s-1)2 (speed)2, so J = kg m2 s-2.
21. Using the definition of force to be force = mass x acceleration, express the unit N in
terms of SI units.
22. Using stress = force ÷ area, express the Pa in terms of SI units.
(Hint: write it down in the units of force and area first, then convert to SI).
Prefixes (useful for small scale chemistry)
m – milli 10-3 (millimole scale, very common in chemistry)
μ – micro 10-6 (published images of materials often use this scale)
n – nano 10-9 (carbon nano-tubles, size of clusters)
p – pico 10-12 (covalent/ionic radii ~ 100pm)
k – kilo 103 (used for masses (kg), enthalpies (kJ))
23. Using the information above, label the following TRUE OR FALSE
a. 1nm = 10-9 m
b. 567 g = 5.67 kg
c. 467 nm = 46.7 μm
d. 9870 pm = 9.87 x 10-9 m
e. 0.07 km = 7 m
f. 876 μm = 8.76 x 106 m.
24. C – H bond length is 1.09 x 10-10 m, write this in nano meters.
Another unit commonly used in chemistry, although not SI, is the Angstrom (Å). This is
equivalent to 10-10 m or 0.1 nm. As you can see from the question above, this is the average
length of a covalent bond.
25. H – H bond length is 0.74 Å what is this in pm?
26. O = O bond length is 121 x 10-12 m, what is this in Angstroms?
27. C – C bond length (in diamond) is 0.154 nm, write this in Angstroms.
We can use all this information to help us convert units too.
28. In the conversion of m s-2 to km s-2, what conversion factor should we use?
29. How would we convert from units of mol dm-3 to mol cm-3?
(Hint: think about the conversion from dm to cm then think about the powers in the
units).
Algebra Manipulation (not as scary as it sounds)
(useful for being able to draw a straight line graph for almost anything):
30. The Beer Lambert law is A = εcl, where A is absorbance, ε is the molar absorption
coefficient, c is the concentration of the solution and l is the path lenth (normally the
width of the cuvette). We can measure the absorbance, the path length and the
concentration of the solution. We want to find the molar absorption coefficient, what
series of experiments could be run to gather the data required to calculate? State
what the variables are.
What graph should be used to find ε?
(Tricky) What are the units of ε?
31. Using a hot plate, we can find the specific heat capacity of a liquid. Using q =
m.Cp.∆T, where q is the heat required, m is the mass of substance, Cp is the specific
heat capacity (at a constant pressure) and ΔT is the change in temperature. The
specific heat capacity is the amount of energy required to raise the temperature of
the substance by one kelvin. If we could measure the temperature, the heat required
and the mass of water being heated, what would be the independent and dependent
variables and how could we find the specific heat capacity of water?
32. Challenging:
𝑘=
𝑘𝐵 𝑇 −𝛥𝐺
𝑒 𝑅𝑇
ℎ
where ΔG is the Gibbs energy, kB is the Boltzmann constant, h is
Planck’s constant, T is temperature, R is the universal gas constant and k is the
reaction rate. ΔG = ΔH – TΔS. If we measured the reaction rate over a range of
temperatures, how could we find the enthalpy and entropy of the reaction?
(Hint: it can be rewritten as 𝑘 =
𝑘𝐵 𝑇 (𝛥𝑠) (−𝛥𝐻)
𝑒 𝑅 𝑒 𝑅𝑇
ℎ
where ΔH is the enthalpy and ΔS is the
entropy.)
(Really stuck? Extra hint: think of how to get rid of exponentials (ln))