Transcript presentation source - Personal Home Pages (at UEL)

Maths and Chemistry for
Biologists
Maths 1
This section of the course covers –
• why biologists need to know about maths and
chemistry
• powers and units
• an introduction to logarithms
• the rules of logarithms
• the usefulness of logs to the base 10
Why do biologists need to know about
maths and chemistry?
The next slide describes a typical experiment in
biology.
It is written in four languages –
common English, biology, chemistry and maths
You need to speak all four to understand it
This part of the course aims to cover those bits of
chemistry and maths that the biologist must
know
Make up 10 ml of a solution of histamine (Mr = 111) at a
concentration of 10 mM
Measure the effect of histamine on the contraction of guinea pig
ileum suspended in 10 ml of buffer solution, pH 7.4, over the
concentration range 1 mM to 1 nM
Present the results as a graph of effect against log (concentration)
Determine the concentration of histamine which gives 50%
maximum contraction
English Biology Chemistry Maths
LEARNING MATERIALS
Workbook - section on Numerical Methods and Chemical
Calculations
Text book – Chemistry for theLife Sciences by R Sutton,
B Rockett and P Swindells (multiple copies in the Library)
Revision material - in Department menu under Chemistry
Tutorials
i) double click on Chemistry Tutorials
ii) in welcome screen select General Course
iii) for general revision take topics in order
iv) for specific topic, browse through Terms (eg covalent
bonds is topic 13 in Term1)
Powers and Units
Chemical and biological systems often involve very
large and very small numbers
There are 602,000,000,000,000,000,000,000
atoms in 12 grams of carbon
Each atom has a radius of
0.00000000000000275 m
These numbers are very inconvenient – easy to
get wrong number of zero’s
This is where powers come in
Powers
Number multiplied by itself several times e.g.
2 x 2 x 2 x 2 written as 24
(spoken as two to the power four)
Special cases 22 is two squared and 23 is two cubed
Powers can be negative e.g. 2-3 (two to the minus three)
1
This means
2x2x2
Special case is 20 = 1
Any number raised to the power zero is equal to 1
How does this help with large and small numbers?
602,000,000,000,000,000,000,000
is the same as 6.02 x 1023
that is, 6.02 multiplied by 10 23 times
(move the decimal point left 23 places)
0.00000000000000275
is the same as 2.75 x 10-15
that is, 2.75 divided by 10 15 times
(move the decimal point right 15 places)
Rules for powers
When terms are multiplied powers are added
so 32 x 33 = 35
When terms are divided powers are subtracted
37
so 4 = 33
3
and
37 x 3-3
= 3(7-3-4) = 30 = 1
4
3
Units
All chemical and physical quantities have units
We could give a length as 0.005 m or 5 x 10-3 m
Or we could give it as 5 mm (5 millimetres)
So we can avoid using powers of ten by changing
the size of the unit
For example, you might buy 1000 g of sugar or
alternatively 1 kg (1 kilogram)
We add a prefix to the unit to change its size
Prefixes to units
The common ones are
Frac
10-3
10-6
10-9
10-12
10-15
10-18
Prefix
milli
micro
nano
pico
femto
atto
Symbol
m

n
p
f
a
Mult
103
106
109
1012
Prefix
kilo
mega
giga
tera
Symbol
k
M
G
T
So 10-6 m = 1 m; 3 x 10-9 g = 3 ng; 5 x 109 V = 5 GV
A word of warning
Do not add, subtract, multiply or divide numbers
with units with different prefixes
e.g. to work out the area of a rectangle 1 m long by
5 mm wide cannot say the area is 5 because the
units are not defined
Change one of the lengths to have same prefix
e.g. 1 m = 103 mm so area is 5 x 103 mm2
or 5 mm = 5 x 10-3 m so area is 5 x 10-3 m2
One for you to do
The universe contains 1011 galaxies and each
galaxy contains 1011 stars
Suppose that 1 in 1000 of those stars has a planet
with conditions suitable for life to develop
Suppose that the probability of life developing on
such a planet is 1 in 1,000,000,000,000
How many planets might have developed life?
Answer
11
10 x10
3
10 x10
11
12
= 1011+11-3-12 = 107
Logarithms
DEFINITION: if a = bc then c = logba
(spoken as log to the base b of a)
Two important cases base 10 and base e
(e is an irrational number equal to 2.71828….)
Base 10: log10 2 = 0.3010
What this means is that 100.3010 = 2
Why are they useful?
Change numbers with powers of 10 into simpler
forms
e.g. log10 5x106 = 6.699
log10 2x10-4 = -3.699
As number goes up by a power of 10 the log goes
up by unity
e.g. log10 5 = 0.699
log10 50 = 1.699
(Note – numbers less than 1 have negative logs;
negative numbers do not have logs)
An example – a dose/response curve
Dose (ng) Log
Resp
(dose)
1
0
0.01
1
0.03
100
2
0.07
1,000
3
0.40
10,000
4
0.90
100,000
5
0.98
1,000,000 6
1.00
response
10
(b)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
log (dose)
6
In the experiment in the previous slide we
plotted the response against the log of the
Dose. This is because the dose covered a
Very wide range of values. We used the
property of logs that as the number goes
up by 10 fold the log goes up by unity.
Try plotting the response directly against
The dose and you will see that you get a
Rather silly looking graph.
Rules and results
If no base is specified then it is assumed to be 10
log (a x b) = log a + log b
log (a/b) = log a – log b
log an = n x log a
It follows from the definition that log 10 =1
so log 10n = n x log 10 = n
e.g. log 106 = 6
log 10-3 = -3
Some for you to do
Without using a calculator, work out
log 1023
log 1.2 given that log 120 = 2.0792
(remember that these are all logs to the base 10)
Answers
log 1023 = 23 x log 10 = 23
log 1.2 = log 120 x 10-2 = log 120 + log 10-2
= 2.0792 + (-2) = 0.0792