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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