Section 3: Introduction to Quantities, Units and Scientific Equations

Transcription

1 Section 3: Introduction to Quantities, Units and Scientific Equations Contents Scientific equations 1:3:2 Significant figures in calculations 1:3:2 Rearranging equations 1:3:8 The rules of rearranging equations 1:3:8 The subject of an equation 1:3:9 Quantities, units and symbols 1:3:16 The SI System of Units 1:3:16 Conventions for deriving and expressing quantities and units 1:3:17 Introduction to prefixes 1:3:18 Non-SI units 1:3:19 Conversion between different units 1:3:20 Using proportions 1:3:20 Using ratios 1:3:21 Units encountered in the laboratory 1:3:22 Measurement of volume 1:3:22 Volume equivalents 1:3:23 Measurement of pressure 1:3:30 Measurement of temperature 1:3:32 Self study revision exercises 1:3:35

2 Scientific equations Significant figures in calculations The value that the calculator returns to a calculation is not necessarily corrected to the appropriate number of significant figures You have to decide for yourself how the final answer to a calculation should be written Note: A final answer should include not more than all figures that are known with certainty The precision of a calculated value can only ever be as good as the least precise measurement that was involved in the calculation The precision of any calculated quantity is determined by considering the precision of each individual data value that is involved in the calculation Methods of calculating, accurately, the precision of a calculated value are beyond the scope of this book However, the following gives a guide to obtaining a best estimate for the precision of calculated values, which you might find useful The overall precision of a calculated measured value depends on the number and type of mathematical operations that are being carried out (a) In calculations which involve only multiplication and/or division, the precision of the answer is limited by the data value that has the least number of significant figures The final answer is therefore quoted to the same number of significant figures as contained in this value Consider the example:, The calculator value gives us a 5 figure answer The smallest number of significant figures, (sig fig), in the data is given by the number,, which has only 2 significant figures So we need to correct our final answer to 2 significant figures also Therefore we can conclude Now consider the example: (correct to 2 sig fig) The calculator value gives us a 2 figure answer Since both numbers are known precisely to three significant figures, we should quote our final answer to three significant figures also So we can conclude (correct to 3 sig fig) (b) In calculations which involve only addition and/or subtraction, the precision of the answer is limited by the data value that has the highest least significant digit Workbook 1 Communication of scientific data 1:3:2

3 Consider the following examples Example 1: In the preparation of a small batch of a drug X, three ingredients were mixed together in a mortar If the mass of each of the 3 ingredients was 255 g, 125 g and 21 g respectively, determine the total mass of the batch mixture Give your answer to an appropriate number of significant figures A calculator would carry out the calculation as The answer quoted in this way indicates that we have a total mass known precisely to 4 significant figures This is incorrect To show why, let us write out the calculation so that we can see where the uncertainty of each number starts Mass/g Precision g g g = g We can see from the calculation that all the figures to the left of the decimal point are known so the overall sum of 47 for these figures is known with certainty Looking at the first column of digits after the decimal point, we can see that there is a digit missing from the third value in our calculation This will introduce a degree of uncertainty in our answer of 4775 Any further figures after the decimal point are even more uncertain so this tells us that these digits are not known to any degree of certainty so they are definitely not significant We can, therefore, only quote our answer with any level of confidence to either the nearest whole number, 48 or to 1 decimal place In order to quote to 1 decimal place, we need to consider the value of the second decimal place figure Since this is 5, we correct our answer to 478 Therefore, we should quote our total mass to be 478 g, where the small character size indicates that there is uncertainty about the accuracy of the 8 When much bigger or smaller numbers are involved in a calculation, we can use the following method to determine the overall precision First, express all the data values in standard form Then convert these to the same power of 10 Carry out the calculation using the method described in section 2 Finally, determine the first digit that contains a level of uncertainty Workbook 1 Communication of scientific data 1:3:3

4 Example 2: Calculate the following and express your answer to an appropriate number of significant figures: By expressing both numbers in terms of, the calculation can be written as:, Since both original numbers are precise to 2 decimal places when expressed as the same power of ten, they are equally precise, so the precision of the final answer may be expressed similarly The calculated answer is therefore correctly expressed as above as the final answer should have 2 decimal places when expressed in terms of the factor Example 3: Now consider the calculation: The calculated value gives us a 3 decimal place answer when expressed in terms of Since the number, is only known to 1 decimal place then this is the less precise of the two values and this limits the number of digits that can be quoted in the final answer to 1 decimal place also So we can conclude (correct to 2 sig fig) Note: Beware of introducing errors into calculations by rounding intermediate values to significant levels Where intermediate values are recorded, they should be written to a minimum of 2 figures greater than the final number of significant figures that will be quoted; usually overall 4 or 5 figures is sufficient This is particularly important if figures are to be re-entered into a calculator for further calculations Always aim to carry out calculations in one single step or to keep intermediate values in full in the calculator memory if they are to be used again When calculations involve multiplication and/or division and addition and/or subtraction, the overall precision is decided by the multiplication rule Using the calculator memory functions Most calculators have two types independent memory storage that can be utilised by the operator These are used for storing intermediate values in calculations or for storing the value of variable(s) in serial calculations These memory functions are accessed using STORE and RECALL operations The mode of operation varies between different calculator manufactures and models Workbook 1 Communication of scientific data 1:3:4

5 A CASIO fx-83w calculator s memory and storage functions are described below The CASIO fx-83w accesses the usable memory storage features via the following separate buttons: Store STO and Recall RCL Note: On many calculators, these are located on the same button with the shift button used for obtain the alternative option Before entering new data always check that the mode is set correctly and that memory is empty: To set the mode and clear the memory: Mode: COMP MODE 1 Memory clear: SHIFT AC Mcl_ = Mcl 0 Storing a variable This calculator can store values in several different variables These are addressed using the buttons marked in red as follows: A B C D E F A single variable is stored using the SYNTAX: number STO variable button = Example: To store the value 123 in A, type 123 then press STO (-) store A To recall the value 123 from A, press RCL (-) recall A Storing in memory The memory storage function is a useful aid when performing many complex or repetitive operations in a calculation It can be used to retain intermediate values that can be recalled for later use The value of the stored value can be used and updated using the memory plus and memory minus options The memory function is accessed using the button marked in red as M A value is stored using the SYNTAX: number STO memory button = Example: To store the value 123 in M, type 123 then press STO M+ store M To recall the value 123 from M, press RCL M+ recall M Workbook 1 Communication of scientific data 1:3:5

6 Record below how data may be stored and recalled using your own calculator Workbook 1 Communication of scientific data 1:3:6

7 Exercise 32 > Calculate the answers to the following problems 1 Complete the calculations below as follows: Estimate the answer Use your calculator to determine the precise answer State your answer to an appropriate number of figures (i) Estimated answer precise answer = correct to 4 sig fig, in standard form (ii) Estimated answer precise answer = correct to 2 decimal places (iii) Estimated answer precise answer = or correct to 4 sigfig 2 A titration was repeated 5 times and the titre volumes were recorded as follows: 1505, 155, 1482, 149, and 15 cm 3 Determine the mean titre value for the titration Total volume = So the mean titre volume = The mean titre volume is 151 cm 3 which reflects the uncertainty of the last experimental value Note: if the last data value had been given as 150 cm 3, then the mean titre value would be quoted as 1505cm 3 Workbook 1 Communication of scientific data 1:3:7

8 Rearranging equations The rules of rearranging equations Rearranging equations involves focusing on one particular element of the equation (usually a letter); thinking about what is being done to that element and then undoing it by performing an opposite operation When more than one operation is being performed on the element then these are undone in the order of reverse BEDMASS, ie starting with negate, then subtractions and additions, multiplication, etc When rearranging equations always follow the three rules below (1) The equation is valid whether read from left to right or right to left eg has exactly the same meaning as (2) Whatever operation you perform on the left of the equals sign must also be performed on the right eg add 7 to both sides of which simplifies to (3) If you divide both sides of an equation by zero, the equation may no longer be valid If you are not familiar with solving quadratic equations at this stage then revisit this section when you have completed the work on quadratic equations in book 2 The following two examples show proof of the statement above Example 1: Consider the equation:, this factorises to Divide both sides by the common factor This gives so so There are no real solutions to the equation In fact, when, then, so and There is one real solution, that is In this case the division by zero has masked the real solution and so no solution seems possible Workbook 1 Communication of scientific data 1:3:8

9 Example 2: Consider the equation: This factorises to Divide both sides by so which suggests, if true then In fact which is impossible and should never have been used as a divisor The subject of an equation To make a letter the subject of an equation we must rearrange for that letter and only that letter to appear on the left of the equals sign and everything else excluding the subject on the right of the equals sign For example, in the following two equations y is the subject: but y is not the subject in the following two equations State below why the subject of these two examples is not y The equation is written in terms of 2y; it should be just y There is a y term on each side of the equals sign It is often necessary to change the subject of an equation One very common but simple type of equation involves only multiplication and division operations Consider the equation that relates the concentration of a solute, c, to the amount of solute, n, and the volume of solution, V The form that is explicit in c is To make n the subject of the equation first reverse the order and write Workbook 1 Communication of scientific data 1:3:9

10 and then multiply both sides by V to give which simplifies to To make the subject of the original equation, first multiply both sides by to give, which simplifies to Divide both sides by c to obtain which may be expressed as or Now let us consider an example containing addition and/or subtraction Make the subject in the following equation We need on the left hand side and everything else on the right, so first we write the equation in reverse with the term on the left and the term on the right Next we add 4 to both sides to give, so Then we divide both sides by 3 to obtain, so When rearranging more complex equations, remember the following guidelines: The subject must be written down on the left of the equation on its own with all other terms on the right of the equals sign The equation is arranged so that it is expressed only in terms of the new subject using the rules of reverse BEDMASS When an equation includes the required subject as part of the denominator of a fraction the equation first needs to be manipulated so that the new subject is NOT on the bottom row of a fraction If there are several terms involving the subject letter, we must manipulate the equation so that the subject appears just once Workbook 1 Communication of scientific data 1:3:10

11 Consider the following example: Make x the subject in the following equation: (linear form ) Notice that there are two x terms; one as part of the numerator of the fraction and the other as part of the denominator Using reverse BEDMASS order, we see that the main structure is a fraction so we need to undo the divide first, ie we must move the x term that is part of the denominator of the fraction Note: we must move the whole term from the bottom of the fraction (we cannot just move the x) We put brackets round the and to preserve these terms in the fraction and then multiply both sides of the equation by : This gives Simplifying and expanding the bracket gives Next we need to get both of the x terms on the same side Since there is already a term containing x on the left hand side of our equation, we leave that where it is and move the 2x term over to the left hand side So we subtract from both sides (Notice that the should not be moved) This gives Therefore Next we move the term by adding to both sides to leave only terms that involve x on the left: This gives Therefore So now we have two our terms containing x terms on the left and everything else on the right The equation then needs to be manipulated so that only x remains Take out x as a common factor This gives Then divide both sides of the equation by to get x as the subject This gives therefore or Being able to rearrange equations quickly and efficiently is an essential skill for a scientist so try the problems set out in Exercises 33 and 34 for yourself Workbook 1 Communication of scientific data 1:3:11

12 Exercise 33 Make r the subject in the following equations; explain your method clearly in each case (i), where A is the surface area of a sphere By reversing the order we can write Dividing both sides by gives so Then by taking the square root of both sides we obtain So, which is more simply written as (ii), where V is the volume of a sphere By reversing the order we can write Dividing both sides by and multiplying both sides by 3 gives so Then by taking the cube root of both sides we obtain Therefore Workbook 1 Communication of scientific data 1:3:12

13 (iii), where A is the surface area of an open cylinder By reversing the order we can write Then dividing both sides by we obtain, (iv), where V is the surface area of an open cone By reversing the order we can write Multiplying both sides by 3 gives so Then dividing both sides by we obtain, Finally taking the square root of both sides gives So or or Workbook 1 Communication of scientific data 1:3:13

14 Exercise 34 (i) In radioactive decay the half-life,, is related to the decay constant, λ, by Rearrange the equation to make λ the subject Multiply both sides by λ to give, Divide both sides by to obtain, which simplifies to (ii) Make the subject in the following equation First reverse the order so that the equation reads Then add to both sides to give or Dividing both sides by 3, using brackets to ensure the correct result, gives:, which simplifies to So, in conclusion, if then Workbook 1 Communication of scientific data 1:3:14

15 (iii) Make x the subject in the following equation Firstly putting brackets round the top and bottom of the fraction then multiplying through by gives which simplifies to Reverse the order and write Adding to both sides and then simplifying gives Then, by subtracting 2 from both sides and simplifying, we obtain, So Taking the common factors of both sides outside of brackets gives On dividing both sides by we find that, so Workbook 1 Communication of scientific data 1:3:15

16 Quantities, units & symbols The SI system of units It is common practice in science to represent quantities and units by using recognised symbols You should become familiar with the symbols that are commonly used in text books or scientific documents without having to look them up You should be able to recall a unit s name from its symbol and write the symbol that represents a unit name Throughout the world there are several different systems of units that are in common use In communications such as research papers, units are usually converted to conform to a standard system of units to avoid ambiguities This is the Système Internationale D Unités, commonly known as the SI system of units, which was devised with the agreement of scientists and technologists from countries all over the world so it is an internationally recognised system of units The system is based around the definition of seven base units (or fundamental units) In this system, the size, name and symbol used to represent each of the base units are clearly defined These unit names and symbols are listed in Table 31 below You can find more on the SI unit system, for example, the definition of these units, at the following website: Table 31: SI base units quantity SI unit unit symbol length metre m mass kilogramme kg time second s temperature kelvin K current ampere A luminous intensity candela cd amount of substance mole mol You must learn the information given in Table 31! Workbook 1 Communication of scientific data 1:3:16

17 Conventions for deriving and expressing quantities and units The units for all other quantities can be expressed in terms of one or more SI base units It is very important that you understand how the SI base units of quantities can be defined and that you quote correct units for quantities; otherwise your calculations may become extremely confused The relationships that lead to these derived units are set down in the basic laws of Physics and Chemistry and these are often expressed in the form of equations The derived units of a quantity can be determined by writing the equation in terms of that quantity and then considering the units of the other terms in the equation It is common practice to use symbols to represent the quantity names as well as units in equations rather than writing expressions in words Whilst quantity symbols are not defined in the SI system, certain symbols have been adopted to represent specific quantities for many years so you should become familiar with and follow these traditions The symbols that are commonly used to represent the quantities listed in Table 31 are given below in Table 32 Table 32: Quantity symbols for SI base units quantity length mass time temperature current luminous intensity amount of substance quantity symbol l m t T I I v n You must learn the information given in Table 32! Whilst preparing reports, extra care needs to be taken to avoid ambiguity when using symbols, as the same symbol may be used to represent several different things It is important that you consider the context in which a symbol is being used in order to interpret its meaning correctly As an example consider the letter m It is used as the symbol to represent: the quantity - mass, the prefix - milli, which has the value of 10-3, the unit - metre, so the symbols mm in text would represent the word millimetre as a prefix symbol always comes directly before the unit symbol, without an intervening space The SI system sets out a number of writing style conventions which should be adopted to ensure the meaning of any symbol is unambiguous Workbook 1 Communication of scientific data 1:3:17

18 The following guidelines should be considered when interpreting quantity information that you are reading and when expressing quantities in documents you are writing The SI conventions for expressing quantities are as follows: In printed text, unit symbols are printed in normal or erect type face whereas quantity symbols are printed in italics as shown in Tables 31 and 32 Spaces are used to distinguish between prefixes and unit symbols The unit symbol always stays in the singular form even when the number is more than one ie a length of five centimetres is written in symbols as 5 cm not 5 cms Make sure that you adopt the following conventions: o Put a space between the number and the letters representing its unit unless the next character is a superscript So we write 298 K but 25 C and 0005 ma but A o o Thus Put a single space between the elements of compound units unless the previous term involves an exponent Whereas Never put a space between a prefix and a unit mn represents the unit of a force, millinewton m N corresponds to the compound unit of torque, metre newton We should write: metres per second as m s -1 rather than ms -1 (per millisecond) and the units of the universal gas constant, R, as J K -1 mol -1 not J K -1 mol -1 Also, remember that any symbol that is used to represent a quantity name should always be defined in the document Introduction to prefixes Different sizes of a unit are indicated by preceding the unit symbol by a scaling factor symbol We will show below that the symbols c and m can be used to represent the scaling factors of one hundredth and one thousandth respectively These symbols can be attached to any unit of choice and they always mean one hundredth and one thousandth of that unit There are many other symbols of this kind that are defined by the SI unit system to represent specific scaling factors These symbols are known as prefixes Later in this book we will: meet other prefix symbols that are used as scaling factors for other sizes of unit; consider calculations in which prefixed units are involved Note that the SI convention for writing prefixed units states that there should be no space between a prefix and its unit symbols When converting between different prefixed units, it is often useful to consider these different ways of expressing numbers Workbook 1 Communication of scientific data 1:3:18

19 For example, the length 1 cm may be represented: in words as one centimetre, or one hundredth of a metre This can also be expressed using digits and the unit symbol: in fraction form as 1/100 m and 1/10 2 m, as one line form as 10-2 m, in standard form as m When expressing quantity data, care has to be taken to read/write information correctly to avoid any ambiguity For example: 1/100 m should be read as the fraction unit ie 1/100 m (one hundredth times 1 metre) = 001 m; This represents a measurement of the quantity, length, l The length 1/100 m can also be written as 1 cm, where the symbol c means one hundredth So the prefix symbol c has the name centi and represents the factor or The prefix symbol and its factor are interchangeable, ie cm m Notice that the prefix c has been substituted for by the expression This type of substitution can prove to be very useful in calculations involving quantities Sometimes you will need to change the prefix in front of a unit We have just seen that we can eliminate the prefix entirely by substituting for its scaling factor Changing the scaling factor to a different size is more complicated, especially when reciprocal units are involved We will consider examples of these later Non-SI units You may come across other units which are often used in conjunction with SI units, but which are not strictly part of the SI system For example, according to the National Physical Laboratory (NPL) it is acceptable to use the units: minute, symbol min, hour, symbol h, day, symbol d and litre, symbol l or L with the metric system In the laboratory you will often use pieces of equipment which are calibrated in non-si units Examples include: scientific glassware, stop clocks, thermometers and gas pressure regulators In order to use data obtained from such equipment in calculations, you will frequently have to convert your raw data to its SI equivalent unit Conversion factors for many non-si units can be found in Chemistry and Physics reference data books Table 33 shows some common examples Workbook 1 Communication of scientific data 1:3:19

20 Table 33: Conversion factors for some quantities quantity non SI unit conversion factor to SI units length Ångstrøm 1 Å = m = 01 nm pressure torr torr N m -2 heat calorie 1 cal = 4184 J volume litre 1 L = m 3 = 1 dm 3 energy electron volt 1 ev = J mass atomic mass unit 1 amu = kg Conversion between different units There are many different ways to set out a calculation that shows unit conversions We will consider different ways to do this Using proportions Conversion factors can be set down in the form of equations, as shown in Table 33 If we remember that if the same operation is carried out on both sides of an equation, then the overall equation remains valid, so we can scale the conversion factor by any factor that we choose Consider the following examples Example 1 Convert the energy 200 ev into joules First, set down the conversion factor between ev and J Given 1 ev = J, then 200 ev = ev = J = J correct to 3 sig fig Notice that we need simply to multiply both sides of the conversion factor equation by the ev energy value that we need to convert in order to obtain the value in J Example 2 Convert 150 torr into N m -2 First, set down the conversion factor between torr and N m -2 Given 1 torr = N m -2, then 150 torr = torr = N m -2 = N m -2 Therefore 150 torr is N m -2 correct to 3 sig fig (or N m -2 ) Sometimes it is necessary to rearrange the conversion factor equation before we apply the scaling factor In the original SI report there was a difference in that 1 L was equal to dm 3 This anomaly was rectified in 1978 and subsequently 1 L 1 dm 3 Workbook 1 Communication of scientific data 1:3:20

21 Example 3 Convert into amu First, set down the conversion factor between kg and amu, Given amu Note: we need to convert kg into amu so we need the equation expressed in terms of By reversing the order we can write amu Dividing both sides by gives: kg, so It follows that amu = amu Therefore is 161 amu correct to 3 sig fig Using ratios In this method, we write two ratios The first ratio is formed using the information detailed in the conversion of unit problem The unknown quantity in terms of the units required is written as the top of the ratio and the quantity in terms of the units that are given is written as the bottom of the ratio The second ratio is formed in similar way but using the conversion factor information The conversion factor quantity in the required units is written as the top and the conversion factor quantity in the units that are given is written as the bottom of the ratio The two ratios must be equal so we can then write the following equation: Workbook 1 Communication of scientific data 1:3:21

22 This can then be rearranged with the as the subject Since all the other values are known, these can be substituted to find the unknown value in the required units Example Convert 100 cal to J The given quantity is 100 cal so we let x J be the unknown quantity Using information given in Table 33, we find that 1 cal = 4184 J So we can write and Since the two ratios are equal then Rearranging gives Therefore 100 calories are 418 joules correct to 3 significant figures Note: We will see other examples later of how this method and other methods can be used We will also see that this method does not work for conversion between different temperature scales Units encountered in the laboratory Measurement of volume Volume is probably the most common quantity that is measured in the chemistry laboratory Conversion of volumes to different sizes and types of units should be simple but many people find this sort of calculation difficult Examples of incorrectly calculated concentrations particularly are commonplace in student reports Think! When these types of mistakes occur in the work place, they can have serious or even fatal consequences for patients Whatever your scientific discipline, you must make sure that you can do calculations of this kind correctly We will consider some different approaches The SI base unit of volume is the metre cubed Imagine a cuboid box with all sides equal to a size of 1 m in length This has a volume of 1 m 3 One metre cubed is not a very convenient unit of volume to use in the laboratory as often we work with much smaller volumes of solution In reports and other scientific documents, it is common to quote volumes in terms of cm 3 and dm 3 Workbook 1 Communication of scientific data 1:3:22

23 Most pieces of laboratory glassware are calibrated in litres, L, or millilitres, ml; although you may come across a few items that are calibrated in microlitres, µl, centimetres cubed, cm 3 and decimetres cubed, dm 3 Note: units such as the centilitre, cl and decilitre, dl are sometimes used to specify the volume of everyday consumer items eg wine, spirits and shampoo When working in the laboratory, you are frequently required to convert between different forms of volume units Conversion between the SI metric units, (cm 3, dm 3 and m 3 ) and the non SI units (litres and millilitres) is easy provided you know the fundamental relationships between the two systems (We will consider converting to other less common sizes of units later in the course) Note: in some texts, the symbol lowercase L, (l) is used to represent the unit litre However, using certain font styles, it is not always possible to distinguish between the number one, 1, lowercase L and capital I The litre will be represented by L throughout this workbook to prevent ambiguity Volume equivalents Converting between ml, L, cm 3, and dm 3 There are many ways that you could perform conversions between different volume units The easiest way is to learn the following basic relationships then you will be able to work out most of the volume conversions that you encounter quite simply REMEMBER 1 ml is exactly the same volume as 1 cm 3, also 1 L is exactly the same volume as 1 dm 3 There are 1000 ml in 1 L also there are 1000 cm 3 in 1 dm 3 LEARN THIS INFORMATION Alternatively you can convert between ml, L, cm 3 and dm 3 using either the proportions method or the ratios method explained previously Workbook 1 Communication of scientific data 1:3:23

24 Using proportions Since we know that litres to millilitres conversion factor Then, by dividing throughout by 1000, millilitres to litres conversion factor We can then convert any volume in millilitres to its litre equivalent or any volume in litres to its millilitre by using direct proportions If we increase the volume on one side of the equation by a scaling factor, x, then we must increase the value on the other side of the equation by the same factor, x, to maintain the equality To find the volume in litres from millilitres, start with the millilitres to litres conversion factor: Since then, So generally, and so on To find the volume in millilitres from litres, start with the litres to millilitres conversion factor: Examples Since then, In general Convert the following volumes using proportions: (i) 01 L to cm 3 Using, then So Therefore and, and so on Workbook 1 Communication of scientific data 1:3:24

25 (ii) 250 ml to L Using, then So Therefore Exercise 35 Convert the following volume units using proportions: (i) 75 ml to dm 3 Using, then So and Therefore (ii) 0075 dm 3 to ml Since 1 dm 3 = 1 L then 0075 dm 3 = 0075 L Also 1 L = 1000 ml, so Therefore 0075 L = ml = 75 ml 0075 dm 3 is 75 ml Workbook 1 Communication of scientific data 1:3:25

26 Using ratios We can write the relationships between volume units in terms of the following ratios of conversion factors: eg Write conversion factors in a similar way for cm 3 to L and ml to dm 3 Examples Use the general equation: to convert the volume 500 cm 3 to ml, L and dm 3 (i) Conversion to ml Let the measured volume be x ml, then Rearranging gives Therefore 500 cm 3 = 500 ml (ii) Conversion to L Let the measured volume be x L, then, therefore 500 cm 3 = 0500 L (iii) Conversion to dm 3 Let the measured volume be x dm 3 cm Therefore 500 cm 3 = 0500 dm 3 Workbook 1 Communication of scientific data 1:3:26

27 What can we see from the previous examples? A volume in ml is equal to that volume in cm 3 and a volume in L is equal to that volume in dm 3 Therefore, to convert between the units ml and cm 3 or L and dm 3, we can simply change the units For example, we can write: 500 ml 500 cm 3 and 25 cm 3 25 ml 25 L 25 dm 3 and 075 dm L So 253 ml 253 cm 3 and 752 cm ml 10 L 10 dm 3 and dm L Exercise 36 Where necessary use the ratio method for the following volume conversions (i) 575 ml into a cm 3 b L c dm 3 a b Let x be the unknown volume in L Using the general equation: Rearranging gives so Therefore c Workbook 1 Communication of scientific data 1:3:27

28 (ii) L into a dm 3, b ml c cm 3 a L = dm 3 b Let x be the unknown volume in ml Using the general equation: Rearranging gives so Therefore c Workbook 1 Communication of scientific data 1:3:28

29 Exercise 37 Table 34 below summarises the relationships between a several volumes expressed in ml, L, cm 3 and dm 3 In the table, each row represents equivalent volumes The first row shows the equivalent volumes for 1 ml Complete the remaining entries in the table to show the equivalent volumes for each row Table 34: Volume unit equivalents ml cm 3 L dm Use the space below for any calculations you wish to make Workbook 1 Communication of scientific data 1:3:29

30 Measurement of pressure Traditionally, pressure has been measured using several different units When using older pieces of equipment or reading past literature, you may come across some of these For example, when measuring gas cylinder pressures, you are measuring very high pressures so you may come across units such as bars, pounds per square inch and atmospheres and when measuring pressures in vacuum systems and instruments you may be measuring very low pressures so you may encounter units such as torrs and millimetres of mercury, mmhg In calculations, you will often need to convert these values into the SI unit, pascal, Pa Make sure that you use the appropriate conversion factor to convert your values to the SI units when required Table 35 shows the conversion factors between various pressure units Table 35 Conversion factors between common pressure units pressure unit unit symbol conversion factor pascal Pa 1 Pa = 1 N m -2 torr torr 1 torr = N m -2 atmosphere atm 1 atm = Pa bar bar 1 bar = 10 5 Pa Example 1: Convert the pressure 1275 bar to pascals To perform this unit conversion, let us use the ratio method In this example, the units required are pascals and the units given are bars From Table 35, we can see that the conversion factor between bars and pascals is 1 bar = 10 5 Pa So using the information from Table 35, we can write Let x Pa = 1275 bar, then It follows that: Rearranging gives: Therefore a pressure of 1275 bar is Workbook 1 Communication of scientific data 1:3:30

31 Example 2: Convert the pressure 1275 bar to pascal Repeat the calculation using the proportions method Using the conversion factor, then Exercise 37 Use the specified method to convert the following pressures to the stated units a 1115 bar to atmospheres using ratios Recall that Let Since x atm = 1115 bar 1 atm = Pa = Pa 1 bar = 10 5 Pa Substituting values gives Rearranging gives bar bar Therefore a pressure of 1115 bar is 1100 atm, (correct to 4 sig fig) b 750 torr to pascal using proportions Since 1 Pa = 1 N m -2 and 1 torr = N m -2, then 1 torr = Pa It follows that: Therefore a pressure of 750 torr is better expressed as This result is correct to 3 sig fig and is Workbook 1 Communication of scientific data 1:3:31

32 Measurement of temperature When measuring the temperature during an experiment, the units of any raw data would normally be recorded directly from the thermometer in degrees celsius, C The temperature in degrees celcius is often represented by the symbol, t In calculations this often needs to be converted to absolute temperature, T, with units of kelvin, K As a scientist, you should be able to convert between the celsius and kelvin scales without reference to data books To convert between C and K you simply need to remember the equation: or by rearrangement Note: For a temperature rise: It therefore follows that a temperature rise, is identical to a temperature rise For example, a temperature rise of 150 C is also a temperature rise of 150 K You are recommended to quote temperature changes in Kelvin in your practical work We can work out conversions between the 2 temperature scales by substituting into the equations above Example 1: Example 2: If then so If then so In words then: C To convert temperatures in degrees celsius to kelvin, add to the numerical value of t To convert temperatures in kelvin to degrees celsius, subtract from the numerical value of T Temperature changes are identical whether stated in C or K K is the preferred choice, Note that when units take the name of a scientist, normally the word is written in full, in lowercase, whereas the symbol is a capital letter Workbook 1 Communication of scientific data 1:3:32

33 Remember: When converting between C and K you must make sure that you use the appropriate number of significant figures To do this, you need to consider the precision of your original temperature measurement Note: For normal laboratory work, mercury thermometers are significantly more accurate than either alcohol thermometers or cheap electronic devices The precision depends on the type being used A typical mercury thermometer should have an accuracy to within ± 05 C and can often be read carefully to a precision of ± 02 C Alcohol thermometers can typically measure a temperature range of between -10 C to 100 C, giving values with an accuracy to within ± 2 C with precision of ± 1 C For most temperature readings, the uncertainty of the measurement is at best ± 05 C so there is no point in using the precise value of to convert to K and it is sufficient to round the conversion factor to 273 in these calculations Examples: Convert the following temperatures from: (i) degrees celcius, C, to kelvin, K When, When, When, When, (ii) kelvin, K to degrees celcius, C When, When, When, When, Workbook 1 Communication of scientific data 1:3:33

34 Exercise 38 1 Convert the following temperature values to the stated units Remember to quote the appropriate number of significant figures in each case (a) To kelvin, K (i) 355 C;, correct to sig fig (ii) 2518 C; K correct to 2 dp (b) To degrees celcius, C (i) 3257 K; (ii) 125 K; = = 525, C correct to 1 dp = C correct to 3 sig fig 2 Convert the temperature change,, values in the following (a) To kelvin, K (i) correct to 2 decimal places (ii) correct to 1 decimal place (b) To degrees celsius, C (i) correct to 1 decimal place (ii) correct to 3 sig fig Workbook 1 Communication of scientific data 1:3:34

35 Self study revision exercises 1 (a) Complete the following table: Table 1 Base SI units quantity quantity symbol base SI unit unit symbol length l metre m mass time temperature amount of substance electric current luminous intensity m kilogram kg t second s T kelvin K n mole mol I ampere A I v candela cd (b) A few examples of SI derived units are given in Table 2 Complete the missing entries in the table Table 2: SI derived units quantity SI base unit unit symbol(s) area square metre m 2 volume cubic metre m 3 speed metre per second m s -1 concentration mole per metre cubed mol m -3 Workbook 1 Communication of scientific data 1:3:35

36 2 (a) Without using a calculator, convert the following temperature values to kelvin (i) 250 C (ii) C (iii) 273 C (iv) 60 C Answers (i) 523 K (ii) 255 K (iii) 546 K (iv) 333 K (b) Without using a calculator, convert the following temperature values to degrees celsius (i) 225 K (ii) 80 K (iii) 500 K (iv) 425 K Answers (i) C (ii) C (iii) 227 C (iv) 152 C 3 (a) Complete the entries in Table 3 below such that each row represents equivalent volumes Write your answers in either regular number or engineering form as appropriate Table 3: Volume equivalents μl ml cm 3 L dm 3 m Workbook 1 Communication of scientific data 1:3:36

37 (b) Convert the following volumes into: litres, L and centimetres cubed, cm 3 Table 4: Volume equivalents in L and cm 3 volume volume/l volume/cm ml ml ml dm ml (c) Convert the following volumes to: millilitres, ml, and decimetres cubed, dm 3 Table 5: Volume equivalents in ml and dm 3 volume volume/ml volume/dm 3 1 L dm cm cm L (a) For the following problems: Estimate the answer Use your calculator to determine the precise answer State your answer in an appropriate form to the correct number of significant figures (i) (ii) (iii) Workbook 1 Communication of scientific data 3:37

38 (b) For the following problems: (i) (ii) (iii) (iv) (v) Use your calculator to determine the precise answer State your answer in an appropriate form to the correct number of significant figures 5(a) Make x the subject in each of the following equations (i) so (ii) (iii) (iv) (v) (vi) 5(b) Rearrange the following equations (1) The equation for the concentration of a solute in solution, c, is c = n /V Express the equation in terms of: (i) the volume of solution, V ; ans: V = n / c (ii) the amount of solute, n ans: n = V c (2) The equation of state for a perfect gas is Express the equation in terms of: (i) the volume of gas, V; ans: (ii) the amount of gas, n; ans: (iii) the pressure, p ans: Workbook 1 Communication of scientific data 338

39 (3) The Beer-Lambert Law states that Absorbance, A = c l Express the equation in terms of: (i) the path length, l; ans: (ii) the concentration, c; ans: (iii) the molar extinction coefficient, ans: 1 2 (4) Show that the equation: Mnc nrt 3 rearranges to: c 3RT M So Workbook 1 Communication of scientific data 339