MATHCHEM 1.0. By Madhavan Narayanan Graduate Student, Department of Chemistry, Temple University

Transcription

1 MATHCHEM.0 By Madhavan Narayanan Graduate Student, Department of Chemistry, Temple University

2 Preface I dedicate this document to my beloved parents and my teachers without whom I would not have had the best thins in life. The motivation to write this document stems from my interaction as a Teachin Assistant with students in the eneral chemistry proram at Temple University. From my interaction I have found that most of the students are uncomfortable with basic alebra and, hence, are scared and not confident with the subject matter. Instead of runnin away from the problem, it is better to take a step towards it and make a enuine attempt to be friends with it. Section of this document is a review of basic alebra. This review is not comprehensive, but covers the most common concepts that the students need to know. This section also includes a discussion of some common mistakes that students usually make in their calculations. Also included in this section is a brief discussion on units and scientific notation. Section explains how to plot a raph between two different quantities on the X- and Y-axes. It also explains how to obtain a best-fit line. This section ends with a brief step by step instruction on how to use Microsoft Excel to plot a XY scatter plot and obtain a best-fit line for the same. Section 3 explains the need for usin moles in chemistry. It also includes a story about how Avoadro s number came into existence. Section 4 deals with makin dilutions from a stock solution. Section 5 reviews makin solutions of required concentration from a bottle of commercially available acid. Section 6 ives a detailed description of titration. In this section, the students are introduced to the concept of ram equivalent and normality. Usin the definitions of ram equivalents and normality, it is shown how normality is related to molarity. Finally, a sample calculation is done to show that calculations can be performed either with molarity or normality as lon as we take into account the ram equivalents or moles involved in the reaction. This document is an attempt to provide a detailed explanation of certain basic concepts that present particular difficulty to students. These concepts are traditionally assumed to be understood by the students, and as a result are not explained clearly. I would be elated to know that the students found this document useful. This document would not have been possible without help from my colleaues and friends. I would like to thank my friends Mr. David Yearsley, Mr. Konshen Yan and Mr. Konstantine Halkidis for their useful feedback on the manuscript. Many thanks to Ms. Rubinder Sinh, Director of Student Services, Collee of Science and Technoloy, Temple University for her useful inputs. I would like to especially thank Mr. Salim Siddiqui for carefully reviewin many versions of the manuscripts and providin me with useful suestions. It is always a pleasure discussin various topics with Prof. David Dalton whose feedback was immensely useful. Many thanks to Ms. Nicole Robinson of Sima-Aldrich for her help in providin the label of perchloric acid. ast but not the least I would like to thank to Mr. Mark Schwartz, Coordinator, General Chemistry Proram, Department of Chemistry, Temple University, for his encouraement in preparin this document. Constructive criticism and suestions are most welcome. Philadelphia Madhavan Narayanan October, 004 chemtamadi@yahoo.com

3 SECTION : MATHEMATICS CONCEPTS REQUIRED FOR GENERA CHEMISTRY ABORATORY Document prepared by: Madhavan Narayanan, Department of Chemistry, Temple University, Philadelphia PA This document contains a list of concepts that you are required to know for the General Chemistry ab. This list is just a uideline and does not reflect all of the different problems that appear in the General Chemistry lab. Mathematics is a tool that is absolutely essential to better understand and appreciate chemistry. It is of utmost importance to have a ood understandin of at least some of the basic mathematical concepts so that you can appreciate the beauty of chemistry. Are you uncomfortable with mathematics? Many students feel uncomfortable with mathematics. The only way to loose this discomfort is to become friends with math. The only way to become friends is to make a sincere attempt to understand and practice usin math until you et comfortable with it. First try to relate the math you learn to your everyday activities. In everyday life, there are plenty of occasions when you need to add, subtract, multiply and divide. For example, if you have set aside $0 for your daily expenses, and if you would like to see how much money you spend in a day, you would add up all your expenditure. If you were tryin to check how much money is left with you at the end of the day, then you need to subtract your expenses from the $0 you had at the beinnin of the day. You can try this as an exercise and see if the balance left makes sense. You can try out similar exercises for multiplication and division. For example let us say you o out shoppin for Christmas ifts for your 0 little cousins and you want to ive them the same ift so that your cousins would not fiht. If the ifts are marked at $5 a piece, then there are two ways to find out how much you will end up payin in total: Method : $5+ $5+ $5+ $5+ $5+ $5+ $5+ $5+ $5+ $5 = $50 Method : $5 0=$50 You can either add $5 ten times and find that the total is $50 or you can multiply $5 0 = $50. In this case, it is easy to add $5 ten times to et $50, but what if we need to buy 00 items. Then, we do not want to be addin $5, 00 times. Instead, we can multiply the price of one item by 00 and find the total. If you were iven the cost of 00 items and you need to find the cost of one item, then we divide the total cost by 00 and we can fiure out the cost of one item. We should keep in mind when dividin that we are tryin to split up a lare number in to small parts. When multiplyin we are tryin to put all these little parts toether to form a lare piece. So you can see that multiplication is the opposite of division. The take home point that the mathematical operations you perform are not abstract operations. They are very useful to us in our everyday life. If you really want to enjoy learnin and usin math, then the first thin to do is to start relatin these mathematical operations to your everyday life. The more you practice usin them consciously, the more you will et familiar and confident in usin them. Sometimes readin up on the history of how and why certain mathematical terms came in to existence miht ive you an understandin of why we are usin these mathematical symbols and operations. You can find this information in some textbooks on the history of mathematics. Unit conversions: Instead of tryin to memorize the conversion between units without havin a feel for the size of these units, it will be a ood idea to try practical exercises to relate different units. For example, you can pick up a wooden stick and try makin different lenth measurements and try relatin different lenth units. Also when travelin in a car, try relatin different lenth units. You can use a bottle of soda or water to try relatin between volume units. You can try a similar practical exercise for relatin between different mass units. 3

4 Calculator: Calculators are useful tools for your calculations. Remember it is a tool and not a maic wand! If you are oin to use your calculators without havin a clear understandin of the oal for which you are doin the math, you will not have any idea if the numbers that your calculator displays are is riht or wron. Do not become totally dependent on your calculators! Try doin simple calculations just usin the paper and pencil. It is very important to know the multiplication tables for numbers to 0. Sometimes students have the most advanced calculators (e.., TI-83, TI-89, etc.,) but have no idea how to use the most basic mathematical functions in the calculator. earnin to use a particular calculator is like learnin a new lanuae. Unless you are familiar with what the functions on your calculator are capable of doin, then there is no point in havin a calculator. It is important that you have read the manual that comes with the calculator. If not at least talk to a friend who is familiar with the type of calculator you have and et familiar with usin the calculator independently. For example if you are tryin to multiply , before even typin these numbers in your calculators, look at the numbers carefully. We can see that the first number is reater than 5, but less than 6, and the second number is reater than, but less than 3. Therefore, the answer we expect should be between 0 (i.e., 5 =0) and 8 (i.e., 6 3=8). By punchin in the numbers in the calculator if you et a number reater than 8 or less than 0 then you know that you are doin somethin wron.. Basic alebra: a. Addition, subtraction, multiplication and division of numbers and variables. (i) Positive + Positive = Positive For example: + 3 = 5 (ii) Positive + Neative = Positive (if the positive number is reater than the neative number in manitude) For example: 3 + (-) = (iii) Positive + Neative = Neative (if the positive number is less than the neative number in manitude) For example: + (-3) = - (iv) Positive Neative = Neative Positive = Neative For example: -3 = - 3 = -6 (v) Positive Positive = Positive For example: 3 = 6 (vi) Neative Neative = Positive For example: - -3 = 6 (vii) Positive Neative = = Neative Neative Positive -3 3 For example: = = (viii) (ix) Positive = Positive Positive 3 For example: 5 = 0. 6 Neative = Positive Neative -3 For example: = b. Multiplication and division of exponential functions: a b a b (i) ( X ) ( X ) = ( X ) For example: ( X ) ( X ) = ( X ) = ( X ) 4

5 a b (ii) ( X ) ( ) a (iii) ( ) a (iv) ( X ) (v) a X X = ( ) b X 3 For example: ( X ) ( ) a b = ( X ) - æ X è X - 3 X = ç = ( X ) 3 Written another way: ( X ) ( ) X m a m = ( X ) For example = ( ) = a m ( X ) X 3 = ( 3 ) 3 ö ø - = ( X ) æ X ö æ X X ö æ ö X = ç = ç = ç 3 è X ø è X X X ø è X ø 6 X = ( X ) - = ( X ) What the above equation mean is, multiplyin ( X ) to itself 3 times (i.e., cubin the X ) is equal to multiplyin ( X ) to itself 6 times, because when you value of ( ) multiply ( ) X to itself 3 times, in writin ( X ) we have already multiplied ( ) itself times (i.e. squared the value of X). n a n = ( X ) a n = ( X ) 3 3 = ( X ) = ( X ) For example: ( ) 3 X The expression ( ) a X n Discussed, we can call ( ) c. Division of fractions: a b a a c a = c = = = c b b b c For example: 3 = = 3 3 is called n-th root of ( ) a X 3 X as 3 rd root or cube root of ( ) X.. In the example we have X to Enterin numbers with exponents in the calculator: Here are few steps if you want to enter in the calculator, Step i: Type in 6.0 Step ii: ook for the key that say EE or EXP (this depends on the make and model of your calculator) and press it. Then 0 and 00 will show up in the display. Step iii: Then enter 3. Step iv: Find the key that has + on it and press it. - The way the above number will be displayed will depend on the make and model of the calculator a bc = 3 = 6 d. oarithms: (i) lo (A B) = lo A + lo B For example: lo ( 3) = lo + lo 3 = = o (6) = (ii) (lo A) (lo B) = (lo A) (lo B) ¹ lo (A B) For example: (lo ) (lo 3) = (lo ) (lo 3) = = (iii) lo ( B A ) = lo A lo B = 3 3 For example: lo ( ) = lo 3 - lo 5 5 lo (0.6) = ~

6 (iv) lo A lo B A ¹ lo( ) B The symbol ¹ means not equal to. In this case the fraction on the left is not equal to the fraction on the riht. lo A lo A = lo B lo B lo3 lo For example: = = = ¹ lo( ) lo5 lo Please refer to the previous case for the value of lo( ) 5 The student should be familiar with calculatin lo values usin a calculator. If you are findin it hard to remember these rules, you can always pick numbers, let s say and 3 and substitute these for A and B and see if the values on the left side of the equation equals the values on the riht. (v) lo ( A ) n = n lo( A) For example: lo( ) 3 = 3lo( ) = lo(8) e. Relationship between n (i.e., natural lo or lo to the base e) and lo (lo to the base 0) n X =.303 lo X For example: n 0 =.303 lo =.303 () f. Takin inverse of the lo (i.e., anti lo) and inverse of n: If lo X = -a X = 0 -a For example: If o X = -.5 X = = If n X = -a X = e -a For example: If n X = -.5 X = e -.5 = The quantities calculated by takin lo or n have no units. For example ph = -lo [H + ]. The above equation means that ph is iven by takin the neative lo of the hydroen ion concentration. Even thouh the concentration of hydroen ion could have units of Molar (M), ph, which is calculated from the lo of hydroen ion concentration, has no units. A historical note: In 594, the Scottish mathematician John Napier of Merchiston had discovered the correspondence between the terms of an arithmetic proression (proceedin by successive additions of a constant quantity) and the terms of a eometric proression (proceedin by successive multiplication by a constant factor). This was the basis for his invention of loarithms, by which multiplication can be achieved by usin addition. He published the first table of loarithms, to the base e. Jost Buri had apparently made the same discovery in 604. Henry Bris (later Savilian professor of eometry at Oxford) in 65 found Napier s loarithm inconvenient for practical calculation (as a consequence of usin base e). Havin stayed a month with Napier in 65, Bris constructed a table of loarithms to base 0 (a chane that Napier had already envisaed), for the first 000 inteers, and with the entries iven to 4 decimal places. 6

7 The familiar loarithm tables are to base 0 ( common loarithms ). Any number can be expressed as a positive or neative power of any fixed number (base) which is not equal to. Thus, = = , for instance; power is the loarithm of the number to the iven base so that lo 0 = The base used by Napier is the denoted by e whose value is equal to (a non terminatin decimal). This apparently strane number is in fact fundamental in mathematics. Strictly speakin, Napier s loarithms were not to the base e thouh very closely related.. When you perform a series of mathematical operations in a function, the order of operation is very important. For example, if you were to calculate the result of {[(3+4) ]} 5, then the order of operation could be: Step i: Calculate =7 Step ii: Calculate 7 = 4 Step iii: Calculate 4 5 =.8 h. Findin square root ( ), cube root ( 3 ) and fourth roots ( 4 )of numbers: There are different ways of findin the roots of numbers. But you have to understand what square root or cube root or etc., mean. You may need to find roots of numbers, when you have an equation where you have a variable (say X) raised to a certain power (say n) on one side of the equation bein equal to some value (say A) (See the equation ()). n ( X ) = A () If we want to find the value of X, then we have to eliminate the n, which is on top of the X. What we are tryin to the find here is, the value of X which when multiplied to itself n times will ive us A. But keep in mind, to keep the equation balanced; whatever operation you perform on one side of the equation should be performed on the other side of the equation. Usin option 4, under section b (multiplication and division of exponential functions), we can eliminate the value of n from the exponent of X. To do that, equation () has to be raised to the power on both sides. Now, equation () will become: n n ( X ) n = ( A ) n Rearranin terms further in equation (), we will et, ( ) n X = ( ) Which means the value of X is equal to the n-th root of A. () A (3) For example if Then, ( X ) = 4 (4) ( ) X = ( ) 4 = 4 = (5) Which means the value of X is equal to the nd root of 4 or the square root of 4. The square root of 4 is. This means that when multiplied to itself times will ive you 4. There are many ways to find the value of X from equation 3. Here are some of them. 7

8 a. Usin a calculator for findin square root, cube root and 4 th root: Different calculators work differently, so you will have to fiure out how your calculator works. Find the symbol for square root ( press ( ) in your calculator. In some calculators you ) first, followed by the number for which you want to find the square root. In some other types of calculators you enter the number first and the operation (symbol) next. The same method works if there is a ( 3 ) symbol on your calculator and you want to find the cube root of a number. The simplest way to determine the 4 th root (i.e., ( 4 ) is to take the square root(i.e., ) twice. Check: By usin your calculator please make sure that you are obtain the square root of 4 as and the cube root of 8 as and the 4 th root of 6 as. b. Usin loarithms to calculate roots: Instead of tryin to find the roots of equation 4, we can use equation (6) to find the solution for a more eneral case. ( ) n X = ( ) A (6) Takin the lo of the equation on both sides of the equation ives n ( X ) lo( A ) lo = (7) Usin the method from section d, part (v), we have ( X ) lo( A) n If we want to find the value of ( X ) from lo ( X ) lo = (8), we will have to use the method iven in section f (Takin inverse of the lo (i.e., anti lo) and inverse of n). Usin this method, ( X ) lo( A) n = 0 (9) For example if we have an equation X 6 = 64 (0) Then we can transform equation 0 to, ( ) 6 X = 64 () If we take the loarithm of equation () on both sides, we have 6 ( 64 ) lo X = lo () Usin the method from section d, part (v), we have, X = lo ( 64) (3) 6 Usin the method iven in section f (Takin inverse of the lo (i.e., anti lo) and inverse of n), we have ( 64 ) 6 lo X = 0 (4) When solvin this equation you may use the followin order of operation: Step : Find the value of lo 64 usin the lo function in the calculator. lo 64=.806 Step : Find.806 lo( 64) = = x Step 3: Find 0 =.0 usin the 0 function in the calculator. 8

9 So the 6 th root of 64 is. This means that multiplied by itself 6 times equal 64. ( = 64). This method can be used if your calculator does not have the option to calculate hiher roots. c) Usin calculator to find hiher roots (roots reater than 3 or 4): Try to find the x in your calculator. If you have this icon then you can find hiher roots on your calculator. et us say you want to find 5 th root of 3 ( 5 3 ). Step i: Enter 5 in your calculator. Step ii: Then press x key on the calculator. Step iii: Then enter 3 and press ENTER. You will see. To see if you understood how to find the roots, try findin the 6 et 3 as the answer. 79 and see if you can. Common Mistakes: a. When takin averae of say numbers (0 and 9) usin a calculator, the averae is iven by: Averae = = 9. 5 The most common mistake happens if you do one of the followin: 9 i) 0 + = ii) 9 + = 9 So you have to make sure that you add 0 and 9 first and then divide the total by. ast but not the least, please check to make sure if the value that you have obtained makes sense. For example, when takin the averae between 9 and 0, even before you start puttin these numbers in your calculator, you should have a fair idea of what the averae value is oin to be. The averae of 9 and 0 should lie some where in between 9 and 0. When you take the averae by puttin the numbers in your calculator and end up with the number which is reater than 0 then you know that you are doin somethin wron. But if you obtain 9.5, then you know it makes sense. b. One more common mistake that happens is this: The ideal as equation is iven by PV = nrt () If we want to find n, at a pressure of.00 atm, volume of and temperature of 73 K, then we need to rearrane the iven equation so that we have just n on one side of the equation and all the other variables on the other side. If we want to do that then we should cancel out all the variables other than n on the riht hand side of the equation. But how do we do this? If we look carefully on the riht hand side of the equation, we see that n, R and T are multiplied toether. If we want to remove/eliminate R and T from the riht hand side, then we should divide the riht hand side of the equation by R and T. But why do we have to divide? Because, division is the opposite of multiplication. So now the equation will become nrt Incorrect PV = () RT 9

10 Before doin the division of the riht hand side of equation () by RT, PV was equal to nrt, that is, the product of P and V was equal to the product of n, R and T. But now that we have divided just one side (the riht hand side) of the equation by RT, the two sides are not equal any more and hence equation () is incorrect. Our oal is to keep both the sides of the equation equal all the time. So in order to keep both sides balanced/ equal, instead of just dividin one of the sides by RT, we will have to divide both sides of the equation by RT. So instead of equation () we should have PV nrt = correct (3) RT RT Now we can cancel off the RT terms in the numerator and denominator of the riht hand side of the equation. We will now be left with PV = n (4) RT So now all we have to do is to substitute the value of P, V, R and T in the equation and fiure out the value of n. After substitutin the values, the equation becomes,.00 atm 5 0 n = atm. mol K -5-73K (5) When calculatin the value of n usin a calculator, this is how the most common mistake happens: Step i: Multiply.00 with and you will et Step ii: Then divide by 0.08 to et Step iii: Multiply by 73 K and et 0.66 for the value of n. Everythin in the above process is correct until step. But in step 3 instead of dividin by 73 K, one ends up multiplyin 73 K to the value. So if you want to avoid this mistake a useful method will be: Step i: multiply.00 with 5 x 0-5 and you will et 5 0-5, write this down on your work sheet. Step ii: multiply 0.08 with 73 and you will et.4. Write this down on your work sheet. 5 0 Step iii: Now your equation to find n will simplify to.4 Step iv: Now it s easy to input this in your calculator and divide the numerator by the denominator. You will see that the answer is moles. You have to be very careful enterin the numbers in your calculator. If you make any mistake here, then rest of your calculation will be incorrect. 3. Units: Never foret to use appropriate units for quantities involved in calculations. Most quantities have units unless otherwise specified. In many cases the quantities or the values you have calculated will have no meanin unless appropriate units are specified. For example, if you were to tell your friend that the heiht of the Bell tower on temple s campus is 50, do not be surprised to see a blank look on your friend s face. Your friend has no idea whether the heiht is 50 m or 50 feet or 50 inches. Just sayin that the heiht is 50 has no meanin unless you put in the appropriate units next to the number. If you were to say it riht, you could say that the heiht of the bell tower is 50 feet. When doin calculations, you will perform various mathematical operations with various quantities. Each of these quantities may have different units. It is your responsibility to keep track of all these units when doin calculations. You can do mathematical operations with the units of quantities like you do with the quantities themselves. When solvin an equation, please make sure that the units are balanced on both sides of the equation. There are scientific reasons for why and how mathematical relationships exist between certain quantities. It is important to keep these reasonin in mind when performin calculations. Knowlede of this reasonin will tell you if the value you have calculated makes sense or not. -5 0

11 Here are some interestin links about measurements and units: Sinificant fiures: Please make sure that you have appropriate number of sinificant fiures in the calculations that you will be doin. Refer to your chemistry textbook or the experiment on measurement and density in your lab manual to refresh your memory about the rules on sinificant fiures. Please use scientific notation when representin very lare and very small numbers. For example, 43,000,000 can be written in scientific notation as The number can be written in scientific notation as Note: When usin calculators, please make sure that the floatin point in your calculator is set to the maximum number. Usually it is 9 for most the calculators. 5. Scientific notation: Decimal notation is the everyday method for expressin numbers. Such notations become cumbersome for very lare and very small numbers (which occurs frequently in scientific work). For example, the number of atoms in one mole is 6,0,000,000,000,000,000,000,000. The mass of one atom of hydroen is Recordin such lare and small numbers is not only time consumin, but also open to error; often too many or too few zeros are recorded. Also, it is impossible to multiply or divide such numbers with most calculators because they cannot accept that many diits. Most calculators accept 8 or 0 diits. A method called scientific notation exists for expressin multidiit numbers involvin many zeros in compact form. Scientific notation is a system in which an ordinary decimal number is expressed as a product of a number between and 0 times 0 raised to a power. The ordinary decimal number between and 0 is called a coefficient and is written first. The number 0 raised to a power is called an exponential term. The coefficient is always multiplied by the exponential term, as in the followin example. Coefficient Exponential term Multiplication Sin Exponent (n) Convertin from decimal to scientific notation: When convertin a number to scientific notation, notice that the exponent n is positive if the number is reater than and neative if the number is less than one. The value of n is the number of places by which the decimal was shifted to obtain a number in scientific notation. 4 For example, the 345 can be written in scientific notation as = The number 0.00 can be written in scientific notation as =. 0-3

12 SECTION : HOW TO POT A GRAPH? Document prepared by: Madhavan Narayanan, Department of Chemistry, Temple University, Philadelphia PA A raph is a pictorial representation of your data. You should take care in the creation of your raph. Even if you have the best data if you don't create the raph correctly your results will be meaninless. Most of the raphs that you will plot in this course will have a set of values on the X-axis (horizontal) plotted aainst a set of values on the Y-axis (vertical). This means that for every X-value you have you will have a correspondin Y-value. If there are 5 X-values, then there will be correspondin 5 Y-values. You should keep the followin items in mind when makin a raph:. Title of the raph. abeled axis with units 3. Pae usae: Choosin appropriate scale 4. Plottin the values 5. Curve or Best-fit line 6. Equation of the best-fit line.. Title of the raph The title of the raph should tell you somethin about what is plotted and how it relates to the experiment. Please look at the sample raph in Graph. The title of the raph says A plot of temperature chane aainst distance from the surface of the earth. From the title of the raph, you can easily fiure out what quantities are bein plotted on the raph.. abeled axis with units The first thin to be done is to assin axes (of the four possible sides) on the raph sheet as X- axis and as Y-axis. Then you can proceed to ive names to the axes (plural for axis), based on the kind of data you are plottin. The kind of data you will be plottin will depend on the experiment. Each axis of your raph should be identified not only with the quantities you are plottin, but also with their appropriate units. In Graph, you can see that the X- axis is labeled as Distance from the surface of the earth, in km and the Y-axis is labeled as Temperature Chane, in C. From the axis title, we et a clear indication of what quantities are bein plotted and on each axes. Also from the axis title, we can clearly determine the units of these quantities. From the axis titles and the title of the raph, it should be easy to understand that by the raph represents how temperature chanes with chane in distance from the surface of the earth. 3. Graph Sheet usae: Choosin appropriate scale The point of makin the raph is to visualize and present data in a pictorial format. Therefore, it is to your advantae to make your raph lare enouh for easy readin. Students often just use the,, 3 approach to number the axis and the raph turns out to be the size of a postae stamp. What was the point of makin the raph if you can't see the details? Select the reion of the number line that encompasses your data points. For example, if your numbers are between 6. and 34.8 there is no point startin at zero, start with 5.0 and end with This reion of the number line startin from 5.0 and endin at 35.0 encompasses your data points; this is your reion of interest. The only complication with this method is if you need to make an extrapolation (i.e., usin the data in the "known reion" and extendin it to an "unknown reion"). In that case you should make the raph so that it also includes the unknown reion. This entails that you have a reasonably ood idea of what information you are tryin to et by plottin this raph. It is important that you identify the reion of interest for both your X and Y-values.

13 3a. Identifyin the reion of interest: i) Carefully o throuh your X and Y-values and identify the minimum and maximum value in your data set. ii) Once you have identified the minimum and maximum, pick a whole number value p, which is less than your minimum value and pick a whole number value q, which is reater than your maximum value. The rane of your data points is iven by (q p). Make sure the p and q values you chose are appropriate for the information that you are lookin to obtain from the raph. Please refer to the example problem to a et a feel for how the reion of interest is identified. 3b. Graph sheet: A raph sheet usually contains a whole bunch of little squares. All these little squares are present with in a boundary that could look like a square or a rectanle. If you carefully look at the raph sheet you will find that it contains some thick and thin lines. These thick lines run both horizontally and vertically. When a horizontal line meets a vertical line it forms a larer square that contains many small squares. We can call these lare squares as major units and the little squares as minor units. You will see in Graph and Graph that the major unit is made up of 0 little squares (i.e., minor units) alon each axis. 3c. Choosin Axis: 3d. Scale: When we bean talkin about raph, there was a brief mention about X and Y-axes. How do we set our axes? Of the 4 sides in a raph sheet, you can assin any one of the sides as your X- axis. The side that s perpendicular (90 ) to the X- axis will be your Y-axis. There are two possible choices for Y-axis; you have to pick one of the two. Scale will be decided based on the reion of interest. Each axis will have its own scale. The scale for both axes may or may not be the same. How is the scale decided? Step i: Count the number of little squares alon a particular axis. et s say its n Step ii: Based on your data points for that axis, choose the p and q values (Refer section 3a). q - p Step iii: Calculate ( ) = t where t is called the scale of the raph. n Step iv: The first point on that axis should be p + t. This point p + t is marked the first line on the axis. The next point should be p+ t+ t or p + t and so on until you reach q. But it is not necessary to indicate the values on each and every point. It is sufficient to show the labels on every 5 th point or say every 0 th point on the axis. Step v: Follow step throuh 3 for the other axis. Please look at Graph. There is a little box towards the bottom riht hand corner of the raph, which has information about the scale of the raph. More detailed calculation on how this scale was obtained is iven on the next pae Note: The numbers that you may need to plot may be positive or neative. To understand how to plot these numbers on the raph, you can use Graph as a reference. In Graph, the sheet has been divided in to 4 quadrants so as to be able to plot both positive and neative numbers, both on the X and Y-axes. Here are some of the possible scenarios: 3

14 Table : Numbers on the X-axis Numbers on the Y-axis The raph will look like Positive Positive Quadrant I Neative Positive Quadrant II Neative Neative Quadrant III Positive Neative Quadrant IV 4. Plottin the values: Once you have calculated the scale and labeled the axis (i.e., numbers, titles and units), you are now set to plot the data points. et us say for a hypothetical case that you have 5 X-values (i.e., x, x x 3, x 4, x 5 ) and correspondin 5 Y-values (i.e., y, y, y 3, y 4, y 5 ) to be plotted. 5. Curve or best line fit Step i: Decide which value (X or Y-value) you are oin to plot first. If you decide to plot the X-value first, then pick the first X value, which is x for our hypothetical case and find it alon the X-axis. Put a faint cross mark at that point on the X-axis. Step ii: Then pick the correspondin Y value, which is y and try to find it alon the Y-axis. Put a faint cross mark at that point on the Y-axis. Step iii: Follow the line on the raph that runs perpendicular (i.e., 90 ) to the X- axis from x and the line that runs perpendicular to the Y-axis from y. See where these two lines intersect. This point of intersection is the first data point (x, y ). Circle this point of intersection and label it (x, y ). It is always a ood idea to label the points. (Please refer to Graph ) Step iv: Follow steps ii and step iii for the other data points (x, y ), (x 3, y 3 ), (x 4, y 4 ) and (x 5, y 5 ). Now you have successfully plotted all your data points on the raph. As a practice, try plottin the raph for the data points iven in table and check if you are able to obtain a raph similar to Graph. Once the data points have been raphed you need to decide how to "connect the dots". This is a fiurative expression. In most cases you will NEVER connect the dots! You will either draw a line that "best fits" the data. This means you will draw a line, not a stock market raph with ups and downs. The best-fit line is an averae of all of the data points. It is possible that none of the data points are actually on the line. On an averae, some points will be below the line while others will be above it. The idea is that you are averain the data pictorially. Best line fit (a. k. a Best-fit line): Why is this needed at all? By drawin a best line fit, we are tryin to see if the data we plotted on the raph fall perfectly on a straiht line. Most of the time you will see that not all the points will fall on this straiht line. This could be due to various factors. The reason for checkin if they all fall on a straiht line is because there is a linear dependence between the quantities that are bein plotted on the Y and X- axis. Most of the experiments for which you will plot a raph are time-tested experiments. The relationship between the quantities involved in these experiments has already been established throuh experiments and theories of many renowned scientists. If the data you have collected has a perfect linear relationship they will all fall on the straiht line perfectly, provided there are no errors in our measurements. 4

15 Since we are tryin to fit the data to a straiht line, the relationship between Y and X should be compared to that of a eneral equation for straiht line, which is iven by Y= mx + C. The straiht line that s a best fit for the data tries to minimize the error in the data. By findin the equation for the best-fit line for the data, we are findin the relationship between the quantities in the Y-axis to the quantities on the X-axis. If we know the equation for the best-fit line then we can predict the value of Y for an unknown X or predict the value of X at an unknown Y without doin an experiment under those unknown conditions. So there is a lot of value in drawin the best-fit line and obtainin its equation. How to obtain a best line fit? Once you have plotted your data points, here are some points to have in mind when drawin the best line fit: 6. How to obtain the equation of the best line fit? (i) There are many ways to obtain a best line fit. We will be usin the simplest method to draw this best-fit line. The simplest method that we are oin to use is not necessarily the most accurate method. But this simple method ives a reasonably accurate fit for our data. (ii) Remember, it s a straiht line that you are tryin to obtain, so do not use a free hand drawin to draw the straiht line. Always use a ruler to draw the best - fit line. (iii) When drawin a best-fit line the usual way is to use a ruler and draw a straiht line throuh the data points such that most of your data points fall on the line (see Graph ). But often none of the points may fall on the line. Under these circumstances, the line can be drawn such that you have points on both sides of the line. Step i: Once you have plotted the best fit line, pick any two points which lie far apart on the line, other than your data points and label them R and S. It is not advisable to pick the same points as your data points. Step ii: Find out the X and Y values of the two points you have picked. Point R will be (x R, y R ) and point S will be (x S, y S ). chane in y Dy ys - yr Step iii: The slope m = = = (See the calculations for chanein x Dx x - x sample problem and Graph ) Step iv: The intercept C is the value on the Y-axis where the best-fit line intersects the Y-axis. Step v: The equation for the best-fit line that we have just drawn will have the form Y= mx + C. So substitute the value of m (from step iii) and the value of C (from step iv) in the equation Y= mx + C. S R 5

16 Sample Problem: Below is an example of how to plot a raph. The followin data represents temperature chane at different heihts from the surface of the earth. Table : X (Heiht above the surface of the earth, km) Y (Temperature chane, C) Thins to keep in mind:. For every X-value, there is correspondin Y-value. If the X-value is 0., the correspondin Y- value is The sins of the numbers on X and Y-axes are different. All the numbers on the X-axis are positive where as the numbers on the Y-axis are neative. We can use this information and see if in raph (table ), any of the quadrants have positive X-values and neative Y-values. You will see that quadrant IV satisfies this condition. So the raph you will plot will have axes that look like quadrant IV (see raph and compare it with quadrant IV in raph ). Step : Count the number of little squares alon the axis. For example, alon X-axis, there are 60 little squares. So n = 60. See raph. Step : To fiure out the scale for X and Y-axes: For X-axis: Identify the minimum value from the 5 X-values iven: 0. Identify the maximum value from the 5 X-values iven:.0 But for calculatin the scale, we need to fix a rane that includes our data points. So the rane has to have a lower limit that is lower than the minimum value and an upper limit that is reater than the maximum value. We are free to choose appropriate upper and lower limits dependin on what information we are lookin to obtain from the raph. Choosin a really wide rane is useless if the information that we are lookin for can be obtained by usin a smaller rane. Also, another disadvantae of choosin a wider rane is that the raph that we are tryin to plot will be left in one corner of raph sheet, with lot of empty space left on the raph. So our oal is to use the space on the raph sheet effectively for the information that we are lookin to obtain from the raph. For the above X-values, we can choose the followin limits: ower limit, which we denoted earlier as p: 0.0 Upper limit, which we denoted earlier as q:. Keep in mind that it s not necessary to choose 0. 0 as the lower limit for all data points. In this case it happens that 0.0 happens to be close to the minimum value. The scale for the X-axis is iven by ( q - p) n = = 0.0 km So each little square alon the x-axis represents 0.0 units. So the first point on the X-axis will be 0.0. The point riht next zero will be 0.0 and the second point will be 0.04 and so on. But we do not have 6

17 to label each and every point alon the X-axis. We can simplify our job by markin only the major unit (dark lines) that appears once every 0 little squares. So the first major unit will be marked (0.0 + (0 0.0)) = 0.). The second major unit will be marked (0.0 + (0 0.0)) = 0.4 and so on. For Y-axis: Identify the minimum value of the 5 Y-values iven: -0.9 (Remember!! for neative numbers, the value farther away from zero is the smallest, for example > - because is farther away from zero than ) Identify the maximum value of the 5 Y-values iven: -0.5 Upper limit q : -0.4 ower limit p : -.0 Alon the Y-axis, there are 90 little squares. So n = 90. See raph. (-.0 - (-0.4)) Scale= = C 90 So each little square alon the Y-axis represents C. The hihest point on the Y-axis will be 0.4. The point riht next to -0.4 (the first minor unit) will be and the second point (the second minor unit) will be and so on. But, we do not have to label each and every point alon the Y- axis. We can simplify our job by markin only the major unit (dark lines) that appears once every 0 little squares. So the first major unit will be marked (-0.4+(0 (0.0067))) = ). The second major unit will be marked ( ( ) = and so on. Step 3: Plottin the values: Once you have calculated the scale and labeled the numbers on the axis, you are now set to plot the points. There are a few important thins to be done before startin to plot the raph. Step (i): Decide which value (X or Y) you are oin to plot first. If you decide to plot the X-value, then pick the first X- value, which is 0., and try to find it alon the X-axis. Put a faint cross mark. Step (ii): Then pick the correspondin Y-value, which is -0.5 and try to find it alon the Y-axis. Put a faint cross mark. Step (iii): Follow the lines on the raph that runs from 0. and 0.5 and see where they intersect. This point of intersection is the first data point (0., -0.5). It is always a ood idea to label the points. See Graph. Step (iv): Follow steps throuh 3 for all the other data points. Data points Place the data points correctly on the raph. Many students miscount the lines of the raph paper and place data point in the wron location. In addition, if there are data points that just don't look like they make sense it could be one of two problems: () You misplaced the data point, or () the data is "bad". If the data is bad you need to think lon and hard as to why. This comes back to thinkin about sources of experimental error. If the data point is truly bad and you don't think you should use it in the analysis you should still place it on you raph. ater in your analysis of your data you may fiure out that your data is not meaninless after all. 7

18 Step 4: How to obtain a best line fit? (i) Remember, it s a straiht line that you are tryin to obtain, so do not use a free hand drawin to draw the straiht line. Always use a ruler to draw the best -fit line. (ii) When drawin a best-fit line the usual way is to use a ruler and draw a straiht line throuh the data points such that most of your data points fall on the line (see Graph ). But often none of the points may fall on the line. Under these circumstances, the line can be drawn such that you have points on both sides of the line. Step 5: How to obtain the equation of the best line fit? In the example we have discussed so far (Graph ), the two points chosen on the best line fit has coordinates: R (0.3, -0.56) and S (0.9, -0.86) The slope of the best-fit line, y - y R (-0.56) ( C) m = = = = x - x ( km) S S R C km Since slope is a measure of chane in Y with chane in X, based on our calculation of slope, we have fiured out that for every km chane in the distance from the earth, there is a chane in temperature of C. Please note that the slope can have units. The Y-intercept, C = -0.4 (see Graph ) Since the eneral equation for the best line fit is Y=mX + C, and now that we know the value of m and C, we can substitute the value of m and C in the equation and we et, Y= X 0.4, which is the equation of best-fit line for the example we discussed. Now that we have the equation for the best line fit, we have found the relationship between the quantities plotted on the Y-axis (temperature chane) to the quantity plotted on the X-axis (distance from the surface of the earth). Usin this equation it is possible to predict the temperature chane at any distance from the surface of the earth. Conversely, you can also use this equation to predict at what heiht from the surface of the earth there will be a certain value of temperature chane. 8

19 Graph : A sample raph showin how inteers are plotted on a raph sheet. Y 4 3 Quadrant II: X values are neative and Y values are positive Quadrant I: Both X and Y values are positive One major unit on the X-axis is made of 0 minor units 0 x x Quadrant III: Both X and Y values are neative - - Quadrant IV: X values are positive and Y values are neative -3-4 One major unit on the Y-axis is made of 0 minor units -5 Y 9

20 Graph : A Plot of Temperature Chane Aainst Distance From The Surface of the Earth Y-intercept C = -0.4 Distance from the surface of the earth, in km (R x ) (S x ) x (R y) (0., -0.5) R ys - yr Slope = m = xs - xr (-0.554) = ( C) = = 0.8( km) C km Temperature Chane, in C (0.4, -0.6) S (0.6, -0.7) (0.8, -0.8) ( S y) Scale: X-axis: major unit = 0. km Y-axis: major unit = C Scale: X-axis: y - y major unit = 0. km R (-0.56) ( C) slope = m = = x x Y-axis: = = S - R ( km) major unit = C S S (.0, -0.9) C km y 0

21 How to obtain a best line fit usin Microsoft Excel? Document prepared by: Madhavan Narayanan, Department of Chemistry, Temple University, Philadelphia PA Here is a list of steps to plot the raph usin Microsoft Excel. This list of step-by-step instructions does not cover all the wonderful thins that you can do with Microsoft Excel. This list of steps is meant to help you do the followin:. Plot a set of X and Y-values. Obtain a best-fit line 3. Obtain an equation for the best-fit line. The requirement is that you should have Microsoft Excel in your computer. Microsoft Excel is part of the Microsoft Office software. The computers on Temple campus libraries may have Microsoft Excel installed in them. Here are the steps:. Open Microsoft Excel in your computer. If you have it as an icon inside a folder, or on the desktop, then double click on the icon. If you have Microsoft Excel in one of the pop-up menus, then sinle click on the icon. Most probably you will be able to find the software if you do the followin. (i) Click on the START button on your desktop. (ii) A menu will pop-up. Find the icon called PROGRAMS in that menu. (iii) Click on the PROGRAMS icon and another menu will pop-up listin the prorams in your computer. (iv) In this list look for Microsoft Excel and sinle click on it to open the proram. If you do not see it there, most probably Microsoft Excel is not installed in that computer. (v) Once you open it you will see an empty excel sheet with boxes (rows and columns) ready to be filled in.. etters A, B, C (on the topmost row) refer to the name of the columns. Numbers,, 3 refer to the name/number of the rows. For example: A means first box (in row ) in column A. 3. If you are oin to plot a set of X and Y-values, you should have X-values in a column to the left of your Y-values. For example if I was oin to plot Concentration (Y-axis) aainst Absorbance (X-axis), I would fill in all the absorbance values in column and then fill in the correspondin concentration values in column. 4. Now you need to select the X and Y-values that you have entered in these two columns. This can be done in any one of the followin ways. a. Put the mouse on your first X-value. eft click the mouse in the same box. Hold the mouse in clicked position and then dra the mouse to the other boxes that you want to select. You will see that the selected boxes will appear rey. For example, if you have 5 X -values and 5 Y-values. Then put the mouse on your first X value and then click and hold the mouse in the first value. Then dra the mouse to your 5 th Y value, with the mouse in the clicked position. Release the click once you have hihlihted all the values that you want to plot. b. Put the mouse on the first X-value. Then press the SHIFT key and hold it. Now put the mouse on the last Y-value. Then release the SHIFT key. Now you will have all your X and Y-values hihlihted. 5. Once you have hihlihted all the values, now click on the CHART WIZARD icon. Chart Wizard icon looks like a miniature version of a bar raph that is colored in blue, yellow and red. 6. Once you click on the chart wizard, a menu will pop-up with different ways of plottin a set of values. You have to choose XY SCATTER. This menu will be on the left.

22 7. Once you chose XY SCATTER, in the same menu, on your riht you will see different ways of plottin the raph in XY scatter. Of the 5 options iven, choose the first option, which will help you plot the points on the raph without joinin them. 8. Press NEXT 9. Press NEXT aain 0. Now enter the title for your raph and the axis titles.. Press NEXT and then press FINISH.. Now you will see your raph with the data points plotted (but not joined). 3. If you are plottin just one set of X and Y- values, then you do not need a leend. The leend appears on the riht of the raph. To remove this, riht click on the leend with your mouse and chose clear. This will remove the leend from the raph. 4. FOR POTTING THE BEST-FIT INE FOR YOUR GRAPH: a) Riht click on any point on your raph. b) From the menu that pops up click on ADD TREND INE. c) Once you click on the add trend line a menu will pop up. d) You will see different ways of fittin the points (linear, loarithmic, exponential, and polynomial). Choose INEAR. (Click on linear). e) One the same menu, if you look at the top of the menu, you will see an icon that says OPTIONS. Click on options. f) In the options menu, if you look towards the bottom, you will see three options that you can choose. (i) Set intercept = 0 (ii) Display equation on chart. (iii) Display R-squared value. Select options (ii) and (iii). Once you select them you will see the little square in front of these options et filled with a tick (ü) mark. Then press OK. ) Now you should see the straiht line and its equation alon with the R-squared value. The equation will have a form Y = m X + C. Where m is the slope and c is the intercept of the straiht line. Here is an example of how to plot a raph usin Microsoft Excel. The followin data represents temperature chane at different heihts from the surface of the earth. Table : X (Heiht above the surface of the, km) Y (Temperature chane, C)

23 The data when filled in the appropriate columns in an Excel spread sheet will look as shown below: A B C D E F G H I J Follow the above instructions and the final raph will look as shown below: Chane in Temperature, in Deree celsius A Plot of Chane in Temperature Aainst Distance from the Surface of the Earth Distance from the Surface of the Earth, in km y = -0.5x R = Now you can compare the raph that was drawn by hand to the one drawn usin Microsoft Excel. The R value. atm shown on the raph is not the same as the square of the as constant R (whose value is 0.08 ). The R value mol. K iven is a statistical parameter that tells us about the quality of the best-fit line. The closer the value of R to, the stroner the linear dependence between X and Y. 3

24 SECTION 3: WHY DO WE NEED MOES? Document prepared by: Madhavan Narayanan, Department of Chemistry, Temple University, Philadelphia PA The word mole in Greek refers to a heap or a collection. In science it refers to a certain collection of atoms or molecules. It is somethin similar to a dozen, which refers to a collection of items. If you have 40 items, you can say you have 0 dozen items. What determines whether you say 40 items or 0 dozens? It is just a matter of convenience and preference. Specifically, a mole refers to a collection of atoms or molecules. Usin moles offers a sinificant convenience. We will see how? If you look at the periodic table, you will see 0 or more boxes filled with symbols of elements or atoms. These atoms have been filled up in the increasin order of atomic number. In each of these boxes you will see two numbers that are relevant to our present discussion. One is the atomic number (whole number) and the other is the atomic weiht (not a whole number). The atomic weiht is the weihted averae of the different atomic weihts of the same element. Atoms of different masses of the same element with different masses are called isotopes. For example, hydroen exists as 3 different isotopes. Their natural abundances are very different. The isotopes of an element contain the same number of protons because they are atoms of the same element. They differ in mass because they contain a different number of neutrons. Table iven below lists the three isotopes of hydroen. Tabe Y : Name Symbol Nuclide symbol Mass (amu) Atomic Abundance No. of protons No. of Neutrons No. of Electrons (in neutral in nature atoms) Hydroen H H % 0 Deuterium D H % Tritium* T 3 H % * No known natural sources; produced by decomposition of artificial isotopes. Y Taken from reference. 3 If you look at the column showin the nuclide symbol, it shows, H, H and H, which are the chemical representations of hydroen, deuterium and tritium, respectively. For any element E, the eneral chemical representation is E, where A is the mass number and Z the atomic number. The term atomic number refers to the number of protons in an atom. In the case of neutral atoms (no positive or neative chare on the atom), the number of protons is equal to the number of electrons. The mass number refers to the sum of the number of protons and number of neutrons in an atom. The mass number is always a whole number. The atomic weiht refers to the mass of the atom. Since an atom is made up of protons, neutrons and electrons, the total mass of the atom should be equal to the sum of the masses of protons, neutrons and electrons. Here is a table that will show you what the masses of these components are: Table * : Electron Proton Neutron Chare Actual mass () Relative mass (based on mass of Electron bein one unit) * Take from reference A Z 4

25 Table shows that the proton and neutron are rouhly 875 times heavier than the electron. Therefore, the electron does not contribute much towards the atomic mass in the atom. Thus, we can simplify our expression for atomic mass as: Atomic mass of an atom = {(No. of protons) (mass of one proton)} + {(No. of neutrons) (Mass of one neutron)} The atomic mass of an atom is not a whole number. For example, the sodium atom with symbol Na, has an atomic 3 number of. Accordin to the literature there are 3 isotopes of sodium. For the isotope Na atomic number = number of protons =, The number of neutrons = mass number atomic number. There fore the number of neutrons = 3 =. So the mass of a sodium atom = {() ( ) } + {() ( )} = = But = Atomic Mass Unit (amu) (refer to the section on atomic weiht scale iven below for detailed explanation) Thus, the mass of the sodium atom is 3.9 amu. Obviously, the mass of an atom is very small. It would be impossible to just 3 weih out one atom, because there is no balance that exists that can measure such a small mass. In the case of Na, the number is How about measurin a mole of sodium atoms? The mass of mole Na atoms is oin to be ( ) = 3.9. We can easily measure 3.9 on an analytical balance. Thus, workin with moles offers us reat convenience in the laboratory. By the way, this is called the Avoadro s number. For more interestin information about the mole feel free to visit The atomic weiht scale and atomic weiht: The atomic weiht scale is based on the mass of the carbon- isotope. As a result of the conventions established by the International Union of Pure and Applied Chemistry (IUPAC) in 96, One amu is exactly of the mass of a carbon - atom. This is approximately the mass of one atom of H, the lihtest isotope of the element with lowest mass. The mass of one mole of atoms of any element, in rams, is numerically equal to the atomic weiht of the element. Because the mass of one carbon- is exactly amu, the mass of one mole of carbon- atoms is exactly rams. What is the relationship between amu and rams? To fiure out the weiht of of carbon atoms in terms of amu, here is how we do it: Step (i) Fiure out how many moles are in of carbon. Step (ii) Convert the calculated moles in to number atoms. We already know that mole of atoms is a collection of atoms. Step (iii) Since we know that the mass of atom of carbon- is amu, we can fiure out the weiht of number of atoms that we have calculated in terms of amu. Followin steps throuh 3 we end up with the followin equation for the relationship between amu and rams: 3 mole 6C C atoms amu 6C = C atoms mole C atoms C atom 6 Now, we have, = amu or amu = amu 5

26 The story of Avoadro s number: In 8, Italian physicist and mathematician Amedeo Avoadro published a hypothesis (also termed Avoadro s law or principle) statin that volume of a as is proportional to the number of molecules of a as. This is represented by the formula V = an where a is a constant, V is the volume of the as, and N is the number of as molecules. Therefore, equal volumes of ases with the same pressure and temperature contain the same number of molecules. Gay-ussac and others also believed such proportionality must exist, but what made Avoadro s hypothesis complete and correct was his new definition of a molecule as the smallest characteristic particle of a substance which may be a sinle elementary atom or a permanent union of several elementary atoms. In a reaction of hydroen with oxyen to form water, Avoadro proposed that some ases, such as hydroen and oxyen, are diatomic molecules and a water molecule could consist of three elementary atoms, two of hydroen and one of oxyen. Avoadro s concepts reconciled Dalton s atomic hypothesis with Gay-ussac s law of combinin volumes. To understand Avoadro s hypothesis, let us consider the followin reaction: N + O NO vol. vol. vols. n molecules n molecules n molecules molecule molecule molecules orenzo Romano Amedeo Carlo Avoadro In the above reaction, if.0 of N and.0 O reacted at same temperature and pressure to produce.0 of NO, then the number of molecules of N will be equal to the number of molecules of O. The number of molecules of NO will be twice the number of molecules of N or O. If the P, V and T of N and O are held constant, then the number of molecules of each should remain the same. Since the volume of NO is twice as that of either N or O, the number of molecules of NO should be twice as much as either N or O. This means that for every molecule of N and O there will be molecules of NO. What are the implications of such reasonin? Since the number of molecules of N and O are equal, if the mass of each as were measured, then the ratio of these masses will ive us the value of the relative atomic weihts. It took almost 50 years for Avoadro s hypothesis to be appreciated. In 860, a scientist named Stanizlav Cannizaro developed a system of atomic weihts based on Avoadro s hypothesis. Additionally, part of the delay in appreciatin Avoadro s hypothesis can be attributed to the confusion that surrounded the definition of a molecule. Additionally Avoadro bein a theorist, did not have experimental evidence to prove his hypothesis. Avoadro did not make any quantitative estimate of the number molecules in a mole. Many other scientists worked on obtainin this number. Finally, after many experimental measurements, it was areed that its value of one mole is 6.0 x 0 3. The best modern values of Avoadro s number are the result of the X-ray diffraction measurement of lattice distances in metals and salts. 6