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Measurements Measurement is the process by which we report on the existing amount of matter and energy in a given system. It is very important to define the nature of the of the system under consideration, so the expression of the language we use should correspond to the real situation of this system. The measurement we carry out is to report the real fact of the system in using numbers with specific units. The used units are related to specific dimensions which are Mass (m), Length (L), Amount of matter (n), Time (t) , Temperature and Energy (E). The use of one or more of these dimensions depends on the nature of nature of the system and the required information to report. For instance, if we talk about the population of a city or a country we use numbers of people, not mass or length. This applies to any living systems or individual structures such as cars, houses, atoms, molecules….In other words we do not talk about masses in kilogram or lengths in meter of houses, cars but about how many of these are present in a town or parking lot. Scale of numbers of unit matter Let us start by exactly defining the subject under investigation. Human being is one of the living systems defined by a mass in kilogram and length in meter is active part in doing the measurement on other units of matter. As mentioned previously, the basic unit of matter is an atom. Atoms form molecules present in our daily life in form of substances and compounds. As a matter of fact, atoms are permanent structures and do not change as function of time, while humans as well as other living systems are made of these atoms and have limited life time, then are transformed to atoms and molecules. Based on this basic fact, it is essential to have a clear idea about the dimension of an atom and a molecule as compared to our dimensions. This question must be divided into two parts: 1. Atoms and molecules deal with each other as structures having the same dimensions. For example, a hydrogen atom reacts with another hydrogen atom to produce a hydrogen molecule. This operation consists of number 1 and 1 1 atoms to yield 1 molecule, and so on. It does not deal with masses or lengths…The same logic applies for humans. All relations and actions ,regardless of its kind, take place between individuals; numbers, not masses or other dimensions. 2. The relation between basic units of matter: Atoms and molecules and human being as individuals, especially that human being is the active part in performing the measurement. In order to understand this process, we consider the following example. Example: i- The mass of on hydrogen atom is = 1 unit mass in its dimension = 1.67 x 10-24 g in our language of mass. ii- If we have one gram of hydrogen atoms. This quantity is equal to 6.02 x 10 23 atoms iii- Suppose that we want to count the number of these atoms which is the right thing to do. If we count one atom per second non stop, then we need : The number of seconds in one year = 365 day x 24 h/day x 3600 s/h = 31536000 seconds or atoms counted per year. Number of years = 6.02 x 10 23 atoms / (31536000 s/year) = 1.9 x 1016 years =19000000000000000 years (long time) In other words, it is impossible to account exactly the number of atoms in a small mass of one gram. Since it is impossible to count all the atoms, then we measure this quantity by weight. Even so, it is impossible to report exactly the real mass of all the atoms present in this sample. Simply because we do not have a common instrument of measuring the mass of all the atoms with an accuracy of one single atom of the order of 10-24 g. iii- From the above example in reporting the quantity of matter in a system, we have to use different measuring devises which yield approximate numbers about it. The accuracy of the reported data is related to the sensitivity of the measuring instrument. In all cases, it will never by exact . The question is how these measurements are close to the real number of unit matter in the system. In conclusion, it is very important to be aware that no matter how good our measurements are, it is always approximate. Consequently, the main 2 question will always be how close the approximate measurement from the reality and how to achieve it. In reporting measurements, two information (terms) concerning these measurements are encountered: Precision and Accuracy. Precision: Signifies how different measurements using the same instrument are close to each other. Example: The reported weights of one sample using a balance are: 2.49 g, 2.48 g, 2.50 g, 2.49 g. These measurements are called precise. Accuracy: Signifies how much different measurements using different instruments are close to the real value. Example. Suppose that the real mass of a sample is 2.70 g. Different balances were employed and the measured values are: 2.69 g, 2.71 g, 2.68 g, 2.70 g, 2.69 g. Note. Precision does not necessary mean accurate. Suppose that one of the balances was not calibrated correctly and gave the values: 2.49 g, 2.48 g, 2.50 g, 2.49 g. It could be said in this case that the measurements are precise but not accurate. In order to report measured data as close as possible to the real value, different instruments should be used. In order to perform a measurement such as weight, temperature, pressure length dimensions measuring instruments should be used. In order for the reported measurement to be as close as possible to the real value in question, the measurement should be precise and accurate as mentioned above. Despite that, there is always an error in the conducted measurement. In this respect and in order to report experimental data which cover the real value, the given number should reflect the scale devise of the used instrument. For example if a balance has a scale in grams, the reported number should be in gram (g). If the scale in milligram, then the reported number should be in milligram (mg) and so on. In all these measurements, the last digit is uncertain and should be included in the reported data. This is called significant figures. Example. Suppose that the real mass of a system is: 56.442324 g, and we have three balances of different scales (sensitivity). The first one (balance 1) has a scale in 3 grams, the second centi-gram (1/100 g) and the third one in milligram (mg). The reported measures using the three balances should be: Balance 1: 56 ± 1 g. This means that the real value is between (55-57) g, which is true. The error in this case in units of gram Balance 2: 56.44 ± 0.01 g. This means that the real value is between (56.43-56.45) g, which is true. The error in this case in units of centi- gram. Balance 3: 56.442 ± 0.001 g. This means that the real value is between (56.441- 56.443) g, which is true. The error in this case in units of milli- gram (mg). From the above results, it is concluded that the reported data reflect the sensitivity of the used instrument by considering the last digit is uncertain. This underline the magnitude of the error (uncertainty) in the available data. In other words, we can recognize the scale of the measuring instrument from the reported number. Thus 56 ± 1 g signifies that the scale of the balance is in gram, while the scale in the case of 56.442 ± 0.001 g, is in mg (more sensitive). The number of significant figures (numbers) in approximate calculations are counted as follows: 1. All numbers are counted as significant figures. However, if the first digit is preceded by zeros on the left (mathematical transformation)*, these zeros are not counted. The zeros on the right side (measurement unit)** are counted. * suppose we have a value of 1 mg. This means that the scale (sensitivity ) of the balance is in mg. This digit could written as 1 mg = 0.001 g . In both equality of the equation, the sensitivity of the balance is in mg. ** In another measurement, the mass of a sample is 10 mg. In this case, the zero is a digit of the measurement (significant). Examples: Number Significant figures 2 1 2.0 2 6.0010 5 0.00212 3 0.002120 4 4 1.002120 7 2. The last digit in any number is always uncertain and the real number should be written as follows: 6.0 ± 0.1 2.102 ± 0.001 5.00001 ± 0.00001 3. As the number of digits in any measurement is large, the measurement is more sensitive ( more sensitive instrument). Examples: If we take the three measurement of a given length using different rules: Measurement (rule) Value (m) Error 1 0.6 ± 0.1 2 0.61± 0.01 1/61 = 0.016 (1.6 %) 3 0.610± 0.001 1/610 = 0.0016 (0.16 %) 1/6 = 0.17 (17%) Obviously, the error in measurement 3 is the smallest because the rule is more sensitive. Numbers Approximation In performing different computations on approximate (experimental data) numbers, the numbers should be rounded if they are followed by a digit greater than or equal to 5. Examples: 1. Approximate the following numbers 5.16 , 4.256, 3.7445 to two significant numbers each. Solution: 5.16 = 5.2, 4.256 = 4.3 , 2. Approximate the following numbers: 3.7445 = 3.7 7.0107 , 3.00454, 2.164436 to four significant numbers each. Solution: 7.0107 = 7.011, 3.00454 = 3.005 , 5 2.164436 = 2.164 Scientific Notation In order to avoid writing very large or very small numbers, it is more convenient to write these numbers as a product of a number between 1 and 9 multiplied by 10 raised to some power. Numbers written in this manner are said to be expressed in scientific notation. Examples: Number Scientific notation Significant figures 2000 2.000 x 103 4 310.789 3.10789 x 102 6 2001.56 2.000156 x 103 7 0.001 1 x 10-3 1 0.010030 1.0030 x 10-2 5 Computation using significant figures: There are specific rules to apply when computations were carried out so the results of these calculations well have the order of error to the component having the largest error. 1. Addition and Subtraction "The error (uncertainty) of the result of an addition or subtraction should have the same error (uncertainty) as the component (term) having the largest error". In other words, it have the same digits after the point as the least one in the calculation. Examples: 1. addition 4.15 + 0.1012 = 4.2512 = 4.25 ± 0.01 2. Subtraction Uncertainty ± 0.0001 the result should be of the largest error ± 0.01 25.0102 - 10.0 = 15.0102 = 15.0 ± 0.0001 ± 0.1 ± 0.1 2. Multiplication and Division 6 "For division or multiplication, the quotient or product should not contain more significant figures than the least term in the calculation". Examples: 1. 2.56 x 4.0 = 10.24 = 10 Significant figs. 2. 3 2 2 3.456 x 7.8 = 26.9568 = 27 Significant figs. 4 2 2 41.7 ÷ 22.124 = 1.88483 = 1.88 3. Significant figs. 3 5 3 Exact Numbers In performing calculations, if it is required to transform numbers from one type of units to another such as gram to kilogram or meter to kilometer these are called exact numbers and do not follow rules of approximate numbers and significant numbers. Examples: 1. Transform the following numbers from grams to kilograms: i- 5 g ii- 500000 g iii- 10 g solution: i- 5 g ÷ 1000g /kg = 0.005 kg = 5 x 10 -3 kg ii- 500000 g ÷ 1000g /kg = 500 kg = 5 x 102 kg iii- 10 g ÷ 1000g /kg = 0.01 kg = 1 x 10-2 kg 2. Transform the following numbers from kilograms to grams : i- 5 kg ii- 1.01 kg iii- 2.0785 kg solution: i- 5 kg x 1000 g /kg = 5000 g ii- 1.01 kg x 1000 g /kg = 1010 g iii- 2.0785 kg x 1000 g /kg = 2078.5 g Mathematical Operations 7 1. Powers (Xa)b = Xab Example. (102)3 = 106 2. Roots (Xa)1/b = Xa/b Example (102)1/3 = 102/3 Example 103/102 = 103-2 = 10 Xa / Xb = Xa-b 3. Division 4. Multiplication Xa . Xb = Xa+b Example 103 x 102= 103+2 =105 5. Addition and subtraction: In the case of additions or subtractions operations of terms having exponential numbers, the exponents (raised powers) should be unified and similar to the largest exponent. Examples: 1. Perform the following calculation: 5 x 103 + 4 x 104 Solution. Before doing the calculations, the exponents should be unified: 0.5 x 104 + 4 x 104 = 4.5 x 104 2. Perform the following calculation: 2 x 10-5 + 3 x 10-3 Solution. 0.02 x 10-3 + 3 x 10-3 = 3.02 x 10-3 Measurements Dimensional Units: International Symbols for Units A coherent system of unites is a system based on a certain set of "basic units" from all "derived unites" are obtained by multiplication or division with out introducing numerical factors: The three metric systems used in science are: 1. The cgs system. This stands for: centimeter (cm),gram(g),second(s). This is a coherent system of units based on three basic units for the basic quantities: length (cm), mass (g) and time (s). 2. The mks system. This stands for meter (m), kilogram (kg), second(s). Here the three basic units are again length (m), mass (kg) and time(s). 8 3. SI-International system of units. This is basically the mks system with certain improvements. IN 1960, The international conference of weights and measures adopted and recommended the use of the SI system of units by all scientists. Table (2-1) The SI system of unites Physical SI units quantity 1-Basic units Name of unit Symbol Length meter m Mass kilogram kg Time second s temperature degree Kelvin K mole mol Amount of substance 2-Derived Units Force Newton (N) kg m s-2 Area square metre m2 9 Volume cubic metre m3 Density kilogram per kg m-3 cubic metre Velocity metre per m s-1 second Acceleration metre per m s-2 second square Pressure Newton per N m-2 = square meter; Kg m-1 s-2 pascal (Pa) Work, energy joule ( J ) Nm= kg m2 s-2 10 Derived Units Quite often, units which are of the dimension atoms and molecules (Atomic scale) are employed. This is practical in a sense that small numbers are used. 1 Angstrom (Å) = 1x10-10 m 1 electron volt (eV) = 1.602 x10-19 J Electron volt is the energy received by an electron when it is accelerated by one volt. The charge of the electron =1.602 x10-19 Coulomb So, 1eV = 1.602 x10-19 C x 1V = 1.602 x10-19 J Coulomb x Volt = Joule EXERCISES 1. How many significant figures are there in the following numbers? A. 0.0100210, B. 1.00120, C. 0.00020, D. 100.002, E. 10.000, F. 0.00300 2. Express the following measurement numbers in scientific notation. How many significant figures does each number contain? a. 0.00066 b. 0.000660 c. 0.00102030 e. 114.809 f. 896500000 g. 17000.12 3. Perform the following computations: a. 325.12 - 7.8432 b. 0.015 - 0.009 c. 0.029 - 0.01 d. 14.2854 - 7.815 e. 4.005 - 2.010 4. Perform the following computations: a. 6.050 + 14.001 11 d. 0.0000020 b. 0.0213 + 0.40 c. 789.0 + 1.5987 d. 432 + 1.3980 e. 0.0010 + 2.01 5. Perform the following computations a. 18.910 ÷ 7.8 b. 2.01 ÷ 1.5 c. 979 ÷ 9.08 d. 8543 ÷ 1.2 e. 17.000 ÷ 1.53241 6. Perform the following computations a. 3.4 x 4.05 b. 18.9 x 0.016 c. 7.020 x 3.01 d. 24.6 x 695 e. 0.146 x 0.0132 7. Perform the following computations a. (8.05 x106) + (9.65 x 106) b. (9.77 x 104) + (3.01 x 105) c. (8.09 x 105) - (7.01 x 104) d. (1.645 x 107) - (8.9 x 106) e. (9.98 x 103) - (7.85 x 102) f. (4.05 x 10-3) - (5.1 x 10-4) g. (2.010 x 10-4) + (5.10 x 10-3) 8. Perform the following computations a. (315.2 + 41.527) + (92.346 - 78.05) b. (7.081 ÷ 3.02) + (6.001 x 3.0) 12 c. (5.003 x 6.01) ÷ (18.04 x 0.0010) d. (7.00030 x 3.101) ÷ (295.008 – 227.15) e. (5.02 x 10-3 ÷ 3.0 x 10-4) x (2.097 x 105 + 9.811 x 104) 9. Which of the following operations concerning the weight of a sample is: Precise, Accurate, Precise and Accurate, meaningless a. The following masses were obtained using one balance: 5.3 g, 4.8 g, 6.2 g ( ) b. The following masses were obtained using one balance: 5.3 g, 5.1 g, 5. 2 g, 5.3 g c. ( The following masses were obtained using two balances: 4.9 g, 5.3 g, 5.1 g, 4.8 g, 5.4 g d. ) ( ) The following masses were obtained using one balance: 5.1 g, 5.2 g, 5.1 g, 5.3 g, 5.0 g ( ) 13 ANSWERS 1. A. (6), B. (6), C. (2), D. (6), E. (5), F. (3) 2. Question Number Scientific Notation Significant Figures a 0.00066 6.6 x 10-4 2 b 0.000660 6.60 x 10-4 3 c 0.00102030 1.02030 x 10-6 6 d 0.0000020 2.0 x 10-6 2 e 114.809 1.14809 x 102 6 f 896500000 8.9650000 x 108 9 g 17000.12 1.700012 x 104 7 3. a. 317.28 b. 0.006 (6 x 10-3) 4. a. 20.051 b. 0.42 c. 790.6 c. 0.02 d. 433 14 d. 6.470 e. 2.01 e. 1.995 5. a. 2.4 b. 1.3 c. 108 6. a. 1.4 b. 0.3 c. 21.1 7. a. 1.77 x 107 8. a. 371.1 b. 3.99 x 105 b. 20 9. a. meaningless d. 7.1 x 103 d. 1.71 x 104 c. 7.39 x 105 c. 1.7 x 103 b. precise e. 11.094 e. 1.93 x 10-3 d. 7.6 x 106 d. 0.3199 c. accurate 15 e. 9.2 x 103 e. 5.2 x 106 d. precise and accurate