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1 Chemistry 4 Chapter 3 Measurement and Chemical Calculations 1 INTRODUCTION TO MEASUREMENT Measurement of a quantity usually can be made by comparison with a standard that serves to represent the magnitude of a unit. Result of a measurement = number x unit There are three major systems of measurement units in wide use: The US Customary System, The British Imperial System, and the International System (SI) ( sometimes called Metric System). The SI units are used almost exclusively for scientific work. 2 EXPONENTIAL (SCIENTIFIC) NOTATION A number, n, may be written in exponential notation as follows: n = c x 10e Where c is the coefficient. In scientific notation ( Standard exponential notation ) the coefficient is equal to or greater than 1 and less than 10 : 1 < c < 10 10e is an exponential. e is the exponent. It is an integer and may be positive or negative. Procedure for writing number in exponential notation: 1) Rewrite the number, placing the decimal after the first nonzero digit. 2) Count the number of places the decimal in the original number moved to its new place. 3) Compare the original number with the coefficient in step 1 If the coefficient is smaller, the exponent is positive If the coefficient is larger, the exponent is negative. Insert a minus sign in front of the exponent (The exponent is positive if the decimal moves left, and negative if it moves right ) For example 34567890 = 3.456789 x 107 0.0000534 = 5.34 x 10-5 To multiply or divide in exponential notation, work with coefficients and exponentials separately and then combine. Remember that 10a x 10b = 10a+b 10a / 10b = 10a-b For example (3.96 x 104 ) ( 5.19 x 10-7 ) = (3.96 x 5.19 ) ( 104 x 10-7) = 20.6 x 10-3 = 2.06 x 10-2 To add or subtract exponential numbers, adjust all exponentials to same power, then add or subtract coefficient, the exponent of the result is the same as those of the numbers being added or subtracted. 2.24 x 10-2 + 1.12 x 10-4 = 2.24 x 10-2 + 0.0112 x 10-2 = (2.24 + 0.0112 ) x 10-2 1 2 = 2.25 x 10 -2 3 DIMENSIONAL ANALYSIS Many chemical problems involving quantities that are directly proportional can be solved by a method called dimensional analysis. When two quantities are directly proportional, we can find conversion factors to calculate the value of one property from the value of the other property. Information required = information given x conversion factor Example 1. Let us consider the problem: How many days are there in 3 weeks? The steps to solve this problem are: 1) Identify the given quantity: 3 weeks 2) Identify the wanted quantity: ? days 3) Find the conversion factor. From equality 7 days ≡ 1 week we have two conversion factors: (7 days/week) and (1 week/7 days) We use the first one because the unit required is days 3 weeks x ( 7days/week) = 21 days Example 2. Let us consider the conversion of units problem : What length in centimeter corresponds to 2.00 inches? The steps to solve this problem are : 1) Identify the given quantity: 2.00 inches 2) Identify the wanted quantity : ? centimeters 3) Find the conversion factor. From the relation 2.54 cm ≡ 1 in. we can identify two factors: (2.54 cm/1 in ) and (1 in/2.54 cm) We use the first one because the unit required is cm. 4) Setup a unit path. Quantity given quantity to be found 2.00 in. ? cm 5) Carry out the calculation by applying the rule mentioned previously ( always use conversion factor that leads to the cancellation of an unwanted unit) 2.00 in. x 2.54 cm = 5.08 cm 1 in. 6) Check the answer to be sure that both the number and the units make sense. The answer (5.08) is reasonable since we have large number ( 5.08 ) for smaller unit (cm) and small number (2) for larger unit (in.). Should we use the second factor, we would have 2 in. x 1 in. = 0.816 in.2/cm 2.54cm It is easy to recognize that this answer is wrong. If you get an answer with nonsense unit, you know you have made a mistake. 4 SI UNITS The International System was adopted for scientific use by the US National Bureau of Standards in 1964. Prefixes The SI system is a decimal system, that is, one in which all derived units are multiple of ten 2 3 Mass Mass is a measure of quantity of mater. The metric unit of mass is gram, g. The SI unit of mass is kilogram, kg. A penny has a mass of 3 grams. Length The SI unit of length is meter. A convenient unit for laboratory work is centimeter. 1 cm = 10-2 m . A penny has a diameter of about 2 cm. One inch is defined as 2.54 cm Volume The volume of an object is the amount of space it occupies. The SI volume unit is the cubic meter, m3. A more practice 3 4 3 unit is the cubic centimeter, cm . It is the volume of a cube with edge of 1 cm. A common unit for volume is the liter, L, which is defined as exactly 1000 cubic centimeters. A milliliter is exactly equal to 1 cm3 . 5 METRIC-USCS CONVERSIONS The relationship between the metric and the English systems is given in table 3 Table 3 Metric and English Conversion Factors Mass Length Volume 1 lb. = 453.6 g 1 oz = 28.3 g 2.20 lb. = 1 kg 1in. ≡ 2.54 cm, definition 1 ft = 30.5 cm 39.4 in.= 1 m 1.09 yd = 1m 1 mile = 1.61 km 1 qt = 0.946 L 1 gal = 3.785 L 1 in.3 = 16.39 cm3 1 ft3 = 2.832 x 104 cm3 Pressure Energy 14.69 lb./in.2 = 1 atm ≡ 760 torr (definition) 1 calorie = 4.184 J 29.92 in. mercury = 1 atm ≡ 760 mm Hg(definition) 1 Btu* = 1.05 kJ = 101.3 kPa * BTU( British Thermal Unit) is the amount of heat required to raise the temperature of one pound of water one degree Fahrenheit. 5 SIGNIFICANT FIGURES A measurement always has some degree of uncertainty. The accuracy of a measurement is it closeness to the true value. The precision of measurements indicates the degree of reproducibility of several measurements of the same quantity. Accuracy is associated with systematic error, precision reflects the random error. Uncertainty of a measurement depends on many factors, one of which is the instrument used. The length of a board is measured by comparing it with meter sticks that have different graduation marks. The result can be 0.6 ± 0.1 m, 0.64 ± 0.01 m, 0.642 ± 0.001 m. One way to express the uncertainty of a measurement is to use significant figures in reporting data. We use the convention that the uncertainty in the last digit on the right is + 1. The number of significant figures is the number of digits that are known accurately plus the first uncertain digit. Rules for significant digits are presented in the following table 4 5 Number number of significant digit Rule 2.36 g 3 All nonzero digits are significant 6.087 g 4 Zeros between significant digits are significant 55.00 g 4 Zeros that end a number after the decimal are significant 0.0295 g 3 Zeros that begin a number are not significant. 123000 g 3 Zeros that end a number before the decimal are assumed not to be significant. 1.230x105 g 4 Exponential notation must be used for large number to show if final zeros are significant Significant figures do not apply to exact numbers. Numbers fixed by definition are exact. One inch is exactly equal to 2.54 centimeter. Counting numbers are exact. Exact numbers have infinite significant digits. Rounding Off. When experimentally measured quantities are added, subtracted, multiplied or divided, the result must be rounded off to obtain the correct number of significant figures. Rules for rounding off are: 1) If the first digit to be dropped is less than 5, leave the digit before it unchanged. 2) If the digit to be dropped is 5 or greater, increase the digit before it by 1. Significant Figures in Calculation Rule for Addition and Subtraction The number of decimal places in the result should be the same as the smallest number of decimal places in the data. 12.4501 g + 2.36 g = 14.8101 g : round to 14.81 g Rule for Multiplication and Division The number of significant figures in the result should be the same as the smallest number of significant figures in any factor. 1.234 x 0.025 = 0.03085 : round to 0.031 6 TEMPERATURE The familiar temperature scale in the US is the Fahrenheit. Temperatures in chemistry are often reported in Celsius scale. The SI temperature scale is the Kelvin scale ( also called absolute temperature scale). The relationship between these scales are shown in table 4. 5 6 Table 4. Temperature Scale Reference Temperature Fahrenheit Scale Celsius Scale Kelvin Scale Boiling point of water 212 0F 100 0C 373 K Freezing point of water 320F 00C 273 K Limit of lowest temperature -4590F -2730C 0K Conversion Formulas TF = 1.8 TC + 32 TC = (TF - 32) x 100 180 T K = TC + 273 To convert from Celsius to Fahrenheit multiply reading by 1.8 and add 32 ( one degree in Celsius corresponds to 1.8 degrees Fahrenheit ) To convert from Celsius to Kelvin add 273 The Celsius and Fahrenheit temperature are the same at -40 0. At all temperature above -400, the Fahrenheit temperature is the larger number. DENSITY Density of a substance is its mass per unit volume. Density ≡ mass/ volume ; D≡ m/V 6 7 Substance with a high density ( Hg, Pb for example) have a much larger amount of matter in a given volume than do substances with low density ( for example Al) The specific gravity of a liquid is the ratio of its density over the density of water. Since the density of water in SI units is 1.00 g/cm3, the specific gravity for a liquid is the same as its density without units. Density of a substance may be used as a factor for conversions between volume and mass. Strategy for Solving Problem There are six steps: 1) List everything that is given, including the unit. 2) List all wanted quantities, including the unit. 3) Identify the relationship between the given and wanted quantities. Decide the method to solve the problem. Algebra is a very convenient method if you know algebraic equations relating given and wanted quantities. If given and wanted quantities are directly proportional to each other, dimensional analysis method can be used. In using dimensional analysis, your verification that everything is correct is that you end up with the correct unit. Some problems can be solved by both methods. Dimensional method has the advantage of minimizing memorization. 4) Write the calculation setup for the problem. Include the unit. 5) Calculate the answer. Include the unit. 6) Check the answer. Be sure that the units are correct and the number is reasonable. Example : The density of an oil is 0.862 g/mL. Find the volume occupied by 196 g of that oil. Dimensional analysis method Given : mass of oil 196 g Wanted Volume of oil in ml Path g ( mass of oil) mL ( volume of oil) Factors 0.862 g/1 mL and 1 mL / 0.862g Set up and calculation 196 g x 1 mL/ 0.862 g = 227 mL Algebraic method Given : mass of oil 196 g Wanted Volume of oil in mL Equation V = m / D = 196 g / (0.862g/mL) = 227 mL 7