Mathematics in General Chemistry

Mathematics in world(a) ChemistryPrep bed by Paul Okweye and Malinda GilmorePurpose of the proveTo acquire knowledge in the bea of units of measurements and learn how to use dimensional analysis to solve word problems that will be used throughout General Chemistry. In addition, students will learn the importance of statistical analysis in General Chemistry laboratory experiments and also rules with respect to exponential musical note and operative figures.Background In nameationMathematics in chemistry is essential. bingle cannot truly perform a chemical experiment without utilizing mathematics in their data and results. Therefore, it is unequivocal that one grasp the concept of the important topics or areas of mathematics that will be utilized throughout General Chemistry and the General Chemistry laboratory. nigh of the areas that will be discussed herein are as followsUnits of criterionStatistical analytic thinkingExponential note of hand and earthshaking elaboratesG raphing and y = mx + bUnits of measuring rodChemistry is all about observing chemical reactions and physical changes. There are two types of observations in Chemistry qualitative observations and quantitative observations. Qualitative observations live of non-numerical observations, much(prenominal) as the color of a substance or its physical appearance. Quantitative observations consist of numerical data, such as the temperature at which a chemical substance melts or its mass. With respect to qualitative observations, in order to record and report measurements, scientist utilizes the metric system. The metric system is used internationally and is called the world-wide Systems of Units (SI). The International Systems of Units are shown below in Table 1.Measured PropertyName of UnitAbbreviationMassKilogramkgLengthMetermTimeSecondsTemperatureKelvinKAmount of substancemolemolTable 1. SI Base UnitsLarger and small quantities are expressed by using appropriate prefixes with the base unit (Table 2).PrefixSymbolExamplegigaG1 gigameter (Gm) = 109 mmegaM1 megameter (Mm) = 106 mkilok1 kilogram (kg) = 103 ghectoh1 hectogram (hg) = 100 gdekada1 dekagram (dag) = 10 gdecid1 decigram (dg) = 0.1 gcentic1 centigram (cg) = 0.01 gmillim1 milligram (mg) = 0.001gmicrom1 microgram (mg) = 10-6 gnanon1 nanogram (ng) = 10-9 gpicop1 picogram (pg) = 10-12 g Table 2. Prefixes used in the Metric SystemMethod for Solving Conversions including Units of MeasurementDimensional analysis is a problem-solving system that uses the fact that any number or expression can be multiplied by one without ever-changing its value. It is a very useful technique.Equation 1.1 (Proportionality Conversion Factor) shows how dimensional analysis can be applied in solving problems in Chemistry. A symmetricalness calculate is a ratio (fraction) whose numerator and deno bet onator have different units but refer to the same thing. A proportionality factor is often called a conversion factor because it enable s us to convert from one kind of unit to a different kind of unit.An typeface of how this can be used is belowExample 1What would be the value of 157 g if you were to convert it to kilograms (kg)? reply 1Conversion factor needed 1000 grams 1 kilogramThe dimensional analysis method can be useful if the by-line techniques for analyzing the problem properly are taken into considerationIdentify the information given, including units.Identify the information needed in the set, including units.Find a relationship between the known information and unknown answer, and plan a strategy for getting from one to the early(a).Solve the problem.Check your work outTable 3 includes some common conversion factors.Mass1 lb = 16 oz = 0.4536 kg1 ton = 2000 lbLength1 in = 2.54 x 10-2 m = 2.54 cm1 ft = 12 in = 0.3048 m1 yd = 3 ft = 36 in = 0.9144 m1 mi = 1760 yd = 5280 ft = 1609 mVolume1 L = 10-3 m3 = 1 dm3 = 103 cm31 L = 1.06 qt1 gal = 4 qt = 8 pt = 3.785 L1 pt = 2 cups = 16 fluid ouncesTime1 min = 60 s1 hr = 60 min = 3600 s1 d = 24 hr = 1440 min = 86,400 sTemperatureoC = K 273.15oC = 5/9 (oF 32)oF = (oC x 9/5) + 32Pressure1 bar = 105 N/m2 = 105 Pa1 torr = 1 mm Hg = 133.322 Pa1 atm = 760 torr = 101,325 N/m2 = 101,325 PaEnergy1 cal = 4.184 JTable 3. Common Conversion FactorsExample 2If an object has a weight of 0.025 ounces (oz), what is its mass in milligrams (mg)?Solution 2Conversion Factor Needed16 ounces (oz) 0.4536 kilogram (kg)1000 grams (g) 1 kilogram (g)1000 milligram (mg) 1 gram (g)Example 3 If the temperature of warm milk was 75oF, what would the temperature be in oC and K?Solution 3Conversion Factor Needed oC First you must(prenominal) convert to oF to oCSecond you must convert oC to KStatistical AnalysisAverageThe most common statistic used to analyze a set of repeated measurements is the symbolize, or average. We calculate the mean by taking the sum, , of the individual measurements, x, and dividing by the number of measurements, n, as shown in Equation 2. Example 4An experiment was performed where one measured the mass of a penny using a balance. The experiment was done 5 times and the results were as follows 6.47 g, 9.24 g, 4.67 g, 6.54 g, 5.55 g. What is the average, or mean of this experiment?Solution 4Mean = measure 5 is the number of trials (n)Experimental ErrorIf you measure a quantity in the laboratory, you may be required to report the error in the result, the fight between your result and the accepted value (Eqn. 3), or the percent error (Eqn. 4).Eqn. 3Eqn. 4Example 5A laboratory experiment was performed ascertain the melting point of pure aspirin. The accepted value of the melting point of pure aspirin is 140oC. Experimentally, you tried to determine that value, but you obtain the temperature value of 134oC, 150oC, 145oC, 140oC and 142oC. a) depend the error in measurement, and b) Calculate the overall percent error.Solution 5Step 1 Determine the average (mean) value from the experiment.Mean =Step 2 Determine the error in measurement.Error in Measurement =Step 3 Determine the percent error.Percent Error = Standard Deviation science laboratory measurements can be in error for two basic reasons. First, in that location may be determinate errors caused by faulty instruments or human errors such as incorrect record keeping. Secondly, indeterminate errors arise from uncertainties in a measurement where the cause is not known and cannot be controlled by the lab worker. One way to judge the indeterminate error in a result is to calculate the standard deviation.The standard deviation (Eqn. 5) of a series of measurements is equal to the upstanding root of the sum of the squares of the deviations for each measurement from the average divided by one less than the number of measurements.Eqn. 5Example 6Using interrogatoryple 4, calculate the standard deviation.Solution 6Standard Deviation = = 1.71gExponential Notation and Significant FiguresExponential notation, also known as standard form or as scientific notation, is a way of writing number that accommodates values too large or small to be conveniently write in standard quantitative notation. Scientific notation has a number of useful properties and is often used in sciences such as chemistry, physics, etc.In scientific notation, all numbers are written like thisa x 10b(a times ten to the power of b), where the exponent b is an integer, and the coefficient a is any real number (number between 1 and 9.999..).Example 7 Express the following number in exponential or scientific notation.0.067 gb) 0.000873 gc) 58923 gd) 112.483 gSolution 76.7 x 10-2 gb) 8.73 x 10-4 gc) 5.8923 x 104 gd) 1.12483 x 102 gIn chemistry, you will often have to use numbers in exponential notation in mathematical operations. The following five operations are importantAdding and Subtracting Numbers Expressed in Scientific NotationWhen adding or subtracting two numbers, first convert them to the same powers of 10. The digit terms are then added or subtracted as a ppropriate(1.234 x 10-3) + (5.623 x 10-2) = (0.1234 x 10-2) + (5.623 x 10-2) = 5.746 x 10-2Multiplication of Numbers Expressed in Scientific Notation(6.0 x 1023) x (2.0 x 10-2) = (6.0)(2.0 x 1023-2) = 12 x 1021 = 1.2 x 1022 particle of Numbers Expressed in Scientific Notation7.60 x 103 = 7.60 x 103-2 = 6.18 x 1011.23 x 102 1.23Powers of Numbers Expressed in Scientific Notation(5.28 x 103)2 = (5.28)2 x 1032 = 27.9 x 106 = 2.79 x 107Roots of Numbers Expressed in Scientific Notation-3.6 x 107 = -36 x 106 = -36 x -106 = 6.0 x 103Significant figures are the digits in a measured quantity that were observed with the measuring device.The rules for determining the amount of significant figures are as followsZeroes between two other significant digits are significant. For example, both 5309 and 50.08 contain four significant figures.Zeroes to the right of a nonzero number and also to the right of a decimal place are significant. For example, in the number 3.70 cm, the zero is significant.Zer oes that are placeholders are not significant. There are two types of numbers that fall under this rule.The first are decimal numbers with zeroes that occur before the first nonzero digit. For example, in 0.0015, but the 1 and the 5 are significant the zeroes are not. This number has two significant figures.The second are numbers with trailing zeroes that must be there to indicate the magnitude of the number. For example, the zeroes in the number 15,000 may or may not be significant it depends on whether they were measured or not. To avoid confusion with regard to such numbers, we shall assume in this book that trailing zeroes are significant when there is a decimal point to the right of the last zero.The rules for using significant figures in calculations are as followsWhen adding or subtracting numbers, the number of decimal places in the answer is equal to the number of decimal places in the number with the fewest digits after the decimal.In multiplication or division, the numb er of significant figures in the answer is unconquerable by the quantity with the fewest significant figures.When a number is rounded off, the last digit to be retained is increased by one only if the following digit is 5 or greater.IV. GraphingThroughout chemistry, graphs will be used when analyzing experimental data with a goal of obtaining a mathematical equivalence (Equation 6) that may help us predict new results. y = mx + b Eqn. 6y = dependent variable m = slope of the phone line x = is the independent variable b = y interceptExample 8Use Figure 1 below to solve this example. In Figure 1, you have a standard solution curve of CuSO4 5H2O. An unknown prototype was analyzed to determine the concentration of CuSO4 5H2O and the wavelength was 335nm. Calculate the concentration of CuSO4 5H2O in the unknown sample using the straight line equation.Solution 8y = 2.8571x + 190.48y = 335 nm and x = x = (y 190.48) 2.8571x = (335nm 190.48) 2.8571x = 50.6 mmolName______________ __________________________Section/Day/Time________________________MATHEMATICS IN GENERAL CHEMISTRY grooming SHEETUnits of MeasurementHow many centimeters are in 1675 ft?If an object has a weight of 0.700 ounces, what is its mass in milligrams? In kilograms?On the average, the moon takes 30 days, 8 hours, and 56.8 minutes to make a complete circuit around the Earth. Express this time in hours? In minutes?Carry out the following conversionsa) 10 m = _____ km = _____ cm = _____ mmb) 5.5 g = _____ kg = _____ ozs = _____gc) 400 cm = _____ ft = _____ in. = _____md) 45 m/sec = _____ ft/sec. = _____ km/hr = _____mile/hre) 9.9 in2 = _____ cm2 = _____ ft2 = _____mm2Aluminum is a lightweight metal (density = 2.70 g/cm3) used in aircraft construction, high-voltage transmission lines, beverage cans, and foils. What is its density in kg/m3?Ethanol boils at 351.7 K. What is this temperature in Celsius? What is this temperature in Fahrenheit?Name______________________________________Section/Day/Tim e_________________________MATHEMATICS IN GENERAL CHEMISTRYHOMEWORK SHEET (pg. 2)II. Statistical AnalysisA General Chemistry Laboratory had 5 students in it. A test was given the actual grade that could be earned on the exam was a 100. The grades were as followsStudent NumberGrade199280379488595Determine the followingAveragePercent ErrorStandard DeviationExponential Notation and Significant FiguresExpress the answers to the following calculations in scientific notation145.75 + (2.3 x 10-2)89,500 / (2.5 x 103)(7.9 x 10-3) (9.0 x 10-5)(1.0 x 105) x (9.9 x 106)Determine the number of significant figures in each of the following measurements5748 mi38 mL60,293 km0.0005 cmName______________________________________Section/Day/Time_________________________MATHEMATICS IN GENERAL CHEMISTRYHOMEWORK SHEET (pg. 3)GraphingUsing the graph below (Figure 2)What is the value of x when y = 32?What is the value of y when x = 5.50?What are the slope and the y-intercept of the line?What is the value of y when x = 6.67?