Mathematics in General Chemistry

Mathematics in General Chemistry Prepared by Paul Okweye and Malinda Gilmore Purpose of the Experiment To acquire knowledge in the area 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 notation and significant figures. Background Information Mathematics in chemistry is essential. One cannot truly perform a chemical experiment without utilizing mathematics in their data and results. Therefore, it is imperative 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. Some of the areas that will be discussed herein are as follows: Units of Measurement Statistical Analysis Exponential Notation and Significant Figures Graphing and y = mx + b Units of Measurement Chemistry is all about observing chemical reactions and physical changes. There are two types of observations in Chemistry: qualitative observations and quantitative observations. Qualitative observations consist of non-numerical observations, such 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 International Systems of Units (SI). The International Systems of Units are shown below in Table 1. Measured Property Name of Unit Abbreviation Mass Kilogram kg Length Meter m Time Second s Temperature Kelvin K Amount of substance mole mol Table 1. SI Base Units Larger and small quantities are expressed by using appropriate prefixes with the base unit (Table 2). Prefix Symbol Example giga G 1 gigameter (Gm) = 109 m mega M 1 megameter (Mm) = 106 m kilo k 1 kilogram (kg) = 103 g hecto h 1 hectogram (hg) = 100 g deka da 1 dekagram (dag) = 10 g deci d 1 decigram (dg) = 0.1 g centi c 1 centigram (cg) = 0.01 g milli m 1 milligram (mg) = 0.001g micro m 1 microgram (mg) = 10-6 g nano n 1 nanogram (ng) = 10-9 g pico p 1 picogram (pg) = 10-12 g Table 2. Prefixes used in the Metric System Method for Solving Conversions including Units of Measurement Dimensional analysis is a problem-solving method that uses the fact that any number or expression can be multiplied by one without 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 proportionality factor is a ratio (fraction) whose numerator and denominator have different units but refer to the same thing. A proportionality factor is often called a conversion factor because it enables us to convert from one kind of unit to a different kind of unit. An example of how this can be used is below: Example 1: What would be the value of 157 g if you were to convert it to kilograms (kg)? Solution 1: Conversion factor needed: 1000 grams → 1 kilogram The dimensional analysis method can be useful if the following techniques for analyzing the problem properly are taken into consideration: Identify the information given, including units. Identify the information needed in the answer, including units. Find a relationship between the known information and unknown answer, and plan a strategy for getting from one to the other. Solve the problem. Check your work Table 3 includes some common conversion factors. Mass 1 lb = 16 oz = 0.4536 kg 1 ton = 2000 lb Length 1 in = 2.54 x 10-2 m = 2.54 cm 1 ft = 12 in = 0.3048 m 1 yd = 3 ft = 36 in = 0.9144 m 1 mi = 1760 yd = 5280 ft = 1609 m Volume 1 L = 10-3 m3 = 1 dm3 = 103 cm3 1 L = 1.06 qt 1 gal = 4 qt = 8 pt = 3.785 L 1 pt = 2 cups = 16 fluid ounces Time: 1 min = 60 s 1 hr = 60 min = 3600 s 1 d = 24 hr = 1440 min = 86,400 s Temperature: oC = K – 273.15 oC = 5/9 (oF – 32) oF = (oC x 9/5) + 32 Pressure: 1 bar = 105 N/m2 = 105 Pa 1 torr = 1 mm Hg = 133.322 Pa 1 atm = 760 torr = 101,325 N/m2 = 101,325 Pa Energy: 1 cal = 4.184 J Table 3. Common Conversion Factors Example 2: If an object has a weight of 0.025 ounces (oz), what is its mass in milligrams (mg)? Solution 2: Conversion Factor Needed: 16 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 3: Conversion Factor Needed: oC First you must convert to oF to oC: Second you must convert oC to K: Statistical Analysis Average The most common statistic used to analyze a set of repeated measurements is the mean, or average. We calculate the mean by taking the sum, ÃŽ £, of the individual measurements, x, and dividing by the number of measurements, n, as shown in Equation 2. Example 4: An 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 4:Mean = Note: 5 is the number of trials (n) Experimental Error If you measure a quantity in the laboratory, you may be required to report the error in the result, the difference between your result and the accepted value (Eqn. 3), or the percent error (Eqn. 4). Eqn. 3 Eqn. 4 ­ Example 5: A laboratory experiment was performed determining 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) Calculate the error in measurement, and b) Calculate the overall percent error. Solution 5: Step 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 Laboratory measurements can be in error for two basic reasons. First, there may be â€Å"determinate† errors caused by faulty instruments or human errors such as incorrect record keeping. Secondly, â€Å"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 square 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. 5 Example 6: Using example 4, calculate the standard deviation. Solution 6: Standard Deviation = ÏÆ' = 1.71g Exponential Notation and Significant Figures Exponential 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 written in standard decimal 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 this: a x 10b (â€Å"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 g Solution 7: 6.7 x 10-2 gb) 8.73 x 10-4 gc) 5.8923 x 104 gd) 1.12483 x 102 g In chemistry, you will often have to use numbers in exponential notation in mathematical operations. The following five operations are important: Adding and Subtracting Numbers Expressed in Scientific Notation When adding or subtracting two numbers, first convert them to the same powers of 10. The digit terms are then added or subtracted as appropriate: (1.234 x 10-3) + (5.623 x 10-2) = (0.1234 x 10-2) + (5.623 x 10-2) = 5.746 x 10-2 Multiplication 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 Division of Numbers Expressed in Scientific Notation 7.60 x 103 = 7.60 x 103-2 = 6.18 x 101 1.23 x 102 1.23 Powers of Numbers Expressed in Scientific Notation (5.28 x 103)2 = (5.28)2 x 1032 = 27.9 x 106 = 2.79 x 107 Roots of Numbers Expressed in Scientific Notation à ¯Ã†â€™-3.6 x 107 = à ¯Ã†â€™-36 x 106 = à ¯Ã†â€™-36 x à ¯Ã†â€™-106 = 6.0 x 103 Significant 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 follows: Zeroes 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. Zeroes 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, only 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 follows: When 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 number of significant figures in the answer is determined 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. Graphing Throughout chemistry, graphs will be used when analyzing experimental data with a goal of obtaining a mathematical equation (Equation 6) that may help us predict new results. y = mx + b Eqn. 6 y = dependent variable; m = slope of the line; x = is the independent variable; b = y intercept Example 8: Use Figure 1 below to solve this example. In Figure 1, you have a standard solution curve of CuSO4 à ¯Ã¢â‚¬Å¡Ã‚ · 5H2O. An unknown sample was analyzed to determine the concentration of CuSO4 à ¯Ã¢â‚¬Å¡Ã‚ · 5H2O and the wavelength was 335nm. Calculate the concentration of CuSO4 à ¯Ã¢â‚¬Å¡Ã‚ · 5H2O in the unknown sample using the straight line equation. Solution 8: y = 2.8571x + 190.48y = 335 nm and x = x = (y – 190.48) à · 2.8571x = (335nm – 190.48) à · 2.8571x = 50.6 mmol Name________________________________________Section/Day/Time________________________ MATHEMATICS IN GENERAL CHEMISTRY HOMEWORK SHEET Units of Measurement How 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 conversions: a) 10 m = _____ km = _____ cm = _____ mm b) 5.5 g = _____ kg = _____ ozs = _____à ¯Ã‚ Ã‚ ­g c) 400 cm = _____ ft = _____ in. = _____m d) 45 m/sec = _____ ft/sec. = _____ km/hr = _____mile/hr e) 9.9 in2 = _____ cm2 = _____ ft2 = _____mm2 Aluminum 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/Time_________________________ MATHEMATICS IN GENERAL CHEMISTRY HOMEWORK SHEET (pg. 2) II. Statistical Analysis A 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 follows: Student Number Grade 1 99 2 80 3 79 4 88 5 95 Determine the following: Average Percent Error Standard Deviation Exponential Notation and Significant Figures Express the answers to the following calculations in scientific notation: 145.75 + (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 measurements: 5748 mi 38 mL 60,293 km 0.0005 cm Name______________________________________Section/Day/Time_________________________ MATHEMATICS IN GENERAL CHEMISTRY HOMEWORK SHEET (pg. 3) Graphing Using 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?