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?

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During the Washington, Adams, and Jefferson administrations (from 1776 to 1807) neutrality was America’s main foreign policy. In determining that neutrality was the overall focus of American diplomacy, one must assess the deviations from, as well as the success, of neutrality. Neutrality was originally implemented by George Washington in order to maintain the young country’s best interests. However, lapses in neutrality occurred when the government was forced to favor one foreign power, either Great Britain or France, through treaties. Furthermore, commercial interests ultimately drove America’s decisions and thus influenced the effectiveness of neutrality. Despite such lapses in success, America’s number one foreign policy remained neutrality. Political leaders found involvement in European affairs to be irrelevant and detrimental to the young nation, thus neutrality was the primary focus of America. George Washington was a strong supporter of neutrality despite his highly opinionated cabinet (Hamilton was pro-British and Jefferson pro-French). In 1793, in response to King Louis XVI’s beheading in the French Revolution, Washington issued the Proclamation of Neutrality which stated that America would â€Å"adopt and pursue a conduct friendly and impartial toward the belligerent Powers†¦ † (D). Washington opted to remain neutral in order to avoid conflict with Britain and France, refusing to side with either one. Furthermore, Washington emphasized the necessity of neutrality in his farewell address in 1796. He explained that the â€Å"great rule of conduct for [Americans] in regard to foreign nations†¦ † was to have â€Å"†¦ s little political connection as possible† (J). Thus, Washington established the importance of neutrality which was also supported by John Adams. During the Revolutionary War (right after the Battle of Saratoga) the Americans formed the Franco-American alliance and gained French support from the king. However, John Adams wrote in his diary in 1775, â€Å"That We ought not to enter into any Alliance with her [France], which should entangle Us in any future Wars in Europe,† (A). Adams supported Washington’s push for neutrality and recognized the consequences of forming an alliance with France. Adams was right about the Franco-American Alliance, which later resulted in France seeking American aide against Britain after Louis XVI was beheaded. On the other hand, Adams approved Jay’s Treaty with Great Britain which upset the French who were at war with the British. Jay’s Treaty settled the return of confiscated goods and ships to the Americans and established that pre-Revolutionary War debts must be paid as stated in the Treaty of Paris in 1783. Americans had avoided these debts even though the Treaty of Paris stated that â€Å"creditors on either side shall meet†¦ o the recovery of the full value in sterling money of all bona fide debts heretofore contracted,† (E). During the Jefferson administration, foreign tensions rose and eventually lead to the War of 1812. Jefferson also enforced neutrality, claiming in his Inaugural Address in 1801 that the nation will maintain â€Å"honest friendship with all nations, entangling alliances with none†¦ à ¢â‚¬  (K). Such ideas were evident in his response to the Chesapeake-Leopard Affair. In 1807, the British warship Leopard fired on an American warship, the Chesapeake, killing three Americans and impressing four others. Impressment was the act of forcing American sailors into the British navy as depicted in document M (M). Instead of declaring war as many American citizens wanted, Jefferson issued the Embargo Act, which cut off all American shipping to Europe in an attempt to cripple the British economy and prevent them from confiscating American goods and impressing their sailors. However, Jefferson’s attempt to maintain neutrality back-fired and crippled the American economy more than Europe’s. Conflicts between Great Britain and France often caused lapses in America’s focus on neutrality. Specifically, during the Washington’s administration and the French Revolution, â€Å"Citizen† Genet, the French minister to America, broke the typical rules of diplomacy by appealing directly to the American people. He called for American support of the French Revolution, pointing to the Franco-American Alliance which stated that the two nations shall â€Å"make all efforts in its Power, against their common enemy [Britain]†¦ † (C). Such support would directly violate America’s primary foreign policy, neutrality. Thus, Washington refused to support the French Revolution and make an enemy of Great Britain. This decision angered pro-French Americans who supported France’s aspiration to establish a republic. As a result, Thomas Jefferson, who was pro-French, resigned from his position of Secretary of State in disagreement with Washington’s Proclamation of Neutrality. The ratification of Jay’s Treaty with Great Britain further infuriated French supporters, failed to solve the issue of British impressment, and even promised that Americans would pay pre-Revolutionary War debt to Britain (F). Moreover, it caused the French to be outraged with the Americans and started French seizure of American ships. Political tensions between the three nations continued during John Adams’ presidency with the XYZ Affair in 1798. Three French agents, known only as X, Y, and Z, demanded a high fee from American delegates to enter negotiations for a treaty regarding the French seizure of American ships. This tested the young nation’s focus on neutrality as outraged Americans called for war with France. As a result, neutrality was pushed aside as the Quasi War, which was never officially declared, broke out. The public anger caused by the war strengthened the Federalist party, who used the power to pass the Alien, Sedition, and Naturalization Acts. These Acts limited Democratic-Republican power by: increasing the number of years required for immigrants to become citizens (Naturalization Act), by authorizing the president to deport aliens deemed a threat to the nation (Alien Act), and by making it illegal for newspaper editors to criticize the president or Congress (Sedition Act). The Convention of 1800 concluded the Quasi War, however impressment continued into Jefferson’s presidency during the Napoleonic Wars, making it more difficult for the young nation to maintain neutrality. Thus, conflicts between Great Britain and France often involved America despite the American focus on neutrality. Commercial interests often times caused disruptions of neutrality and competed as a priority of the young nation. Because commerce was of high importance to the growing nation, neutrality, in order to maintain trade, was necessary. Thomas Paine even stated, â€Å"Our plan is commerce,† and emphasized the importance of neutrality to the American economy by saying â€Å"we ought to form no political connection with any part of it [Europe],† (B). For instance, Jay’s Treaty negatively affected American commerce and political relations with France. Additionally, James Madison criticized the treaty as being â€Å"ready to sacrifice†¦ the dearest interests of our commerce†¦ † (G). The treaty not only affected political diplomacy with France, but also affected relations with Spain and their North American colonies. Spain saw the treaty as a sign that the Americans were building an alliance with Britain and felt a need to strengthen their American territories. This lead to Pinckney’s Treaty, in 1795, which set the boundary line between the United States and the Spanish territories, as well as allowed the use of the Mississippi River and the Louisiana sea port to both the Americans and the Spanish (H). Pinckney’s Treaty protected American commerce along the Mississippi and resulted in peaceful relations with Spain. Later, during Adams’ presidency, the seizure of American merchant ships by France raised tensions that resulted in the Quasi War. The disruption of American commerce forced the nation to disregard neutrality in order to protect their trading rights. The Convention of 1800 concluded the war and promised that â€Å"Property captured†¦ shall be mutually restored†¦ † (I). The Convention revealed the evident commercial intentions of the Quasi War and set the American economy as a priority equal to that of neutrality. However, Great Britain continued to seize American ships and impress American sailors. Instead of engaging in war, Jefferson opted to pass the Embargo Act of 1807 in an attempt to sustain neutrality. The Act prevented trade with any European nation, attempting to damage Britain’s economy and preclude impressment. Unfortunately, the plan backfired and devastated the American economy. Thus, Americans felt a greater need to deviate from neutrality in order to better their commerce. Although America faced disruptions in neutrality, ultimately, it remained the nations overall primary foreign policy. From 1776 to 1807, neutrality kept the young nation out of unnecessary European conflicts and served to maintain the country’s best interests.

Nietzsche’s Ubermensch Essay

Nietzsche’s Ubermensch stands for homo superior, overman or super man. Nietzsche philosophy of Ubermensch is based on ‘Will to Power’. All doctrines are dependent on ‘power to will’. Many of his philosophies are misinterpreted before but there still a lot of disagreement exists in his philosophy of power to overcome everything by willpower. He describes man to be Ubermensch when he gains such power and then he can do anything. This power can be destructive or constructive or by self-improvement. Though there is some truth in the fact that whatever we do or whatever we want to do we can attain it by willpower, but our willpower is limited. It is not a self-created giant that will overcome all obstacles in life. God has granted us will to choose between right and wrong and will to perform and overcome difficult situations. But our willpower is limited as we experience in day-to-day life. Because in whatever environment we live we’re surrounded by a number of situations and it is not always possible to tackle all the situations simultaneously. A number of examples can be cited for this. A person goes out of his house to perform a particular task. He’s determined that he has five jobs to complete during the day. But he’s not aware of an unknown incident that might occur in his way and block him to do his jobs. Similarly a sports man who is very well prepared for the competition and full determined that he’ll win the competition may loose because of any sports injury or simply bad luck! His philosophical ideas about truth that our minds have faith in the beliefs because they’re very convincing not because they are truth is again controversial. Nietzsche’s philosophical claim cannot be agreed upon. The fact that we do believe in things that are true. And only those beliefs are convincing that have universal truth in it. For example, we do believe in some kind of ancient saying that â€Å"Green tea has power to cure†. It is due to the fact that green tea has been used for several years and it has proved to be good in maintaining health. Now scientific research has also proved that it possess certain good polyphenoids that are required by the body in maintaining health and combating disease. Truth forms the basis of several facts that we believe. Hence, Nietzsche’s notion cannot be said as valid. He has extraordinarily overstressed the on the use of ‘power’. His statement is baseless when he says in philosophy we only have power and no higher truth. On the contrary, in science we have facts based on truth. Though our willpower does affect our inner strength and determination, but our power is limited as there is a divine unlimited power of one God ruling and final decisions rests on His willingness and permission. Men do strive hard to attain a goal but it is in the hands of God to grant us that or not. Man’s power is always limited and controlled. Nietzsche stresses on personal character and inner strong determination for power. However, it is not reasonable to solely judge the occurrence of events on the basis of strong determination. Under many circumstances in daily life we come across things that are not wanted by our own self. And our determination breaks when the outcomes of our thinking and our willpower are just the opposite. References Thus Spake Bart by Mark T. Conrad, Thus Spoke Zarathushra the Three Metamorphoses

The No Miracles Argument For Scientific Realism - 1894 Words

When answering the question What is the â€Å"No Miracles† Argument for scientific realism? We first need to look at what is scientific realism. The definition of which is ill- defined, one definition is; Scientific realism is a positive epistemic attitude towards the content of our best theories and models, recommending belief in both observable and unobservable aspects of the world described by the science. (Anjan Chakravartty,2011). Also When looking at scientific realism it is said that it can be split in three different kinds; theory realism or epistemological, realism, entity realism or semantically realism and natural kind realism or metaphysical realism. that can all be justified by the no miracles argument. A scientific realist believes that the aim of science is knowledge and finding out the truths in the universe, this allows scientist to predict and create things with the products of the theory. The no miracles argument claims that scientific realism is the best explanation of the successes of science, which would otherwise be seen as ‘miraculous’; meaning if scientific theories are incorrect why is there evidence that supports the theories that is successful such as with DNA and electrons that can correctly be predicted and experimented upon. It also means that too have a successful theory it must tell the truth about the universe. Within this essay I will also be looking at the counter arguments that try and disprove the no miracles argument and scientificShow MoreRelatedRealism And Its Impact On The Mind Independent Existence Of The World917 Words   |  4 PagesTraditionally, realism generally is associated with any position that endorses belief in the reality of something. 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