1.5 Measurement Uncertainty, Accuracy, and Precision 1.6 Mathematical Treatment of Measurement Results
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1 1.5 Measurement Uncertainty, Accuracy, and Precision 1.6 Mathematical Treatment of Measurement Results Let s recap what we saw in lab: o A measurement is a quantitative observation; a measurement is never exact. o We assume the absolute uncertainty (au) of a measurement is 1 in the position of the last significant figure. o Measurements are affected by random and systematic errors. Random errors affect precision (closeness to each other). Systematic errors affect accuracy (closeness to a known value). o We need to know how to count sig figs in measurements. Use rules to report sig figs in calculated answer; answer should not be more precise than any individual measurement. Separate sig figs from non-sig figs with a dashed line ; keep two non-sig figs when possible, and use unrounded answer later. o The ppt RAD is the Chem 180L/185L gold standard to assess the precision of our measurements; we convert to a relative result by dividing by the measurement. average measurement = x = N i=1 x i N average (absolute) deviation = d = N i=1 x x i N ppt RAD = 1000 d x o We will see a related formula for calculating the ppt relative error (re) when we assess accuracy. Most of our problems will be solved using dimensional analysis or a formula Dimensional Analysis (aka the factor label or unit equation approach) is a problem-solving strategy which starts with an initial value and multiples it by conversion factor(s) to yield the final result. o Dimensional Analysis must be set up on one line for full credit! 1 Chem 180-Spring 2019
2 Convert a volume of 3482 ft 3 to cubic meters. Use only these equivalences: 1 ft = 12 in; 1 in = 2.54 cm; 1 cm = 10 2 m. When using a formula, symbolically solve for the unknown and then plug in the given data. What is the volume (ml) of a mercury sample that has a mass of g? The density of mercury is 13.8 g/ml. Concentrated sulfuric acid assays 98% by weight sulfuric acid (the rest is water) and has a density of 1.84 g cm 3. Calculate the mass of pure sulfuric acid in cm 3 of concentrated acid. The kinds of info I will provide on a quiz or test: o Formulas (temperature conversion, ideal gas law). o Equivalences other than pure SI. 2 Chem 180-Spring 2019
3 This has been kept brief because I expect you have solved problems like this already. In any case, the homework will give you practice. Here are some tips for your homework. o Print out the homework assignment and answer the questions in the space provided. Don t do this, and you will get zero points. o OpenStax has a free student solutions manual for odd-numbered problems available that you can download. So my assignments will be even-numbered problems! However, problems are often paired, meaning adjacent odd and even-numbered problems are similar. So if you are having trouble, try taking a look at the odd-numbered solution for ideas on solving the assigned problem. o Get in the habit of using the dashed line in your answers. The book seems to provide only rounded answer; I want you to provide the unrounded and boxed rounded answers. This will be especially important in lab where unrounded answers are frequently used in later calculations! 3 Chem 180-Spring 2019
4 Chapter 2: Atoms, Molecules, and Ions 2.1 Early Ideas in Atomic Theory Chemistry is a foreign language with all the pieces we encounter when studying a new language: o Alphabet: the element symbols o Vocabulary (words): formulas of chemical compounds o Sentences: chemical equations o Semantics: the meaning of a chemical equations The development of this language spans thousands of years! The Greek Democritus et al. proposed that all matter was composed of small, indivisible particles called atoms. Later on, Aristotle supported the idea that all matter consisted of earth, air, fire, and water combined in varying proportions. o The ancients had no experimental tradition, and merely reasoned their way to conclusions. o While Democritus was closer to the answer, the Aristotelean view held for over 2000 years. Why? A new way to investigate nature, and a growing body of chemical data occurred from the early 1600 s to the late 1700 s, culminating in Dalton s atomic theory. (The great physicist Richard Feynman considered the atomic theory the most important concept in science. It provides the ideas from which all other science can flourish.) 4 Chem 180-Spring 2019
5 The postulates of Dalton s Atomic Theory are: o Matter is composed of exceedingly small indivisible particles called atoms. An atom is the smallest unit of an element. o An element consists of only one type of atom, which has a mass that is characteristic of the element and is the same for all atoms of that element. o Atoms of one element differ in mass (and properties) from atoms of all other elements. o A compound consists of atoms of two or more elements combined in a small, whole-number ratio. In a given compound, the numbers of atoms of each of its elements are always present in the same ratio. The same elements can combine in different ratios to yield different compounds. o Atoms are neither created nor destroyed during a chemical change, but are instead rearranged to yield substances that are different from those present before the change. Dalton s theory supported existing laws, and was the basis for a new one: o Law of Conservation of Mass (Lavoisier): the total mass of substances present after a chemical reaction is the same as the total mass of substances before the reaction. 5 Chem 180-Spring 2019
6 o Law of Constant Composition (aka Law of Definite Proportions; Proust): All samples of a compound have the same composition the same proportion by mass of the constituent elements. The mass percent of the elements in a compound are constant and do not depend on the source or size of the sample. o Law of Multiple Proportions (Dalton): If two elements form more than a single compound, the masses of one element combined with a fixed mass of the second are in the ratio of small whole numbers. Brown chloride of copper Green chloride of copper Throughout the 1800 s, Dalton s theory was a powerful tool for predicting and explaining what happens in chemical reactions. o But around 1900, experimental evidence was accumulating which was not consistent with Dalton s theory. Some new thinking was needed! 6 Chem 180-Spring 2019
7 We will consider three historic experiments which illustrated that atoms were not indivisible, but contained subatomic particles characterized by: o their charge, o their mass, and o their location within the atom. 2.2 Evolution of Atomic Theory J. J. Thomson Discovers the Electron (1897) Thomson studied the beam created in a cathode ray tube (CRT), and how the path of this beam could be deflected by electric and magnetic fields. Using the physical laws of electricity and magnetism, he concluded that o The beam consisted of negatively charged particles, electrons. o The charge to mass ratio of these particles was a constant, q = constant m 7 Chem 180-Spring 2019
8 R. A. Millikan Determines the Charge on the Electron (1909) In his famous oil drop experiment, Millikan created negatively charged oil drops which fell inside an evacuated chamber. He balanced this force of gravity with a variable electric field which repelled the drops upward until they were stationary. Knowing that the forces on the stationary drops were equal but opposite, he o Calculated the negative charge on each oil drop. o Observed that these charges were always the integer multiple of a single charge. o Deduced the charge of an electron. Combining Millikan s electron charge with Thomson s q/m ratio, the mass of the electron could be calculated: kg. Now if neutral atoms contained negative electrons, they must also contain positively charged particles. The next big question facing scientists was: How are these particles organized in atoms? 8 Chem 180-Spring 2019
9 E. Rutherford Discovers the Nucleus (1911) Rutherford was a New Zealander who placed second for a fellowship to study at Cambridge. But the fellowship winner decided to remain in New Zealand, and Rutherford went on to establish himself as the greatest experimentalist of his time. Some early models of the atom were Thomson s plum pudding model (1903), and Nagaoka s planetary model (1904). Both models suffered from their charge distributions. Rutherford s gold foil experiment provided the evidence that allowed him to propose the modern theory of the atom. Based on his observations, Rutherford concluded that: o Most of the volume occupied by an atom is empty space. o A small, massive, positive nucleus is at the center of the atom. The source of the positive charge was protons. 9 Chem 180-Spring 2019
10 One more subatomic particle was needed to complete the picture. o Frederick Soddy realized that an element could have types of atoms with different masses that were chemically indistinguishable. These different types are called isotopes atoms of the same element that differ in mass. o James Chadwick (1932) discovered the neutron, an uncharged particle with about the same mass as the proton; protons and electrons are found in the nucleus. o Isotopes of an element have the same number of protons in the nucleus (the atomic number), but different numbers of neutrons. Thus we can have atoms of the same element with different masses. 10 Chem 180-Spring 2019
11 2.3 Atomic Structure and Symbolism By the early 1930 s, the modern view of the atom was essentially complete. o The number of protons in the nucleus of an atom is its atomic number (Z). This is the defining trait of an element: the atomic number determines the identity of the atom. o We will see later how quantum mechanics provides a description of how the electrons are organized outside the nucleus. There are 118 known elements; they are organized by increasing atomic number in the periodic table. o Each element has a unique symbol consisting of an uppercase letter, or an uppercase letter followed by a lowercase one. I am picky about this! 11 Chem 180-Spring 2019
12 12 Chem 180-Spring 2019
13 Atoms may gain or lose electrons to form charged atoms, or ions. o The charge is written to the upper right of the element symbol. o Negatively charged anions are formed when electrons are gained. o Positively charged cations are formed when electrons are lost. Isotope information is written to the left of the element symbol. where: A ZX chaaaa A = the mass number = number of protons + number of neutrons Z = the atomic number = number of protons = the element charge = the ionic charge = number of protons number of electrons In a neutral atom (no charge is shown), the number of protons = the number of electrons. The number of neutrons = A Z = mass number atomic number. 13 Chem 180-Spring 2019
14 14 Chem 180-Spring 2019
15 The atomic mass (or atomic weight) listed for an element in the periodic table is the weighted average of the naturally occurring isotopes of the element. 15 Chem 180-Spring 2019
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