Chemistry Lab reports, Lab Reports of Chemistry

Download Chemistry Lab reports and more Lab Reports Chemistry in PDF only on Docsity! 1.1 Experiment 1 Measurement and Data Analysis Introduction Chemistry is a quantitative science and advances in chemistry, as well as in all other sciences, depend on measurements which involve the reporting of quantitative information. The purpose of this exercise is to introduce the concept of measurement and the mathematical techniques used in the analysis of the resulting data. Units of Measurement All measurements contain three elements: a number which indicates the magnitude of the quantity being measured, unit(s) which provide us with a basis for comparing the measured quantity to a known reference and an indication of the magnitude of the error in the measurement. There are several systems of units, each of which contains units of measurement for properties such as distance, volume, mass, energy, and many others. Scientists use le Système International d’Unités or the International System of Units (SI) which was formally established by international agreement in 1960. There are seven mutually independent base units, as shown below, and all units must derive from this set of base units. In other words, any physical quantity we wish to measure may be expressed in terms of one of these seven basic units or an appropriate combination of two or more. Table 1: Base Units in le Système International d’Unités Physical Quantity Name of Unit Symbol mass kilogram kg length meter m time second s thermodynamic temperature Kelvin K amount of a substance mole mol electric current ampere A luminous intensity candela cd 1.2 Base units1 1. Mass: The SI unit is the kilogram (kg) and it is equal to the mass of the international prototype of the kilogram. 2. Length: The SI base unit is the meter (m). The meter is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second. 3. Time: The SI base unit is the second (s). The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom. 4. Temperature: The SI base unit of thermodynamic temperature is the kelvin (K). The kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. 5. Amount of a substance: The SI unit for the amount of a substance is the mole (mol). The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in exactly 0.012 kilogram of carbon 12. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles. 6. Electric current: The SI base unit is the ampere (A). The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 × 10!7 newton per meter of length. 7. Luminous Intensity: The SI base unit of luminous intensity is the candela (cd). The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. Derived Units Base units can be combined in various ways to obtain derived units. Some of these derived units will be familiar to you. Some derived units do not have special names or symbols. Table 2: Some Derived SI Units without Special Names Derived quantity Name Symbol area meter squared, square meter m2 volume meter cubed, cubic meter m3 speed, velocity meter per second m @ s!1 acceleration meter per second squared m @ s!2 wave number reciprocal meter m!1 mass density kilogram per meter cubed kg @ m!3 1NIST Special Publication 330 (SP 330) 1.5 Note that both conversion factors are equal to unity because the quantites in the numerator and denominator are equal. Consequently, when we multiply by either factor, the value of the original measurement does not change. Note also that the answer was obtained by multiplying two numerical quantities to give 4.26 and then multiplying the units and treating them algebraically: Multiplying by the second factor leads to a nonsensical result. Any number of conversion factors of this type can be strung together. Exponential Notation Chemists often deal with extremely large or extremely small numbers. Occasionally the numerical values are so large or small that even with the SI prefixes the numbers become cumbersome. For example, in one liter of water, there are about 33, 420, 000, 000, 000, 000, 000, 000, 000 H2O molecules, each with a mass of about 0.000 000 000 000 000 000 000 0299 g These numbers are inconvenient to work with and so in order to circumvent this problem we use exponential (scientific) notation. In exponential notation, a number is expressed as the product of two numbers: A × 10n The first number, A, is called the mantissa and is always greater than or equal to 1 and less than 10. That is, 1 # A < 10 The second number is called the characteristic and is 10 raised to some integral power. 1.6 The expression 10n is read as “10 raised to the nth power” or simply “10 to the nth.” In this expression, the number n is referred to as an “exponent.” This expression implies that 1 is multiplied by 10 a total of n times. For example, 105 = 1 × 10 × 10 × 10 × 10 × 10 = 100,000 A negative exponent, such as 10!2, implies division of 1 by the number 10 two times. 10 (or any other number) raised to the power of 0 is 1. 100 = 1 The power of 10 used in exponential notation is the number of places the decimal point must be moved in order for the mantissa, A, to be between 1 and 10. If the decimal point is moved to the left, n is positive, but if the decimal point is moved to the right, n is negative. Therefore, 33, 420, 000, 000, 000, 000, 000, 000, 000 = 3.342 × 10 25 and 0.000 000 000 000 000 000 000 0299 g = 2.99 × 10!23 g Powers and Logarithms Taking a logarithm and its inverse, raising a number to a power, play a role in many problems in chemistry. Numbers may be raised to non-integral powers. Scientific calculators have a 10x or “INV LOG” key that can be used to calculate non-integral powers of 10. A number raised to a power is referred to as a base, and numbers other than 10 can serve as a base. Scientific calculators usually have a yx key that will give the result of any number y raised to any power x. An extremely important base is the number e (2.7182818...) Whenever a base is raised to a power that is the sum of two numbers, the result is equal to the base raised to the power of the first number times the base raised to the power of the second number. That is 10(a + b) = 10a × 10b 1.7 The logarithm of a number is the exponent to which a base must be raised to obtain that number. The most common bases for logarithms are 10 and e. Therefore, if 10x = y then log10 y = x Base 10 logarithms are referred to as common logarithms and usually represented by “log.” Base e logarithms are referred to as natural logarithms and are usually represented by “ln.” Since logarithms are exponents, they obey the same rules as other exponents. log (a × b) = log a + log b The log of a number raised to a power is equal to the log of the number multiplied by that power. log ab = b log a When the log of a number such as 1, 10, 100, etc. is taken, the result is a whole number. Logs of other numbers are decimal fractions. For example, log (1.25 × 102) = 2.097 The decimal point divides the logarithm into two parts. The mantissa is to the right of the decimal point and the characteristic is to the left of the decimal point. If we take the log of a larger number, for example log (1.25 × 109) = 9.097 This log has the same mantissa but a different characteristic than the previous one. This should come as no surprise in light of the rules for exponents given above. log (1.25 × 102) = 0.097 + 2 = 2.097 and log (1.25 × 109) = 0.097 + 9 = 9.097 It is clear that the mantissa in a log is determined by the mantissa of the number while the characteristic in a log is determined by the characteristic in the number. Error Analysis Earlier we stated that all measurements contain three elements: a number which indicates the magnitude of the quantity being measured, units which provide us with a basis for comparing the measured quantity to a known reference and an indication of the error in the measurement. Errors fall into three classes: gross errors, systematic (determinate) errors and random (indeterminate) errors. 1.10 Table 6. Sample Data in the Determination of the Density of Chromium measurement (i) measured value (xi, g/mL) deviation deviation squared 1 7.12 0.04 0.0016 2 7.07 !0.01 0.0001 3 6.44 !0.64 0.4096 4 7.33 0.25 0.0625 5 7.29 0.21 0.0441 6 7.25 0.17 0.0289 Ó = 42.50 Ó = 0.5468 The smaller the value of the standard deviation the greater the precision of the measurement. When the true value for a measurement is known, the relative error of the measurement (typically expressed as a percent error, can be calculated. ( 1.4) Therefore, if the measured value is smaller than the true value, the % error is negative, but if the measured value is greater than the true value, the % error is positive. In either case, the smaller the magnitude of the % error, the more accurate the determination. 1.11 Dealing with Questionable Data Occasionally we might encounter a set of data in which one datum appears to lie far away from the rest of the data. An outlier is a result which does not belong to the same distribution as the rest of the data. An outlier is always found at one of the extremes (either the largest or smallest value), and is the result of systematic error. If an outlier were to be included in the calculation of the mean and standard deviation of the individual results, these values would not be reflective of the true mean and standard deviation of the population. Outliers, therefore, should be identified and excluded from further statistical analysis. We may not, however, simply discard a questionable datum no matter how inconsistent it may appear. There must always be some justification for omitting a result from statistical analysis, and even when the result is omitted from statistical analysis, the result must be reported. The Grubb's test is a relatively simple and straightforward method for identifying an outlier which is likely to be the result of determinate error. Consider again the data in Table 6. In principle, the result of each measurement should be identical. In practice, however, there is likely to be some variation from one measurement to the next, due to random errors. As was previously mentioned, small random errors are more probable than large ones. Notice how most of the values in Table 6 seem to cluster around 7.2 g/mL. The result of the third determination (6.44 g/mL) is significantly smaller than the rest and should be investigated as a potential outlier. In a Grubb's test, the average and standard deviation of a data set are calculated. The Grubb's statistic, G, defined below, is then calculated. (1.5) If Gcalculated is greater than the critical G, taken from a table, the questionable datum may be discarded. Table 7 gives values for Gcrit at 95% confidence. Table 7: Critical Values of G for Rejection of an Outlier at 95% Confidence n Gcrit n Gcrit n Gcrit n Gcrit 3 1.15 10 2.18 28 2.88 56 3.17 4 1.48 12 2.41 32 2.94 60 3.20 5 1.72 14 2.51 36 2.99 70 3.26 6 1.89 16 2.59 40 3.04 80 3.31 7 1.94 18 2.65 44 3.08 90 3.35 8 2.03 20 2.71 48 3.11 100 3.38 9 2.11 24 2.80 52 3.14 120 3.45 Extreme caution should be exercised when applying the Grubbs test to small data sets. When looking at a suspected outlier, reexamine all of the data used to obtain the value to determine whether a gross error could be responsible. The importance of keeping an orderly and properly formatted laboratory notebook with careful notation of all observations should be apparent. If possible, the degree of precision that can be reasonably expected from the procedure should be estimated to determine whether the questionable result is, in fact, an outlier. If time allows and enough sample is available, the analysis should be repeated. Agreement of the new data with the non-questionable data from the original analysis may provide 1.12 additional support for rejection of the questionable data. On the other hand, if the Grubbs test requires that the questionable datum be retained, the additional data will lessen the effect of the questionable datum on the mean. The Grubbs is a test for a single outlier, and should never be applied more than once to a single data set. Gexp > Gcrit for 6 observations at the 95% confidence limit, so the questionable datum may be discarded. In other words, there is at least a 95% probability that the difference between the questionable datum and the remaining data is not the result of random error. If Gexp < Gcrit then there is at least a 5% probability that the difference between the questionable datum and the rest of the data is the result of random error and the questionable datum must be retained. Significant Figures Since all measurements contain a finite amount of uncertainty or error, we must never report a value which implies a precision that is better than the equipment or techniques used to obtain the result would allow. For example, we could measure the length of a piece of paper two different rulers, A and B. Using ruler B we can determine, with certainty, that the paper is between 4 and 5 cm long, and that it is closer to 5 cm than to 4 cm. Furthermore, we can estimate how much closer it is to 5 cm than to 4 cm. If we imagine that the finest graduation on the ruler is further divided into 10 equal increments, we can estimate one additional digit. Therefore, using ruler B, we report the paper’s length as 4.6 cm. Using ruler A, we can determine, with certainty, that the paper is between 4.5 and 4.6 cm long, and that it is closer to 4.6 cm than to 4.5 cm. Again, we can estimate one digit beyond the finest graduation on the ruler. The end of the paper appears to be about 7/10 of the way between 4.5 and 4.6 cm, so we report its length as 4.57 cm using ruler A. When using an analog scale, such as the rulers above or a graduated cylinder, we report the result of the measurement to one digit beyond the finest graduation on the scale. That is, we report all the digits we know with certainty plus one additional digit which represents an estimate. When using a digital device, such as a balance or digital barometer, we report all the digits and only those digits given. 1.15 are, for example, exactly 60 seconds in one minute. We can therefore state that there are 762 seconds in 12.7 minutes, even though the conversion factor is written as 60 seconds per minute. In addition, quantities that result from a counting operation are exact numbers. For example, if I count the number of coins in my pocket I can say that I have exactly 3 quarters, or if I open a new box of pens, I can say that it contains exactly 12 pens. Graphical Treatment of Data A common experiment is to monitor changes in one variable (the dependent variable, y) as the value of a second variable (the independent variable, x) is systematically changed. For example, we can watch how the volume of a sample of gas changes with temperature, or how the pressure changes with volume. The data thus obtained are sometimes presented in a table which does not permit general trends in the data to be readily observed. The first step in the analysis of these data therefore often entails plotting a variety of graphical relationships between the two variables until a simple relationship is observed. The simplest form of a straight-line relationship occurs when two variables, x and y, are directly proportional, i.e., y = mx where m is a constant which may be either positive or negative. If y = mx, the value of y must be zero when x is equal to zero. Thus, plots of y versus x for y = mx must pass through the origin. Often, a linear relationship is observed where y does not vanish when x becomes zero. Instead, y is equal to some constant (b) when x is equal to zero. Relationships of this kind obey the equation y = mx + b Once again, m is a constant, and b is the value of y when x is equal to zero. A straight-line relationship of the type y = mx + b is shown in the figure to the right. The variable y is plotted on the vertical axis (ordinate) and the variable x is plotted on the horizontal axis (abscissa). The equation which generates this graph is derived by determining the values of m and b. The constant b, referred to as the y-intercept, is found by determining the value of y when x is equal to zero. y = mx + b = m × 0 + b = b The constant, m, referred to as the slope of the line, can be found by comparing any two points on the line. y2 = mx2 + b y1 = mx1 + b 1.16 If we subtract the second equation from the first we obtain y2 ! y1 = m (x2 ! x1) or If we define Äy as the change in y and Äx as the change in x Äx = x2 ! x1 Äy = y2 ! y1 then Often, a plot of y versus x does not yield a straight line. The variables might be inversely proportional, in which case, xy = m or, and a plot of y versus x does not lead to a straight line. However, a plot of y versus 1/x (or 1/y versus x) does yield a straight line. Thus, if the first attempt to obtain a straight line fails, it is often possible to redefine either the y or the x axis. A plot of y versus x for y = 4x2, for example, would produce a curved line. However, a plot of y versus x2 would yield a straight line of the form: y = m( x2 ) + b, where m = 4 and b = 0. Consider the dependence of the wavelength (ë) of lines in the helium emission spectrum on the frequency (í). The frequencies and wavelengths of several lines are given below. 1.17 Table 8. He Emission Spectrum Wavelength ( m ) Frequency ( s!1 ) 7.065 × 10!7 4.243 × 1014 6.678 × 10!7 4.489 × 1014 5.876 × 10!7 5.102 × 1014 5.015 × 10!7 5.978 × 1014 4.921 × 10!7 6.092 × 1014 4.713 × 10!7 6.361 × 1014 4.471 × 10!7 6.705 × 1014 In graphing, it is important to keep several points in mind. 1. The scale should be chosen so that the data fill as much of the available space as possible. 2. It is not necessary that the origin (0,0) appear on the graph. Leaving the origin off the plot may allow the data to be expanded to fill the available space without wasting space. 3. Once the scales are chosen and labeled with both the quantity and units, the data are plotted one point at a time. 4. Due to error in the measurements, the curve may not pass through all of the points. Don't force the curve to pass through each point or try to “connect the dots.” A smooth curve that approximates the data should be drawn A plot of wavelength versus frequency for the helium spectrum is shown to the right. Although the first three points and the last four points appear to lie on straight lines, it should be clear that all seven points do not lie on the same straight line and we can conclude that frequency and wavelength are not directly proportional. 1.20 Name______________________________ Section ____________________ Report Sheet: Measurement and Data Analysis 1. Use the necessary conversion factors and dimensional analysis to perform the following conversions. Show all your work and pay attention to significant figures! a. Convert 1.5 × 109 electron-volts to calories. b. Convert 624 cm3 to m3. c. Convert 745 torr to mbar. 2. Use dimensional analysis to answer the following problems. Show all your work and pay attention to significant figures! a. The volume of seawater on Earth is approximately 3.3 × 108 mi3. Seawater is approximately 3.5% sodium chloride by mass and has a density of 1.03 g @ mL!1. What is the approximate mass of sodium chloride in tonnes (1 tonne = 1000 kg) dissolved in the oceans and seas of Earth? {1 mile = 1609.33 m} 1.21 b. A piece of aluminum is 24.8 cm long, 0.456 dm wide and 13.8 mm thick. If the density of aluminum is 2.70 g @ cm!3, what is the mass of the piece of aluminum? Show all your work and pay attention to significant figures! c. If the speed of light in a vacuum is exactly 299,792,458 m @ s!1, how many miles does light travel in 8.75 hour? Show all your work and pay attention to significant figures! 3. pH is defined as the negative log10 of the hydrogen ion activity, aH+, although general chemistry texts commonly express it in terms of the hydronium ion concentration. Thus, pH = !log10 [H3O +], where [H3O +] represents the concentration of H3O + in units of moles per liter (M). Calculate the pH or [H3O +], as required, for the following solutions. Pay attention to significant figures! a. [H3O +] = 5.79 × 10!4 M b. [H3O +] = 1.0 × 10!11 M c. pH = 3.6 d. pH = 9.000 1.22 4. A student carried out several different determinations of the concentration of a sodium hydroxide solution. The student’s results were as follows: Experiment 1 2 3 4 5 Concentration (M) 0.1123 0.1087 0.1139 0.1112 0.1189 a. Calculate the mean and standard deviation for the data. Show all your work and pay attention to significant figures! b. Use the Grubbs test to determine whether the fifth result is an outlier. Can the fifth determination be discarded on the basis of the Grubbs test at the 95% confidence limit? Show all your work! c. If the true concentration is 0.1109 M, what is the percent error in the mean value? Show all your work and pay attention to significant figures!