Errors in Chemical Analysis PDF

Chapter 5 Errors in Chemical Analysis Modified by Dr. Mohammed Rasheed  Data quality  Measurements invariably involve errors and uncertainties.  Uncertainties can never be completely eliminated, measurement data can give us only an estimate of the “true” value.  Reliability can be a...

Chapter 5 Errors in Chemical Analysis Modified by Dr. Mohammed Rasheed  Data quality  Measurements invariably involve errors and uncertainties.  Uncertainties can never be completely eliminated, measurement data can give us only an estimate of the “true” value.  Reliability can be assessed in several ways:  Experiments designed to reveal the presence of errors  Compared with the known compositions  Consult to the chemical literature  Equipment Calibration  Statistical tests Representative Data  Chemists usually carry two to five portions (replicates) of a sample through an entire analytical procedure. – Replicates are samples of about the same size that are carried through an analysis in exactly the same way.  One usually considers the “best” estimate to be the central value for the set: – Usually, the mean or the median is used as the central value for a set of replicate measurements. – The variation in data allows us to estimate the uncertainty associated with the central result The Mean and Median The most widely used measure of central value is the mean,. The mean, also called the arithmetic mean, or the average, where xi represents the individual values of x making up the set of N replicate measurements. The median is the middle result when replicate data are arranged according to increasing or decreasing value.  Median can be preferred when there is an “outlier” - one reading very different from rest. Median less affected by outlier than is mean. Ex: Calculate the mean and median for the data: 19.4, 19.5, 19.6, 19.8, 20.1 and 20.3 Because the set contains an even number of measurements, the median is the average of the central pair:  Illustration of “Mean” and “Median” Results of 6 determinations of the Fe(III) content of a solution, known to contain 20 ppm: Note: The mean value is 19.78 ppm (i.e. 19.8ppm) – The median value is 19.7 ppm Some Important Terms  Precision – A measure of the REPRODUCIBILITY of measurement.  Accuracy – A measure of how CLOSE a measured value is to the “TRUE” value. Figure 5-2 Note that we can have very precise results (upper right) with a mean that is not accurate and an accurate mean (lower left) with data points that are imprecise. Precision  Describes the reproducibility of measurements  Or, the closeness of results that have been obtained in exactly the same way.  Three terms are widely used to describe the precision of a set of replicate data:  standard deviation; later  variance; later  coefficient of variation; later  Deviation from the mean: Accuracy Indicates the closeness of the measurement to the true or accepted value Expressed in terms of either absolute or relative error. Absolute error: – where xt is the true or accepted value of the quantity – We retain the sign in stating the absolute error. Relative error Accuracy Absolute Error * The absolute error of a measurement is the difference between the measured value and the true value. If the measurement result is low, the sign is negative; if the measurement result is high, the sign is positive. Relative Error The relative error of a measurement is the absolute error divided by the true value. Relative error may be expressed in percent, parts per thousand, or parts per million, depending on the magnitude of the result. x x E  i t  100% r x t  Results can be precise without being accurate and accurate without being precise.  Each dot represents the error associated with a single determination.  Each vertical line labeled (xi - xt) is the absolute average deviation of the set from the true value. precise, accurate imprecise, accurate precise, inaccurate Imprecise, inaccurate Figure 5-3 Absolute error in the micro-Kjeldahl determination of nitrogen Types of Error in Experimental Data (1) Random (indeterminate) Error Data scattered approx. symmetrically about a mean value. Affects precision - dealt with statistically (see later). (2) Systematic (determinate) Error Several possible sources - later. Readings all too high or too low. Affects accuracy. (3) Gross Errors; lead to “Outlier” These differ from indeterminate and determinate errors. They usually occur only occasionally, are often large, and may cause a result to be either high or low. They are often the product of human errors. Gross errors lead to outliers, results that appear to differ markedly from all other data in a set of replicate measurements. 5B Systematic errors  have a definite value,  an assignable (one definite) cause, and  are of the same magnitude for replicate measurements made in the same way.  They lead to bias in measurement results. Bias affects all of the data in a set in the same way and that bears a sign There are three types of systematic errors: Instrumental errors. Method errors Personal errors Instrumental Errors are caused by nonideal instrument behavior, by faulty calibrations, or by use under inappropriate conditions Pipets, burets, and volumetric flasks may hold or deliver volumes slightly different from those indicated by their graduations. Calibration eliminates most systematic errors of this type. Electronic instruments can be influenced by noise, temperature, pH and are also subject to systematic errors.  Errors of these types usually are detectable and correctable. Method Errors  The nonideal chemical or physical behavior of the reagents and reactions on which an analysis is based often introduce systematic method errors.  Such sources of nonideality include the slowness of some reactions, the incompleteness of others, the instability of some species, the lack of specificity of most reagents, and the possible occurrence of side reactions that interfere with the measurement process.  Errors inherent in a method are often difficult to detect and hence, these errors are usually the most difficult to identify and correct. Personal Errors  result from the carelessness, inattention, or personal limitations of the experimenter.  Many measurements require personal judgments.  Examples include estimating the position of a pointer between two scale divisions, the color of a solution at the end point in a titration, or the level of a liquid with respect to a graduation in a pipet or buret.  Judgments of this type are often subject to systematic, unidirectional errors.  A universal source of personal error is bias.  Number bias is another source of personal error that varies considerably from person to person.  The most frequent number bias encountered in estimating the position of a needle on a scale involves a preference for the digits 0 and 5.  Digital and computer displays on pH meters, laboratory balances, and other electronic instruments eliminate number bias because no judgment is involved in taking a reading. 5B-2 The effects of systematic errors on analytical results  Systematic errors may be either constant or proportional  Constant errors:  The magnitude of a constant error stays essentially the same even if the size or the quantity of the sample is changed.  With constant errors, the absolute error is constant with different sample size, but the relative error varies when the sample size is changed; the relative error from this source becomes more serious as the total volume decreases. 5-2. Suppose that 0.50 mg of precipitate (unknown) is lost as a result of being washed with 200 mL of wash liquid. (0.5 mg loss (constant error) regardless the sample quantity or size)  If the ppt weights 500 mg, the relative error due to solubility loss is (499.5- 500)/500 × 100% = – 0.1%  If the ppt weights 50 mg, the relative error due to solubility loss is (49.5 -50)/50 x 100% = -1.0%  The sample size changed (500 and 50mg) then the relative error is changed. While the absolute error is always 0.5mg)  One way of reducing the effect of constant error is to increase the sample size until the relative error is acceptable. Example- Constant Error x x Relative error: E  i t  100% r x t (50.04-50.00)/50.00 x 100% = 0.08% (10.04-10.00)/10.00) x 100% = 0.4%  Although a constant error of 0.04 ml regardless the total volume of titrant, the relative error is great for the less amount of the titarnt  Or Proportional errors: measured quantity decrease or increase in proportion to the size or quantity of the sample but the relative error is independent on the sample size.  A common cause of proportional errors is the presence of interfering contaminants in the sample. Example:  Determination of Cu+2 is based on it’s reaction with KI to give I2 The quantity of iodine is then measured and is proportional to the amount of Cu+2.  if Fe+3 present, it gives also I2 from KI. high results are observed for the %Cu+2 because the iodine produced will be a measure of Cu+2 and Fe+3 in the sample.  The size of this error is fixed by the fraction of Fe+3 contamination, which is dependent of the size of sample taken.  If the sample size is doubled, the amount of iodine liberated by both the copper and the iron contaminant is also doubled.  Thus, the magnitude of the reported percentage of copper is independent of sample size. Then the absolute error is the same e.g: If 1g sample gives 0.1 g of I2 then for example the % of Cu in the sample increased by 10% If 2g sample gives 0.2 g of I2 then for example the % of Cu in the sample increased by 10% 5B-3 Detection and Elimination of Systematic (Instrumental and Personal) Errors 1. Periodic calibration of equipment is always desirable because the response of most instruments changes with time as a result of component aging, corrosion, or mistreatment. 2. Most personal errors can be minimized by careful, disciplined laboratory work. 3. It is a good habit to check instrument readings, notebook entries, and calculations systematically. 4. Errors due to limitations of the experimenter can usually be avoided by carefully choosing the analytical method or using an automated procedure. 5B-4 Detection of Systematic (Method) Errors  Bias in an analytical method is particularly difficult to detect.  Estimate the bias of an analytical method by  Independent Analysis: using different method to determine the analyte  Blank Determinations  Variation in sample size: can be used if there is constant error  Best by analyzing Standard reference materials (SRMs). Analysis of Standard Samples using standard reference material  The best way of estimating the bias of an analytical method is by the analysis of standard reference materials:  Materials that contain one or more analytes at known concentration levels.  Can sometimes be prepared by synthesis.  Can be purchased from a number of governmental and industrial sources. Ex: National Institute of Standards and Technology (NIST); Sigma Chemical Co.  Independent Analysis & Variation in Sample Size  A second independent and reliable analytical method can be used in parallel with the method being evaluated.  A statistical test must be used to determine whether any difference is a result of random errors in the two methods or due to bias in the method under study. Variation in Sample Size As the size of a measurement increases, the effect of a constant error decreases. Constant errors can often be detected by varying the sample size. Blank Determinations  A blank contains the reagents and solvents used in a determination, but no analyte.  Many of the sample constituents are added to simulate the analyte environment, often called the sample matrix.  In a blank determination:  All steps of the analysis are performed on the blank material.  Blank determinations reveal errors due to interfering contaminants from the reagents and vessels used in the analysis.