山大《醫(yī)學(xué)基礎(chǔ)化學(xué)》雙語課件10Titrimetry

1、Inorganic chemistryOrganic chemistryPhysical chemistryAnalytical chemistryChemistry 分析方法的分類分析任務(wù)定性分析 qualitative定量分析 quantitative結(jié)構(gòu)分析 structural分析對象無機分析 inorganic有機分析 organic方法原理化學(xué)分析 chemical儀器分析 instrumental微量分析 micro試 樣 用 量常量分析 macro半微量分析 semi-micro超微量分析 ultra-micro 分析目的例行分析 routine仲裁分析 arbitral1 分

2、析任務(wù)定性分析（qualitative analysis）： 鑒定物質(zhì)由哪些元素、原子團(tuán)或化合物組成定量分析(quantitative analysis)： 測定物質(zhì)中有關(guān)成分的含量結(jié)構(gòu)分析(structural analysis)： 研究物質(zhì)的分子結(jié)構(gòu)或晶體結(jié)構(gòu)2 試樣用量 方法試樣質(zhì)量試液體積常量分析半微量分析微量分析超微量分析0.1g0.010.1g0.110mg10mL110mL0.011mL 1%）、 微量組分（0.01%1%）、 痕量組分（ 0.01%） 3 分析目的例行分析（Routine analysis） ： 一般化驗室日常生產(chǎn)中的分 析，也稱常規(guī)分析仲裁分析（Arbitra

3、l analysis） ： 對分析結(jié)果有爭議，要求權(quán)威部門用指定的方法、準(zhǔn)確測定，進(jìn)行裁判原分析結(jié)果是否正確，也稱裁判分析。仲裁分析對分析方法和分析結(jié)果準(zhǔn)確度要求高。Quantitative analysis(定量分析) Gravimetric analysisChemical analysis Volumetric analysis SpectrometryInstrumental analysis Chromatographic method Electrochemical analysis Volumetric analysis (volumetry, titrimetry)Acid-b

4、ase titration (neutralization titration) Oxidation-reduction titration Coordination titration (EDTA titration) Precipitation titration10-1 General Principles of titration10-2 Acid-Base Titration10-3 EDTA Titration10-4 Analytical Error and Significant FiguresA titration is a standard laboratory metho

5、d of chemical analysis which can be used to determine the concentration of a unknown reactant.In a titration two reagents are mixed, one with a known concentration and one with an unknown concentration. There is some way to indicate when the two reagents have reacted essentially completely, and at t

6、he end of the titration the unknown solutions concentration can be calculated. General Principles of titration Titration consists of adding carefully metered volumes of the solution of known concentration to the other solution of unknown concentration to reach an end-point. Typically, one reagent is

7、 a solution and is added from a buret. This solution is called the titrant or standard solution. The solution from the buret is added to a conical flask that contains either a measured volume of a solution or a weighed quantity of solid that has been dissolved. The buret has graduations that are use

8、d to read the volume of titrant added to the conical flask.Titration: Titrant (standard solution) Analyte (sample) Stoichiometric point Indicator Error of titration End point of titrationAnalyte(sample) in conical flaskindicatorTitrant in burettitration : fill buret remove bubble fill pipette(、) tit

9、ration titration The requirement of a titration are as follows:1. The reaction should be stoichiometric. 2. The reaction should be rapid. 3. There should be no side reactions, and the reaction should be specific. 4. There should be a marked change in some property of the solution when the reaction i

10、s complete. 5. The point at which the reaction is observed to be complete is called the end point. 6. The reaction should be quantitative. Operation system in titrametric analysis1. Prepare the standard solution2.standardization of the standard solution3.determine the content of the sampleStandard S

11、olution Standard solution The concentration of a standard solution must be known within the limits of accuracy necessary in the actual analysis for which it is being used. Significant figure Four figures are necessaryPrepare the standard solution1. Direct method: used primary standard substance Na2C

12、O3 standard solution 2. Indirect method: NaOH standard solution HCl standard solution2.間接配制法（標(biāo)定法） 先配成一種接近所需濃度的溶液， 然后用基準(zhǔn)物質(zhì)（或用已經(jīng)用基準(zhǔn)物質(zhì)標(biāo)定過的另一標(biāo)準(zhǔn)溶液）來標(biāo)定它的準(zhǔn)確濃度。這種操作過程叫做“標(biāo)定”。間接法配制標(biāo)準(zhǔn)溶液。Standardization of standard solution HCl by Na2CO3 , NaOH by HCl2.標(biāo)定： 準(zhǔn)確稱取基準(zhǔn)準(zhǔn)物，溶解后滴定。1.配制溶液： 配制成近似所需濃度的溶液。 3.確定濃度 由基準(zhǔn)物質(zhì)的質(zhì)量或標(biāo)準(zhǔn)

13、溶液的 濃度以及所取試液的體積計算之。 primary standard substance A primary standard substance should fulfill these requirements: 1. It should be 100.00% pure 2. It should be stable to drying temperatures 3. It should be readily available. 4. it should have a high formula weight.(large mole weight) 5. Reaction quickly

14、 titre Determine the content of the sample毫升標(biāo)準(zhǔn)溶液(A)相當(dāng)于被測物(B)的質(zhì)量。 表示為 TB/A = gmL-1 或 %mL-T (Fe /K2Cr2O7) = 0.005260gmL-1，表示 1 mL K2Cr2O7標(biāo)準(zhǔn)溶液相當(dāng)于0.005260g Fe，即1 mLK2Cr2O7標(biāo)準(zhǔn)溶液恰好能與0.005260g Fe2+反應(yīng) .若滴定消耗21.45mL K2Cr2O7標(biāo)準(zhǔn)溶液，試樣中含F(xiàn)e量為多少？0.005260gmL-121.45mL = 0.1128g10-2 Acid-Base titration 1 Titration of a

15、 Strong Acid with a Strong Base the chemical reaction is the neutralization H3O+(aq) + OH- (aq) 2H2O(l) titrant: 0.1000 mol/L NaOH (in buret) titrate: 0.1000 mol/L HCl 20.00ml (in conical flask). Initially, V =0.00ml NaOH added H3O+= 0.1000mol/L pH = 1.00pH = 2.28 . V=18.00ml NaOH mol/L added mmoles

16、 acid left unreacted=2.000-1.800 =0.2000 volume solution =20.00+18.00=38.00ml . V=19.80ml NaOH mol/L added mmoles acid left unreacted = 2.000-1.9800 = 0.02000 volume solution = 20.00+19.80 = 39.80mlpH = 3.30. V=19.98ml NaOH mol/L added mmoles acid left unreacted = 2.000 - 1.9800 =0.02000 volume solu

17、tion = 20.00+19.98 = 39.98mlpH = 4.3. V=20.00ml NaOH mol/L added It is arrived at the equivalence point, the point in this titration: H3O+ = OH- H30+ = 1.0010-7 (mol/L) pH = 7.00. V=20.02ml NaOH mol/L added after the equivalence point mmoles base unreacted = 2.002-2.000 = 0.02000 volume solution = 2

18、0.00+20.02 = 40.02ml pOH = 4.30 pH =14-4.3=9.7.V=20.20ml NaOH mol/L added pOH = 3.30 pH =14-3.3=10.7 .V=22.00ml NaOH mol/L added pOH = 2.32 pH =14-2.32 =11.68.V=40.00ml NaOH mol/L added pOH = 1.5 pH =14-1.5=12.51020Titration jump Figure 10-1: A titration curve for the titration of a strong acid by a

19、 strong base. The magnitude of the titration jump will depend on both the concentration of the acid and the concentration of the base. Usually, the concentration of standard solution should be 0.10000.5000mol/Lcurve1. 100ml 0.1 mol/L HCl versus 0.1mol/L NaOHcurve2. 100ml 0.01 mol/L HCl versus 0.01mo

20、l/L NaOHcurve3. 100ml 0.001 mol/L HCl versus 0.001mol/L NaOH Titration curves at different concentrationtitration jump p4.39.7 5.45.38.7 3.4 3 6.37.7 1.4 突躍范圍與酸堿濃度的關(guān)系 溶液的濃度愈稀，滴定突躍范圍愈窄；若酸堿濃度低于104 mo1L1時，沒有明顯的滴定突躍。酸堿溶液的濃度高時，滴定突躍范圍就寬。但每滴溶液中所含的酸堿溶液的量增多，在計量點附近多加或少加半滴（0.02ml）標(biāo)準(zhǔn)溶液而引起的誤差較大。 所以在分析工作中，通常采用0.1

21、0.5 mo1L1的酸堿標(biāo)準(zhǔn)溶液。 2、Titration of a Weak Acid with a Strong Base titrant: 0.1000mol/L NaOH (in buret)titrate: 0.1000mol/L HAc 20.00ml (in conical flask). 2-1. V =0.00ml NaOH HAc + H2O Ac- + H3O+ 2-2. Less than the equivalent volume of NaOH has been added HAc(aq) + OH-(aq) = H2O(l) + Ac-(aq) 1). V = 1

22、8.00ml of 0.1000mol/LNaOH has been added 2).V = 19.98ml of 0.1000mol/LNaOH has been added: 3. V =20.00ml The equivalent volume of NaOH has been added Ac- + H2O = HAc + OH- at equivalence point: at equivalence point pOH = 5.27 pH = pKw pOH =14-5.27 = 8.734 V=20.02ml More than the equivalent volume of

23、 NaOH has been added . pOH = 4.30 pH = pKw-pOH =14.00-4.30 = 9.70Figure10-2: A titration curve for the titration of a weak acid by a strong base. titration jump 7.759.7 p 1.9512Figure10-2: A titration curve for the titration of a weak acid by a strong base. the titration curve with the acid dissocia

24、tion constant of HA. Figure: Calculated curves showing the titration of 0.1000mol/LNaOH with 0.02mol/LHA . As the acid becomes weaker, the change in slope at the equivalence point becomes less abrupt.濃度相同 強度不同的弱酸： Ka愈大，酸性愈強，突躍范圍愈大；反之亦然。 實驗表明，弱酸 c Ka10-8 條件下，能夠準(zhǔn)確被滴定。 作為判斷弱酸能否被滴定的依據(jù)。If Ka 9), there is

25、 no titration jump, so the weak acid can not be titrate by strong basethe requirement of a weak acid can be titrated: accuracyTitration of a Weak Acid with a Strong Base (中文書 p246)Detection of the end point: indicatorAn acid-base indicator is itself an weak acid or weak base that is highly colored.

26、The color of the ionized form is markedly different from that of the un-ionized form.These substances are usually composed of highly conjugated organic constituentsAssume the indicator is a weak acid HIn H+ + In-Base colorAcid color甲基橙此式說明：當(dāng)選擇的指示劑一定，pKHIn一定，溶液的pH值決定指示劑 的比值，使其在不同酸度的溶液中顯示不同的顏色。 (red)

27、(yellow)The ratio of the concentration of the two form: 1) Un-ionized form 2) Ionized formIndicators require a transition range變色范圍: Indicators require a transition range: change two pH unitsDuring this transition the observed color is a mixture of the two colorsThe indicator changes colore over a p

28、H range， generally only one color is observed pH=pKHIn (transition point 變色點of indicator )溴甲酚綠溴百里酚蘭溴酚蘭麝香草酚蘭茜草黃8.0Choosing an indicator The pKHIn of the indicator should be close to the pH of the equivalence point. we seek an indicator whose transition range overlaps the steepest part of the titratio

29、n jump as closely as possible. Some colors are easier to see than others8.0201.指示劑的用量： 在不影響終點顏色觀察的前提下，少加為宜； 使其在終點時的顏色變化便于觀察。 2.溫度： 引起指示劑解離常數(shù)的變化，變色范圍因此而改變。影響酸堿指示劑變色范圍的因素少加: 顏色沖淡，肉眼難以分辨和觀察。多加: 不易產(chǎn)生顏色突變，會遲鈍。 （本身消耗部分滴定劑， 帶來誤差）。 混合指示劑（縮小變色范圍的方法） 利用顏色的互補作用使顏色變化敏銳，易觀察。如混合指示劑甲基紫+溴甲酚綠1.pH 指示劑惰性染料（色不隨pH變）2.兩種

30、不同變色范圍的pH指示劑范圍窄了pH4.0 pH5.6 pH4.2 pH6.3 黃 藍(lán) 紅 紫 紅 綠 溴甲酚綠 甲基紫 pH5 5.210-3 Coordination Titrations Complexometric titration (coordination titration or EDTA titration): A titration based on formation of a complex ion is called . Chelating agent: An organic agent that has two or more groups capable of c

31、omplexing with a metal ion is called a chelating agent. Chelate: the complex formed is called a chelate. Ligand: the chelating agent is called the ligand.EDTA: Ethylenediaminetetraacetic acid (EDTA)is by far the most widely used chelator in analytical chemistry. HOOCCH2 CH2COOH H+NCH2CH2N+H HOOCCH2

32、CH2COOH Hexaprotic system H6Y2+ Ionization of EDTA H6Y2+ H5Y+ + H+ Ka1 = 1.3610-1 H5Y+ H4Y + H+ Ka2 = 2.510-2 H4Y H3Y- + H+ Ka3 = 1.010-2 H3Y- H2Y2- + H+ Ka4 = 2.1610-3 H2Y2- HY3- + H+ Ka5 = 6.9210-7 HY3- Y4- + H+ Ka6 = 5.5210-11Acid-Base properties pK1=0.0 pK2=1.5 pK3=2.0 pK4=2.66 pK5=6.16 pK6=10.2

33、4 各型體濃度取決于溶液pH值: pH 1 強酸性溶液 H6Y2+ pH 2.676.16 主要H2Y2- pH 10.26堿性溶液 Y4-Characteristics of EDTA complexes 1) It forms strong 1:1complexes with most metal ions. 2) Metal complexes is easy to dissolve. 3) Stability. octahedral compoundHNCMOMn+Y4- MYn-4離子lgKf離子lgKf離子lgKf離子lgKfLi+2.79Eu3+17.35Fe2+14.32Cd2

34、+16.46Na+1.66Tb3+17.67Fe3+25.10Hg2+21.70Ba2+9.30Dy3+18.30Co2+16.31Al3+16.30Mg2+8.70Yb3+19.57Co3+36.00Sn2+22.11Ca2+10.69Ti3+21.30Ni2+18.62Pb2+18.08Sr2+8.73Cr3+23.40Cu2+18.80Bi3+27.94La3+15.50MoO2 + 28.00Ag+7.32Th4+23.20Sm3+17.14Mn2+13.87Zn2+16.50U()25.80表 EDTA配合物的lg Kf（I= 0.1, 2025）主反應(yīng)： M + Y = MY Kf

35、副反應(yīng)： Y + H+ HY、H2Y、H6Y EDTA的酸效應(yīng) M + L ML、ML2、MLn 金屬離子的配位效應(yīng) 1 酸度的影響 EDTA的酸效應(yīng) 由于酸度（H+）的影響導(dǎo)致EDTA的配位能力降低的現(xiàn)象。 則酸效應(yīng)越強，EDTA的有效濃度越小，其配位能力越弱。 2.金屬離子的配位效應(yīng) 由于配位劑L的存在導(dǎo)致主反應(yīng)能力降低的現(xiàn)象。 影響EDTA配合物穩(wěn)定性的因素Figure (10-7) gives the minimum pH needed for titration of many metal ions. 配位滴定中酸度的控制EDTA的酸效應(yīng)曲線問題：Mg2+ Zn2+ 混合滴定分別測定

36、含量？Mg2+ Zn2+Zn2+ 調(diào)節(jié)pH=4 滴定Mg2+ 調(diào)節(jié)pH=10 滴定2. 其它配位劑的影響金屬離子的配位效應(yīng)主反應(yīng)： M + Y = MY Kf M + L ML、ML2、MLn 金屬離子的配位效應(yīng) 由于配位劑L的存在導(dǎo)致主反應(yīng)能力降低的現(xiàn)象。Zn2+ Al3+ EDTAAl3+ + 6F- = AlF63-NH4F(掩蔽劑)問題: Zn2+ 滴定含量測定？ MLn的穩(wěn)定性越高，配位效應(yīng)越強 Kf ? 用EDTA： 滴定金屬離子M， 當(dāng)?shù)味ǖ接嬃奎c附近時， 溶液pM值產(chǎn)生突躍。配位滴定曲線如EDTA滴定Ca2+， 通過計算過程中各點的pM值，繪制出一條曲線（右圖）2040V 1金

37、屬指示劑配合物與金屬指示劑本身色差要大。 2指示劑與金屬離子的顯色反應(yīng)要靈敏、迅速、可逆。 3金屬指示劑配合物及金屬指示劑本身要易溶。 4金屬指示劑配合物的穩(wěn)定性要適宜。A metal ion indicator is a compound whose color changes when it binds to a metal ion.Metal Ion Indicators金屬指示劑應(yīng)具備的條件:特點：(與酸堿指示劑比較) 金屬離子指示劑通過M的變化確定終點 酸堿指示劑通過H+ 的變化確定終點Metal Ion Indicators Eriochrome Black T is a typi

38、cal indicator.It contains three ionizable protons, so we will represent it by H3In. -H+ -H+ H2In- HIn2- In3- pKa2=6.3 pKa3=11.6pH12 orangepH=810 blue鉻黑T1) Before EDTA was added Mg2+ + HIn2- MgIn- + H+ (blue) (red)2) Among titration Mg2+ + HY3- MgY2- + H+ (colorless) (colorless)3) End point MgIn- + H

39、Y3- MgY2- + HIn2- (red) (colorless) (colorless) (blue) KMgY2 - KMgIn- This can be used for the titration of Mg2+ with EDTAOf couse, the metal indicator complex must be less stable than that of the metal-EDTA complex, or else the EDTA will not displace it from the metal.on the other hand, it must not

40、 be too weak,or the EDTA will stare replacing it at the beginning of the titration, and a diffuse end point will result. In general, the metal indicator complex should be 10 to 100 time less stable than the metal-titrant complex.Application of EDTA titration Determine overall hardness of waterHard w

41、ater and soft waterOverall hardness of water is amount of Ca2+ and Mg2+ in water. 1= 1mmol/L Ca2+ and Mg2+ Titrant : EDTA (Na2H2Y) pH = 8 10 (NH3-NH4Cl buffer) Indicator : EBTCaY2- MgY2- MgIn- CaIn- 1) Before EDTA was added (pH 10) Mg2+ + HIn2- MgIn- + H+ (blue) (red) 2) Among titration (pH =8 10) C

42、a2+ + HY3- CaY2- + H+ Mg2+ + HY3- MgY2- + H+ 3) End point (pH8) MgIn- + HY3- MgY2- + HIn2- (red) (colorless) (colorless) (blue)10-4 Analytical Error and Significant Figures定量分析中的誤差: 受方法、儀器、試劑、操作人員等的多種因素的影響，分析過程中的測量誤差是客觀存在、不可避免的。 了解分析過程中誤差產(chǎn)生的原因及其出現(xiàn)的規(guī)律，以便采取相應(yīng)的措施減小誤差，以提高分析結(jié)果的準(zhǔn)確度。Systematic Error or Det

43、erminate Error: It can in principle be discovered and corrected. Errors of the method Operative errors Instrumental errors Reagent errorsAnalytical Error定量分析誤差產(chǎn)生的原因1. 系統(tǒng)誤差 由某種固定因素引起的誤差，是在測量過程中重復(fù)出現(xiàn)、正負(fù)及大小可測，并具有單向性的誤差?？赏ㄟ^其他方法驗證而加以校正。分為： 方法誤差 由所選擇的方法本身決定，無法避免 操作誤差 操作者本人所引起，可消除或減少 儀器及試劑誤差 由儀器性能或試劑性質(zhì)決定 個人

44、誤差 由分析人員的主觀原因造成 Indeterminate Errors or Random Errors: which represent the experimental uncertainty that occurs in any measurement.糾正方法：“多次測量，取平均值” 以減少偶然誤差。偶然誤差又稱不定誤差或隨機誤差，由于一些難以察覺的或不可控制的隨機因素導(dǎo)致的誤差。系統(tǒng)誤差與偶然誤差之比較Accuracy And Precision Accuracy is the degree of agreement between the measured value and t

45、he true value.Precision is defined as the degree of agreement between replicate measurements of the same quantity. Ways of Expressing Accuracy 1. Absolute Errors (E)2. Relative Error (RE)準(zhǔn)確度(accuracy)在一定測量精度的條件下分 析結(jié)果與真值的接近程度。常以絕對誤差(E)和相對誤差(RE)來表示。1. Absolute Errors (E):2. Relative Error (RE) The abs

46、olute error expressed as a percentage of the true value is the relative error. RE = E/T100% The difference between the true value (T) and the measured value (x)。 E = x T(測定值與真值之差)（絕對誤差在真值中所占的百分率）例真值稱得量絕對誤差(x T)相對誤差 (x T)體重62.5kg62.4kg0.1kg買白糖1kg0.9kg0.1kg抓中藥0.2kg0.1kg0.1kg從表中的例子中你看出了什么問題T用相對誤差比絕對誤差表

47、示準(zhǔn)確度結(jié)果要好Sample of Weigh by Analysis BalanceW1 =0.0021gW2=0.5432gAbsolute Errors (E)?E1= E2 0.0001RE = E/T100%RE1 =1 / 21100% RE1 RE2 =1 / 5432100% RE2 誤差是相對真值而言的。而真值是客觀存在的數(shù)值，我們是不知道的。T = ? 實際工作中如何獲得真實值?_以測定多次的平均值來表示。 Absolute deviation (d)Relative deviation(Rd)Absolute average deviation (d )Relative

48、average deviation (Rd) 多次重復(fù)測定某一量時所得測量值的離散程度， 常以偏差來表示。精密度(precision)偏差的表示方法絕對偏差相對偏差 相對平均偏差（精密度）測定值與平均值之差 絕對偏差占平均值的百分率 平均偏差占平均值的百分率化學(xué)定量分析（常量分析）要求精密度在0.1% 0.3%之間。以平均偏差和相對偏差表示精密度比較簡單。在一系列測定結(jié)果中，小偏差占多數(shù)，大偏差占少數(shù)，如果按總的測定次數(shù)計算算術(shù)平均值偏差，所得結(jié)果會偏小，大偏差得不到應(yīng)有的反映。為了凸顯較大偏差的影響引入標(biāo)準(zhǔn)偏差(standard deviation) 。n:為測定次數(shù)relation sta

49、ndard deviation(RSD)coefficient of variation（CV） 是標(biāo)準(zhǔn)偏差占平均值的百分?jǐn)?shù)滴定分析為常量分析，相對平均偏差應(yīng)小于0.2%舉例：滴定管的讀數(shù)誤差為0.02mL。計算消耗滴定劑 2mL，20mL，和40mL 時的相對誤差。解：當(dāng) V2mL 時， RE1 0.02/2.00 100% =1%； V20mL 時，RE2 0.02/20.0 100% =0.1% V40mL 時，RE30.02/40.0 100% = 0.05% 若要求滴定分析的相對誤差在0.2以內(nèi)，消耗滴定劑的體積應(yīng)不小于多少毫升？分析： 若要保證整個滴定分析的相對誤差在0.2以內(nèi)，

50、讀數(shù)時的相對誤差必須在0.1以內(nèi)。 V（E/ RE）1000.02/0.1%=20 ml 答：為使滴定分析的相對誤差在0.2以內(nèi)， 滴定劑的體積應(yīng)不小于20ml。XXddd/XS39.8739.9440.1039.7439.9039.8839.905-0.035+0.035+0.195-0.165-0.005-0.0250.0771.920.117相對標(biāo)準(zhǔn)偏差 也稱變異系數(shù)(CV)，其計算式為： CV =S / x 100%本例中 CV = 0.29%例:在分析某一樣品中Cl含量為：39.87,39.94,40.10, 39.74,39.90,39.88，求各偏差值減免誤差的途徑1、選擇合適的分析方法 容量分析的準(zhǔn)確度高。儀器分析靈敏度高。2、減少測量誤差 應(yīng)減少每個測量環(huán)節(jié)的誤差，天平稱量應(yīng) 取樣0.2克以上，滴定劑體積應(yīng)大于20毫升。3、增加平行測定次數(shù)，減小偶然誤差 分析化學(xué)通常要求在3-5次。提高分析結(jié)果準(zhǔn)確度的方法 1). 對照試驗以標(biāo)準(zhǔn)樣品代替試樣進(jìn)行的測定，以校正測定過程中的系統(tǒng)誤差。 方法有標(biāo)準(zhǔn)樣比對法或加入回收法(用標(biāo)準(zhǔn)樣品、管理樣、人工合成樣等)、選擇標(biāo)準(zhǔn)方法(主要是國家標(biāo)準(zhǔn)等)、相互校驗(內(nèi)檢、外檢等)。2). 空白試驗不加試樣但完全照測定方法進(jìn)行操作的試驗，消除由干擾雜質(zhì)或溶劑對器皿腐蝕等所產(chǎn)生的系統(tǒng)誤差