Mass and volume in analytical chemistry (IUPAC Technical Report)

2.2.1 Mass: kilogram

Mass, quantity symbol m, dimension symbol M, which reflects the amount of matter within a body regardless of its volume or of any forces acting on it.

Mass is not to be confused with weight, which is the measure of the force of gravity acting on that body. There are two principal ways of referring to mass, depending on the law of physics defining it: gravitational mass and inertial mass.

The gravitational mass (weight) of a body is measured by comparing the body on a beam balance with a set of standard masses; in this way the gravitational factor is eliminated (except for air buoyancy corrections for differing densities – see 3.2.2).

Inertial mass is defined as the mass of a body as determined by its momentum. Inertial mass is obtained from the measured acceleration, a, when the object is subjected to a known force, F, by applying Newton’s Second Law, m=F/a.

The International Prototype of the Kilogram (IPK) is the artefact whose mass defines, at present (2017), the SI unit of mass. It is a cylinder with diameter and height of about 39 mm, made of an alloy of 90 % platinum and 10 % iridium. The IPK has been conserved at the BIPM since 1889, when it was sanctioned by the 1^st General Conference on Weights and Measures (CGPM). Initially the IPK had two official copies; over the years, one official copy has been replaced and four others have been added, so that there are now six official copies. There are numerous calibrated copies throughout the world. The kilogram is still the only base unit defined by a prototype of limited accessibility, prone to be damaged or destroyed, and which undergoes drift in mass which cannot be measured directly. (In fact, the German National kilogram prototype, which was stored in Berlin, was never located following World War II.) Unlike definitions of the other base units the IPK is not linked to an unvarying property of nature. Because of these concerns the kilogram is now being redefined [7, 8]. Three of the other base units rely on the definition of the kilogram – the ampere, the mole, and the candela.

Present definition: The kilogram, symbol kg, is the unit of mass; it is equal to the mass of the international prototype of the kilogram.

Proposed draft definition [9]: The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.626 070 040×10⁻³⁴ when expressed in the unit J s, which is equal to kg m² s⁻¹, where the metre and the second are defined in terms of c and ∆ν_Cs.

c is the speed of light in a vacuum and ∆ν_Cs is the hyperfine splitting frequency of caesium. In the “New SI” all units will be defined in terms of a set of seven reference constants, to be known as the “defining constants of the SI”, namely the caesium hyperfine splitting frequency, the speed of light in vacuum, the Planck constant, the elementary charge (i.e. the charge on a proton), the Boltzmann constant, the Avogadro constant, and the luminous efficacy of a specified monochromatic source. This is known as the explicit-constant formulation. In the new formulation the definition of the kilogram is dependent on the definitions of the metre and the second.

An alternative to explicitly using the Planck constant in this scheme would be to express the kilogram in terms of the Avogadro constant and the mass of a carbon 12 atom m_a(¹²C) [10].

At the 25^th CGPM meeting in November 2014, four criteria were announced for the quality of measured data in the New SI project that would have to be fulfilled before the CGPM could adopt the revised SI. One criterion was a suitably precise value of the Planck Constant obtained using Watt balances. The name of a Watt balance comes from the fact that the weight of the test mass is proportional to the product of a current and voltage, which is measured in units of watts. The Watt balance links the Planck constant h to the mass of the IPK by measuring the mass of a cobalt-based superalloy referenced to the “best standard Pt-Ir alloy”, i.e. it measures the ratio h/m_IPK. The goal is to attain a measurement uncertainty of 2 parts in 10⁸, that is 20 μg in 1 kg. As of mid-2017 the criteria are fulfilled, with a value of 6.626 069 934×10⁻³⁴ kg m² s⁻¹ with relative uncertainty 1.3×10⁻⁸ reported by NIST, and it is expected the New SI will be announced at the 26^th CGPM in 2018.

IUPAC has confirmed its support for the redefinition project [11].

2.2.2 Volume and time: metre and second

Volume, quantity symbol V, dimension M³, of an object measures the amount of space occupied by that object. The unit for the quantity is cubic metre, m³=10³ dm³=10³ litres (l or L), a unit derived from the base unit of length.

Time, quantity symbol t, dimension symbol T, is measured in units of second, symbol s, which is defined as 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 caesium–133 atom at 0 K.

Originally there were two competing approaches to the definition of the metre: the length of a pendulum having a half-period of one second, or one ten-millionth of the length of the earth’s meridian along a quadrant (one fourth the circumference of the earth (See Table 2.2-2). (The uncertainties in this table represent the uncertainty of the reported realization, i.e. the range in which the true value is presumed to be.) The latter was chosen based on the fact that the period of the pendulum is affected by the force of gravity which varies slightly over the surface of the earth. Thus, the metre was intended to equal 10⁻⁷ or one ten-millionth of the length of the meridian through Paris from the North Pole to the Equator. However, the first prototype was short by 0.2 mm because researchers miscalculated the flattening of the Earth due to its rotation. Still, this length became the standard, Mètre des Archives, in 1799. In 1889, a new international prototype was made of an alloy of platinum with 10 percent iridium that was to be measured to within a fraction of 0.0001 at the melting point of ice. In 1927, the metre was more precisely defined as the distance, at 0 °C, between the axes of the two central lines marked on the bar of platinum-iridium kept at the BIPM, and declared Prototype of the Metre by the 1^st CGPM. This bar was subject to standard atmospheric pressure and supported on two cylinders of at least one-centimetre diameter, symmetrically placed in the same horizontal plane at a distance of 571 mm from each other.

The 1889 definition of the metre, based upon the artefact international prototype of platinum-iridium (still kept at the BIPM under the conditions specified in 1889), was replaced by the CGPM in 1960 using a definition based on a wavelength of krypton–86 radiation. This definition was adopted in order to reduce the uncertainty with which the metre may be realised. In turn, to further reduce the uncertainty, in 1983 the CGPM replaced this latter definition with the current definition:

Metre (unit symbol m) is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.

This definition fixes the value for the speed of light in vacuum at exactly 299 792 458 m s⁻¹.

As a consequence of the New SI, the wording of the definition of the metre is proposed [9] to be:

The metre, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum c to be 299 792 458 when expressed in the unit m s⁻¹, where the second is defined in terms of the caesium frequency ∆ν_Cs.

2.2.3 Amount of substance: mole

In 1971, after lengthy discussions between physicists and chemists, the 14^th CGPM brought the number of base units to seven by adding the mole as the base unit for amount of substance, quantity symbol n, dimension symbol N. The mole will also be redefined in the new SI, following a campaign to measure the value of the Avogadro constant to the highest possible accuracy [12]. After consultation with its stakeholders, IUPAC [11] has proposed a new definition of the mole to be submitted for consideration by the Consultative Committee on Amount of Substance (CCQM) [13].

Present definition: mole, unit symbol mol, is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12.

In addition, “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.”

Proposed definition [13]: The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly 6.022 140 76×10²³ elementary entities. This number is the fixed numerical value of the Avogadro constant, N_A, when expressed in mol⁻¹, and is called the Avogadro number.

The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles.

The stipulated Avogadro number is the fixed numerical value of the Avogadro constant, N_A=6.022 140 76×10²³ mol⁻¹, which is provided by the CODATA Task Group on Fundamental Constants [14].

3.1.1 Introduction

The measurement of mass is a central point of the quantification of material substances. A balance measures mass by sensing the weight force that presses an object down on the balance pan. Weight is the force exerted on a body by the gravitational field of the earth, and is measured in the unit force newton, N. The weight force acting on 1 kg mass depends on geographic and cosmic factors. However, for mass measurements using mechanical balances, the weight of the unknown object is equilibrated at the same place and same time as the weight of an object of known mass (i.e. of a standard). For high precision measurements, the buoyancy caused by the surrounding air must be taken into consideration. This correction can easily be calculated if the density of the known and unknown mass and that of the air is known. See 3.2.2.

3.1.2 Types of balances

Balances are available with different capacities and sensitivities. The standard balance for many operations in analytical chemistry is the Analytical Balance. Electronic analytical balances can be purchased with different weighing ranges and readabilities. A standard analytical balance typically has a maximum capacity of 300 g and a readability of 0.1 mg. Semi-microbalances (capacity up to 200 g and readability 0.01 mg), microbalances (capacity up to 50 g, readability 1 μg,), and ultra-microbalances (capacity up to 20 g, readability 0.1 μg) are commercially available. Quartz crystal microbalances with 100 μg range that can detect 1 ng changes are available. These balances utilise a thin quartz crystal disk oscillating at, for example, 10 MHz. The frequency of oscillation changes with any change in mass, and this reading is converted into mass units. See Section 3.3.

Table 3.1-1 shows a classification of the types of balances used in chemistry and medicine [15] that are based on the OIML standard [16]. The typical maximum capacity is taken from manufacturers’ descriptions of the ranges of balances.

Electronic balances and their accompanying weight sets are calibrated on the basis of conventional mass. The conventions are: the air density is 1.2 kg/m³ and the density of weights pieces is 8000 kg/m³ [17, 18]. Details for performing calibrations are given below.

The calibration certificate and the manufacturer’s recommendations on measurement uncertainty estimates (instrumental measurement uncertainty [VIM 4.24]) take into account three contributions: measurement repeatability [VIM 2.21], resolution of a displaying device [VIM 4.15] (also called readability) in relation to the balance scale, and any datum measurement error [VIM 4.27]. (See Section 7.2 and [19]).

3.2.1 Standard weights for calibration

Standards agencies specify tolerances for standard masses. The International Organization of Legal Metrology Recommendation OIML R-111 applies to weights with nominal values of mass from 1 mg to 5000 kg in the E1, E2, F1, F2, M1, M1–2, M2, M2–3, and M3 accuracy classes [18]. The class designation of a weight or weight set meets metrological requirements intended to maintain the mass values within specified limits. The error in a weight used for the verification of a weighing instrument shall not exceed one third of the maximum permissible error for an instrument [16].

Tolerance limits [20] (maximum permissible measurement error [VIM 4.26]), T, for weights below 10 g of nominal mass W g are calculated from the equation:

where the unit of T is mg. Commercially available weights for calibration have masses metrologically traceable [21] to national standards and ultimately the SI.

3.2.2 Mass in air and a vacuum

Determination of mass on a common laboratory balance gives the mass in air. The object displaces its volume in air, being buoyed up by the mass of air displaced, according to Archimedes’ principle. The density of air is taken as 0.0012 g cm⁻³. If the density of the object being weighed and the density of the balance weights are the same, they will be buoyed the same amount, and the recorded mass will be that in a vacuum. The density of a typical weight is about 8 kg m⁻³, or 8 g cm⁻³. If the densities are markedly different, the differences in buoyancy will lead to a small error in the recorded mass. Examples are the weighing of very dense objects (e.g. platinum, 21.4 g cm⁻³ or mercury, 13.6 g cm⁻³) or light objects (e.g. water, 1.0 g cm⁻³). For very careful work, a correction should be made for this error. An example is found in the calibration of glassware by measuring the mass of water or mercury delivered or contained by the glassware (see below 3.2.3).

The mass of an object in air (m_air) can be corrected to its mass in vacuum (m_vac) by

where ρ_o=density of object, ρ_w=density of standard weights, and ρ_air=the density of air (taken as 1.225 kg m⁻³ or 0.001225 g cm⁻³). The density of steel weights is 7.8 g cm³ (kg m⁻³).

In many analytical measurements, a correction is not necessary, because buoyancy errors will cancel out in percent composition calculations. Also, the corrected value does not differ significantly from the non-corrected value for typical uses.

The concept of conventional mass aims to simplify the determination of the mass of weights under conditions in ambient air. It implies that conventional mass does not differ from the physical quantity mass by more than certain acceptable values when determined under specified limits of air density and the material densities of the weights. See [17, 22, 23, 24].

3.2.3 Volumetric glassware tolerances and calibration

EN ISO in Europe, and ASTM and USP (United States Pharmacopeia) in the USA publish tolerances for different volumetric glassware. The highest standard is Class A, with USP usually having the most exacting standards. These are sufficiently accurate for most quantitative measurements. Table 3.2-1 lists tolerances for Class A volumetric glassware.

For more accurate measurements, glassware that has been certified by standards agencies may be purchased. Alternatively, one can calibrate glassware to an accuracy generally as good as the standards specifications, correcting for weight in vacuum, glassware expansion or contraction, and water expansion or contraction. Table 3.2-2 lists the calculated volumes for one gram of water in air at atmospheric pressure at sea level for different temperatures, corrected for buoyancy with stainless steel weights of density 7.8 kg m⁻³. The unit of 7.8 g cm⁻³ is used in the formulas. The glass volumes are also calculated for the standard temperature of 20 °C, with small adjustments for borosilicate glass expansion or contraction with temperature changes. See also [25, 26].

Table 3.2-2:

Spreadsheet calculation for calibration of glassware volume (See Supplementary Information for a copy of the spreadsheet).

This spreadsheet (Table 3.2-2) may be reproduced to perform calculations. Alternatively, see Supplementary Information for a copy of the spreadsheet. To run, enter the weight of water at temperature T to calculate volume at temperature T and at 20 °C.

Tolerance intervals for other types of volumetric measuring apparatus, such as displacement pipettes in the microlitre range, are available from manufacturers. They may be calibrated gravimetrically, as with glassware, or by delivering and measuring a reagent whose concentration is accurately known.

3.4.1 conventional mass, m_c

Mass of a body that is balanced by a standard mass under conventionally chosen conditions.

• Note 1: The Organization of Legal Metrology recommends that the conventionally chosen conditions are: reference temperature, t_ref =20 °C, density of air as a reference value, ρ₀=1.2 kg m⁻³; reference density of standard weight ρ_c=8000 kg m⁻³.

• Note 2: The unit of the kind of quantity “conventional mass” is the kilogram.

Source: Adapted from [17].

4.1.1 Solutions prepared by dilution

Solutions are often prepared by diluting a known volume, V_conc, of concentrated solution, C_conc, by the addition of solvent up to a higher volume, V_dil. Since the amount of solute is the same before and after dilution,

This is only valid for initial concentrated solutions prepared by volume.

4.1.2 Preparation of standard solutions

Standard solutions are often prepared by dissolving an accurately measured mass of a solute of certified purity in a known volume of solvent. Such a solution might fulfil the requirements of a primary measurement standard [VIM 5.4], namely that its property value is established by a method that requires no measurement standard [VIM 5.1] for a quantity [VIM 1.1] of the same kind [VIM 1.2] (known as a primary reference measurement procedure [VIM 2.8]). If the concentration of an intended standard solution is obtained by measurement, for example by titration with a standard solution, it is known as a secondary standard [VIM 5.5].

In addition to certified purity features of a material chosen to prepare a standard solution, features also include:

1. Stability in presence of air

2. Absence of any water of hydration which might vary with changing humidity and temperature.

3. Readily soluble to produce stable solutions in solvent of choice

4. A larger rather than smaller molar mass

5. Ready availability

These conditions are met by few materials. Anhydrous sodium carbonate, silver nitrate, and potassium hydrogen phthalate are a few materials that meet these conditions. Certified reference materials (CRM) [VIM 5.14] for preparation of standard solutions are available from providers including the National Institute of Standards and Technology (NIST) and other National Measurement Institutes. For example, NIST SRM 84L, potassium hydrogen phthalate, is a CRM sold as an acidimetric primary standard. The certified property is “mass fraction of total acid (replaceable H⁺) expressed as KHP” and is (0.999 934±0.000 076) g g⁻¹, where 0.000 076 g g⁻¹ is a GUM expanded uncertainty for a level of confidence of approximately 95 % [31].

Some examples of primary standards:

1. Arsenic trioxide for making sodium arsenite solution for standardisation of sodium periodate solution

2. Benzoic acid for standardisation of waterless basic solutions: ethanolic sodium and potassium hydroxide, tetrabutyl ammonium hydroxide (TBAH), and alkali methanolates in methanol, isopropanol, or dimethyl formamide (DMF)

3. Potassium bromate (KBrO₃) for standardisation of sodium thiosulfate solutions

4. Potassium hydrogen phthalate for standardisation of aqueous bases and perchloric acid in acetic acid solutions

5. Sodium carbonate for standardisation of aqueous acids: hydrochloric, sulfuric, and nitric acid solutions (but not acetic acid solutions)

6. Sodium chloride for standardisation of silver nitrate solutions

7. Sulfanilic acid for standardisation of sodium nitrite solutions

8. Zinc powder, after being dissolved in sulfuric or hydrochloric acid, for standardisation of Na₂H₂edta solutions

9. Disodium oxalate for standardising potassium permanganate.

A solid standard is dissolved in a solvent (usually water) and the concentration may be expressed as a mass concentration (mass/volume) or amount concentration (amount/volume). Typical volume units are L, dm³, m³.

Two widely used standardised solutions are those of hydrochloric acid, HCl, and potassium hydrogen phthalate, KHC₈H₄O₄. Three others which have some instability, but with appropriate precautions have proven to be excellent standard solutions, are sodium thiosulfate, Na₂S₂O₃ (light sensitivity, susceptible to bacterial oxidation), silver nitrate, AgNO₃ (light sensitivity), and potassium permanganate, KMnO₄ (water oxidation catalysed by light, heat, Mn²⁺, and MnO₂).

For a standard solution prepared by dissolving a known mass m_A of component A (molar mass M_A) and known purity P in a known volume V_sol, the amount concentration of A in the solution c_A is calculated by

See section 7.4 for a discussion of uncertainties in preparing standard solutions.

4.1.3 Stoichiometry and equivalence in titrations

Chemical reactions in solution take place between different chemical species according to proportions defined by well-established equations (known as stoichiometry [32]); for example:

a is the number of entities of species A (the analyte) of molar mass M_A in one solution (the sample solution) and the b is the number of entities of a species B of molar mass M_B contained in a second solution which reacts with the first solution according to equation (8). Stoichiometric numbers (v) [32] of reacting entities are the coefficients of equation (8) for products and the negatives of the coefficients for reactants [33], v_A=−a and v_B=−b reacting in the theoretically predicted v_B/v_A proportion.

The concept of ‘equivalence’ between the amounts of reacting substances plays an important role in quantitative chemistry; v_A is equivalent to v_B when the masses v_A×M_A and v_B×M_B of the two species are equal. When volumes V_A and V_B of these solutions of concentrations c_A and c_B are mixed, the reaction is stoichiometric when v_A×c_A×V_A=v_B×c_B×V_B. Historically, v×c was known as the ‘normality’ of the solution designating the number of ‘gram equivalents’ per litre. A one-normal solution contained one equivalent per litre, and was written ‘1 N’. This use is discontinued and deprecated [34].

4.2.1 equivalent (in volumetric and gravimetric analysis)

Numerical value of the amount of substance providing one mole of a specified reacting species.

• Example 1: 1 mol of sulfuric acid (H₂SO₄) has 2 mol of H⁺ to be titrated with a strong base. Therefore, one equivalent of H⁺ is provided by 0.5 mol of sulfuric acid.

• Example 2: 1 mol of potassium permanganate (KMnO₄) has 3 equivalents of transferable electrons when reduced to MnO₂ in alkaline solution and 5 equivalents of transferable electrons when reduced to Mn²⁺ in acid solution.

Source: [25] section 6.2.

4.2.2 solution

Liquid or solid phase containing more than one substance, when for convenience one (or more) substance, which is called the solvent, is treated differently from the other substances, which are called solutes.

• Note 1: When, as is often but not necessarily the case, the sum of the amount fractions of solutes is small compared with unity, the solution is called a dilute solution.

• Note 2: A superscript ∞ attached to the quantity symbol for a property of a solution denotes the property in the limit of infinite dilution.

Source: [28] with minor change

4.2.3 standard solution

Solution of accurately known concentration, prepared using a reference measurement standard [VIM 5.6].

Source: [25] with minor change.

4.2.4 standardization

Measurement of the concentration of a component of a solution by titration with a measurement standard for the preparation of a secondary measurement standard [VIM 5.5]

4.2.5 stock solution

Solution prepared by weighing an appropriate portion of a solid or by measuring out an appropriate volume of a liquid and dissolving it in the weighed mass of solvent or by adding solvent to a given volume.

• Note: A stock solution is usually of a greater concentration than is needed for the chemical purpose and is diluted to give the required concentration before use.

Reference: [26] Chapter 2.

5.1.1 Introduction

Historically, gravimetry has formed the basis of analytical chemistry and remains an attractive method when it can be used, because it is considered a primary measurement procedure and is highly accurate, with relative uncertainties of about 0.05 % even for routine measurements. In fact, gravimetric analysis was used to determine the atomic masses of many elements to six-figure accuracy.

5.1.2 Gravimetry by direct weighing of the analyte

In a simple example, gas mixtures are prepared by successively adding each component of a mixture to a gas cylinder, which is weighed first empty and then after each addition, The concentration of each component is usually expressed as a mass fraction or mole fraction [37].

Total suspended solids (TSS), the mass of filterable solids in 1 L of a water sample, is measured gravimetrically. A known volume of water is filtered, and the collected solids are weighed.

5.1.3 Gravimetry of precipitates

Analysis by precipitation of a highly insoluble solid is typified by the measurement of chloride concentration by the addition of silver nitrate, precipitating silver chloride that is filtered, washed, dried, and weighed. Knowledge of the stoichiometry of the reaction (Cl⁻+Ag⁺→AgCl (s)) allows for a calculation of the amount of chloride precipitated and therefore the chloride content of the test solution.

Steps for gravimetric analysis after precipitation:

1. Prepare the solution. Separate or mask interfering materials. Adjust conditions for low solubility. Adjust pH.

2. Perform the precipitation. In order to minimise supersaturation and obtain larger crystals, precipitate from dilute solution, add the precipitating agent slowly, with stirring, to a hot solution, and then cool. Avoiding local excess of precipitating agent keeps Q, the concentrations of mixed reagents before precipitation, low, and heating keeps S, the equilibrium solubility of the precipitate, high. See von Wiemarn Ratio.

3. Digest the precipitate. This process, also called “Ostwald ripening”, improves the purity and crystallinity of the precipitate by allowing it to stand in contact with the mother liquor for a period of time. Fine particles tend to dissolve and re-precipitate on larger ones.

4. Filter and wash the precipitate. The wash solution should contain a volatile electrolyte to avoid peptisation, the formation of colloids. The filter is chosen to trap the precipitate; smaller particles are more difficult to filter. Depending on the procedure followed, the filter might be a piece of ashless filter paper in a fluted funnel, or it might be a filter crucible. Filter paper is convenient because it does not typically require cleaning before use; however, filter paper can be chemically attacked by some solutions (such as concentrated acid or base), and may tear during the filtration of large volumes of solution. Following filtration, the filter paper is charred off, leaving the precipitate.

5. Dry or ignite the precipitate. A precipitate may be ignited to a weighable form, for example, hydrous ferric oxide, Fe₂O₃·xH₂O is ignited to Fe₂O₃.

6. Cool and weigh the precipitate. An ignited precipitate is cooled in a desiccator to avoid adsorption of atmospheric water.

7. Calculate the amount of analyte from the mass of precipitate and the gravimetric factor.

5.1.4 Gravimetry of vapours

It may also be possible to convert the analyte into a substance in a vapour form that can be collected and measured directly, or the sample weighed before and after the reaction; the difference between the two masses gives the mass of analyte lost. For example, water in a solid can be vaporised. The loss in mass can be measured. The water vapour may also be collected on an adsorbent, which is weighed. The analytical precipitate may be decomposed with loss of a gas. Ammonium ions may be determined by precipitating with H₂PtCl₆ as (NH)₂PtCl₆ and igniting the precipitate to platinum metal: (NH)₂PtCl₆→Pt+2NH₄Cl (g)+2Cl₂ (g).

5.1.5 Thermogravimetric Analysis

Thermogravimetric Analysis (TGA) or thermogravimety (TG) involves continuously measuring the mass of a sample as a function of its temperature in which changes in physical and chemical properties of materials are determined. The TGA instrument continuously weighs a sample as it is heated to temperatures of up to 2000 °C and plots the mass as a function of temperature (a thermogram).

Suitable samples for TGA are solids which either lose a volatile species (mass loss) or react with a gaseous species (mass increase). An example is the dehydration of copper sulfate pentahydrate. There are two points of mass loss, one due to the loss of four water molecules and the second to the loss of one. Such plots can be used to qualitatively identify a compound or to quantify it. The composition of a novel compound may be elucidated.

Differential thermal analysis (DTA), in which the difference in temperature between a sample and a reference is measured as they are heated, is more widely applicable than TGA, because it is not limited to reactions involving a change in mass.

A comprehensive terminology of thermal-based methods has been published by the International Confederation for Thermal Analysis and Calorimetry and IUPAC [38].

5.2.1 Co-precipitation

Simultaneous precipitation of a normally soluble component with a macro-component from the same solution by the formation of mixed crystals by adsorption, occlusion, or mechanical entrapment.

Source [39]

• Note: In gravimetry involving precipitation, co-precipitation of an impurity is usually not desired. Proper washing may remove adsorbed impurities.

5.2.2 electrogravimetry

Gravimetry in which the material to be weighed is obtained by electrochemical reaction.

• Note 1: A typical measurement involves the electrodeposition of a metal from its ions in solution.

• Note 2: Measurement of current using a silver coulometer is an example of electrogravimetry.

5.2.3 gravimetric analysis

gravimetry

Methods of analysis based on the measurement of mass.

• Note 1: Gravimetry may be operated as a primary reference measurement procedure [VIM 2.8], as the mass standard that calibrates a balance is not a mass of analyte.

• Note 2: The analyte to be determined is separated from the sample in a weighable form (e.g. by precipitation), and its mass or amount of substance is calculated from the mass of the weighed compound whose stoichiometric composition must be exactly known.

Source: Adapted from [40]

5.2.4 gravimetric factor, g_F

In gravimetry, mass of analyte per unit mass of precipitate.

g F = M A M P × v A v P , where M_A, M_P are molar masses of analyte and precipitate, respectively, in g mol⁻¹ and v_A and v_P are stoichiometric coefficients in the precipitation reaction.

• Note 1: Historically, the gravimetric factor is termed GF, but to conform to the IUPAC convention [33] it is recommended that quantities should have a single symbol, g_F.

• Note 2: The gravimetric factor is used to calculate the mass fraction of an analyte in a sample by w=mPmsample×gF

• Example 1: Sulfur trioxide (M_SO3=80.0640 g mol⁻¹) is precipitated as barium sulfate (M_BaSO4=233.390 g mol⁻¹), SO₃→BaSO₄. Therefore, gF=80.063233.390×11=0.343044.

• Example 2: Silver oxide (M_Ag2O=231.736 g mol⁻¹) is dissolved and precipitated as silver chloride (M_AgCl= 143.321 g mol⁻¹), Ag₂O→2 AgCl. Therefore, gF=231.736143.321×12=0.808451.

Reference for molar masses [41]

5.2.5 homogeneous precipitation

Precipitation in gravimetry in which the precipitating agent is produced from a homogeneously dissolved precursor to minimise local excesses and supersaturation.

Source: [42]

5.2.6 nucleation

Initial formation of small crystals of precipitate, from which larger crystals will grow.

• Note: In gravimetric analysis involving precipitation, supersaturation should be minimised to avoid large numbers of nuclei and small crystals. See von Wiemarn Ratio.

Source: Adapted from [43]

5.2.7 precipitation (in chemistry)

Sedimentation of a solid material (precipitate) from a liquid solution in which the material is present in amounts greater than its solubility in the liquid.

Source: [37] p 2207 with minor change.

5.2.8 solubility, s

Equilibrium concentration of a component of a saturated solution at a given temperature.

• Note: The units of solubility are mol m⁻³, but may also be expressed in any units corresponding to quantities that denote relative composition, such as mass fraction, amount fraction, molality, volume fraction, etc.

Source: [25] p84.

5.2.9 Super-saturation

State of an unstable system which has a greater concentration of a material in solution than would exist at equilibrium.

Source: [37] with minor change.

5.2.10 thermogravimetric analysis (TGA)

thermogravimetry (TG)

Technique that monitors the mass of the sample as a function of time or temperature while the temperature of the sample, in a specified atmosphere, is heated or cooled in a controlled manner.

• Note 1: It is common to use the same abbreviation (TGA) for both the technique and a thermogravimetric analyser.

• Note 2: Very often, TG is used in combination with DTA, Fourier transform-infrared spectroscopy (FT-IR), gas chromatography (GC), or mass spectroscopy (MS) in so-called hyphenated techniques

Source: [44]

5.2.11 Von Wiemarn ratio

relative supersaturation

Ratio of the difference in concentration of mixed reagents before precipitation (Q) and the concentration of the precipitate at equilibrium (S), and the concentration of the precipitate at equilibrium.

Q − S S

• Note: The particle size of a precipitate is inversely proportional to the relative supersaturation, so Q should be kept low and S high during gravimetric analysis by precipitation (see section 5.1.3).

Source: Adapted from [45] p420.

6.1.1 Kinds of titrations

Titrimetric analysis covers a quite large group of methods with a long tradition in quantitative analysis. Owing to their advantageous characteristics, they are still used in laboratories as definitive methods, especially when they can be used with instrumental end-point detection. These are the titration methods with electrochemical reagent generation (coulometric titration) and/or electrochemical detection (potentiometric, amperometric, conductometric), thermoanalytical, optical, or radiochemical detections. See the corresponding sections of [3, 25].

Terms for varieties of titration methods can reflect the nature of the reaction between analyte and reagent. Thus, there are acid-base, precipitation, complexometric, and oxidation-reduction (also redox) titrations (or titrimetries).

Alternatively, the term can reflect the nature of the titrant: acidimetric titration (acidimetry), alkalimetric titration (alkalimetry), and iodometric titrations (iodometry), or, additionally, coulometric titrations, in which the titrant is generated by electrochemical reaction, rather than being added as a standard solution.

6.1.2 Requirements for a titration

There are a number of desirable characteristics of all titrations that are listed below.

1. The reaction should be of known stoichiometry

2. The reaction should be rapid.

3. The reaction should be specific. Interfering substances must be removed or masked. For example, in acid-base titrations, carbon dioxide in air can be prevented from dissolving in the base by performing the titration under argon. Oxygen in air should also be excluded from some redox titrations.

4. There should be a marked change in some property of the solution when the reaction is complete, to mark the end-point, the observed completion of the reaction. The end-point should coincide with the equivalence-point, the point at which a stoichiometric amount of titrant is added.

5. The reaction should essentially go to completion, i.e. the equilibrium of the reaction ‘should be shifted far to the right, towards the products, to obtain a sharp end-point.

6.1.3 Direct and back titration

When the term titration is used without qualification, it usually indicates a direct titration.

A back titration is generally a two-stage analytical technique:

1. Analyte A of unknown concentration is reacted with excess reagent B of known concentration.

2. A direct titration is then performed to determine the amount of reagent B remaining (i.e. in excess).

Back titrations are used when:

1. one of the reactants is volatile, for example ammonia.

2. an acid or a base is an insoluble salt, for example calcium carbonate

3. a particular reaction is too slow

4. direct titration would involve a weak acid – weak base titration

5. the end-point of the direct titration is difficult to observe

The titrant, containing the reacting species with which a titration is made, is either a prepared standard solution, or it has been standardised, i.e. the concentration of the active agent is either known or measured by titration with a standard solution of accurately known concentration.

Provision must be made for some means of recognising (indicating) the point at which essentially all of A has reacted with B, the equivalence-point or stoichiometric or theoretical end-point. In order to recognise the end-point of the titrations, indicators can be used. These may be visual or instrumental. In the former, the indicator is a substance which participates in the titration reaction so as to give a visual change (colour, fluorescence, precipitate, or turbidity) at or near the equivalence-point of a titration.

6.1.4 Visual indicators

6.1.4.5 Adsorption and precipitation indicators

An adsorption indicator is a dye in ionised form (usually anionic) that is adsorbed on the precipitate near the equivalence-point as a counter ion to adsorbed titrant primary ion. The adsorbed dye changes color. For example, fluorescein is adsorbed on the surface of AgCl at the first excess of Ag⁺ titrant. A precipitation indicator forms a colored precipitate with the titrant at or near the equivalence-point. An example is the Mohr method for titrating Ag⁺ with Cl⁻ using K₂CrO₄ indicator. Red Ag₂CrO₄ forms near the equivalence-point.

6.1.5 Instrumental indicators

6.1.5.1 pH meters and ion selective electrodes

In an acid-base titration there is a rapid change in pH at the equivalence-point, particularly for strong acid – strong base titrations. This is the principle of colour indicators for such reactions. A pH electrode gives a potential which is proportional to pH and so, suitably calibrated, can track the progress of the titration. Differentiation of the pH /titrant volume curve, and then second differentiation, can give an accurate value of the equivalence-point. See Fig. 6.1-1.

Fig. 6.1-1:

Strong base/weak acid titration (top) pH v fractional end-point volume (f); (middle) First differential of pH curve; (bottom) Second differential of pH curve.

Similarly, potentiometric measurements with ion-selective electrodes as indicator electrodes can be used to detect end-points for titration reactions in which the titrant or analyte concentration is sensed by the electrode, with a logarithmic dependence of potential on concentration. A silver ion selective electrode could be used for titrations with silver nitrate.

6.1.5.2 Measurement of potential in redox titrations

The equivalence-point in a redox titration can be identified with a redox indicator or via an electrical measurement, such as the potential of an electrode that can monitor the relative concentrations of redox active species against a reference electrode. For example, Fe(II) titrated by Ce(IV) has equilibrium amounts of Fe(II)/Fe(III) before the equivalence-point, at which essentially all Fe(II) is converted to Fe(III) and then to Ce(IV)/Ce(III) after the equivalence-point. The potential of a platinum wire in the titration solution follows Fe(II)/Fe(III) according to the Nernst Equation

(11) E = const + R T F ln ( [ Fe III ] [ Fe II ] )

And then Ce(IV)/Ce(III)

(12) E = const + R T F ln ( [ Ce IV ] [ Ce III ] )

A large difference in the standard electrode potentials of these redox couples (E⁰(Fe^III/II)=+0.68 V; E⁰(Ce^IV/III)=+1.44 V) leads to a clear and sharp change in potential at the equivalence-point. See Fig. 6.1-2.

Fig. 6.1-2:

Redox titration of Fe(II) by Ce(IV) using a platinum indicator electrode.

Some titrations may not exhibit an inflection point at the equivalence-point, for example when the redox reaction is unsymmetrical. An example is the titration of Fe(II) with potassium permanganate:

5 Fe 2 + + MnO 4 − + 8 H + = 5 Fe 3 + + Mn 2 + + 4 H 2 O

An unsymmetrically-shaped titration curve is produced, with the equivalence-point near the top of the potentiometric break.

6.2.1 Measurement of alkalinity of natural waters

The alkalinity of water is its acid-neutralizing capacity, and is considered the sum of all titratable bases. Two titrations with a strong acid are made to pH 4.5 (methyl orange, or methyl red end-point) and to pH 8.3 (phenolphthalein end-point). For many surface waters, alkalinity values are primarily a function of carbonate, hydrogen carbonate, and hydroxide content. The measured values may also include contributions from borates, phosphates, silicates, or other bases, if these are present. Phenolphthalein end-point alkalinity, A_P, attributed to all the hydroxyl and carbonate, AP=[CO32−]+[OH−], is also known as composite alkalinity. Methyl red (methyl orange) end-point alkalinity,AT=[HCO3−]+2[CO32−]+[OH−], is also known as total alkalinity [51]. Alkalinity is often expressed as an equivalent mass concentration of calcium carbonate, mg/L as CaCO₃.

6.2.2 Mohr method for chloride

Chloride concentration is measured by titration by Ag⁺ with a small amount of K₂CrO₄ added to the solution. The end-point is detected by the formation of a reddish-brown precipitate of Ag₂CrO₄. The pH should be between 6 and 10. Below pH 6, chromate is present as HCrO₄⁻ instead of CrO₄²⁻ and the Ag₂CrO₄ end-point is delayed. Silver hydroxide is precipitated above pH 10. For titrations at pH≤6, a back titration is recommended (Volhard method) for the excess Ag⁺. The second titrant is KSCN and the indicator is Fe³⁺. The end-point is given by the reddish-coloured Fe(SCN)²⁺ complex.

6.2.3 Measurement of water hardness

Water hardness, the concentration of titratable calcium and magnesium, is measured by complexiometric titration using the blue dye Eriochrome Black T (ErioT) as the indicator. When added to water, the indicator reacts with Ca²⁺ and Mg²⁺, exhibiting a wine-red colour. Upon reaction with the titrant EDTA (ethylenediaminetetraacetic acid) the colour becomes blue. Water hardness is expressed as an amount concentration of calcium and magnesium or an equivalent mass concentration of calcium carbonate or calcium oxide.

6.2.4 Karl Fischer titration of water

Traces of water may be determined by the Karl Fischer titration, using iodine as a titrant. In the classical Karl Fischer titration, the sample is dissolved in anhydrous methanol and is titrated with the Karl Fischer reagent, which contains iodine, sulfur dioxide, and pyridine dissolved in methanol. The iodine and sulfur dioxide exist as addition compounds with pyridine (C₅H₅N) and an overall reaction can be written:

C 5 H 5 N ⋅ I 2 + C 5 H 5 N ⋅ SO 2 + H 2 O + CH 3 OH + C 5 H 5 N → 2 C 5 H 5 NH + I − + C 5 H 5 NH + CH 3 OSO 3 −

Thus, each molecule of iodine is equivalent to one molecule of water.

The end-point corresponds to the first appearance of iodine. This can be detected visually by the change of the pale yellow of the reaction mixture to a permanent reddish-brown tinge with free iodine, or a few drops of methylene blue in methanol. More often, an electrometric (amperometric) detection of the iodine is utilised. Commercial coulometric titration instruments are available in which the iodine is generated coulometrically at a constant current and Coulomb’s law is used to calculate the quantity of iodine generated.

The Karl Fischer titration can be used for the direct determination of water in a wide variety of organic substances, including alcohols, unsaturated hydrocarbons, acids and acid anyhydrides, esters, ethers, amines, sulfides, and ntiroso and nitro compounds.

6.3.1 adsorption indicator

Visual indicator adsorbed or desorbed, with concomitant colour change, at the end-point of the titration.

• Example: The yellow dye fluorescein added to chloride titrations, which adsorbs on the silver chloride precipitate at the end-point, giving it a pink colour.

Source: [25] p 48 (with minor change).

6.3.2 alkalinity, A

Measure of the capacity of aqueous media to react with hydrogen ions.

• Note 1: Alkalinity is used to assess the buffering capacity of natural waters and is measured by titration using methyl red (pH 4.5, total alkalinity) or phenolphthalein (pH 8.3, composite alkalinity).

• Note 2: Alkalinity is often expressed as an equivalent mass concentration of calcium carbonate, for example alkalinity=3.2 mg L⁻¹ as CaCO₃.

Source: [51] (with minor change).

6.3.3 back titration

Titration of remaining reagent after addition of excess reagent to the analyte.

Source: Adapted from [52] and [25]

6.3.4 colour indicator

Visual indicator that changes colour at the end-point of a titration.

• Note: A one-colour indicator changes between colourless and a colour (e.g. phenolphthalein and many complexometric indicators). A two-colour indicator changes between two colours (e.g. litmus, red-acidic: blue-alkaline).

Source: Adapted from [25]

6.3.5 composite alkalinity, A_P

Alkalinity measured by titration with phenolphthalein as the visual indicator.

• Note: The pH of the end-point is about 10.0 and therefore AP=[CO32−]+[OH−].

Source: [51] (with minor change).

6.3.6 coulometric titration

Titration in which the reactant is generated by an electrochemical reaction and the time of titration at a fixed electric current is measured.

• Note 1: For a redox reaction A+ne→B where species B is the reactant in the titration, the amount of B generated by passage of current I A for time t s is It/nF mol, where F is the Faraday constant.

• Note 2: Coulometric titration fulfils the criterion for a primary measurement procedure, as it does not require a standard of the quantity being measured.

Source: Adapted from [25] p 47.

6.3.7 direct titration

See titration.

6.3.8 end-point

Indication [VIM 4.1] of the equivalence-point of a titration.

• Note 1: The conditions of the titration should be chosen so that the end-point is as close as possible to the equivalence-point.

• Note 2: The difference between the end-point and equivalence-point is the titration error.

Source: Adapted from [52].

6.3.9 end-point error

Difference in the amount of titrant, or the corresponding difference in the amount of substance being titrated, between the end-point value and equivalence-point value.

• Note: End-point error is usually expressed as a volume of titrant solution.

6.3.10 equivalence-point

Stage of a titration at which the titrant has completely reacted with the receiving solution according to the stoichiometry of the reaction.

Source: Adapted from [25] p 47.

6.3.11 visual indicator

indicator

Substance which interacts with species in a titration, giving a visual change at the end-point.

• Note 1: Visual indicators may be described by the nature of the indication (colour indicator, one-colour indicator, two-colour indicator, adsorption indicator) or by the nature of the reaction (acid-base indicator, redox indicator).

• Note 2: If the indicator takes part in the reaction, for example an acid base indicator, it must be added in sufficiently small amount to avoid disrupting the equivalence-point through its interaction with the reacting species.

• Note 3: When there is no ambiguity, the term ‘indicator’ may be used for visual indicator.

• Example: phenolphthalein changes from colourless to pink in the pH range 8.2–10.0 and so may be used as an indicator for a strong base-weak acid titration.

6.3.12 indicator consumption error

Systematic error arising from the reaction of a visual indicator with a reactant in a titration.

• Note 1: The magnitude and sign of this error depends on the amount of indicator used and the nature of the interaction between the indicator and the analyte.

• Note 2: A significant compensation of this error usually takes place because the standardization of the titrant is carried out in similar conditions to the analysis titration.

6.3.13 mixed indicator

Visual indicator containing a supplementary dye selected to heighten the overall colour change at the end-point of a titration.

• Example: A solution of bromocresol green (1 g/L) and methyl red (0.2 g/L) gives a sharper colour change for acid titrations of the weak base morpholine.

6.3.14 precipitation indicator

Visual indicator precipitated, with concomitant colour change, by reaction with excess titrant at the end-point of a titration.

• Example: Mohr method for chloride analysis using chromate. (See 6.2.2)

6.3.15 redox indicator

Visual indicator that reacts with excess titrant or analyte with a colour change at the end-point of a titration.

• Example: Titration of iodine by thiosulfate uses a starch solution to produce a strong blue colour. At the end-point, when all iodine is reacted, the solution becomes colourless.

See 6.1.4.4.

6.3.16 redox titration

Titration depending on the oxidation or reduction of the analyte by the titrant.

• Note: The end-point may be observed using a redox indicator.

• Examples: Titrations using iodine as an oxidising agent or as the product of oxidation of iodide are known as iodometry.

6.3.17 titrand

Reactant solution being titrated.

Source: Adapted from [25] p 47.

6.3.18 titrant

Reactant solution added in titration.

Source: Adapted from [25] p 47.

6.3.19 titration

Addition of a reactant A (known as the titrant) in measured increments to a solution containing a second reactant B with provision for some means of recognizing (indicating) the end-point at which essentially all of B has reacted. The stoichiometry of the reaction and the amount of one reactant (A or B) is known, allowing the calculation of the amount of the other reactant.

• Note 1: The titrant can be added by volume (volumetry) or by mass.

• Note 2: If the titrant reacts directly with the analyte, the process is known as a direct titration. In a back titration a known, excess amount of reactant is added, followed by titration of the remaining reactant.

• Note 3: If the volume of the solution containing the analyte is known, the concentration of the analyte solution may be calculated.

• Note 4: The term ‘titre’, meaning a measure of the concentration of the solution of known concentration or the titration volume, is considered obsolete, and its further use deprecated.

Source: Adapted from [25] p 47.

6.3.20 titration error

indicator error

Sum of end-point error and indicator consumption error.

6.3.21 titrimetric analysis

titrimetry

Methods of analysis employing titration.

• Note: The titrant added is usually measured by volume (volumetry) but it can be weighed.

Source: Adapted from [25] p 47.

6.3.22 total alkalinity, A_T

Alkalinity measured by titration with methyl red as the visual indicator.

• Note: The pH at the end-point is about 4.8 and therefore AT=[HCO3−]+2[CO32−]+[OH−].

Source: [51] (with minor change).

6.3.23 transition potential (of a redox indicator)

Potential at the end-point of a redox indicator.

• Note: The transition potential is a function of the formal redox potential, the total concentration of the indicator (especially for one colour indicators), the depth of the colour layer, the minimal observable absorbance change (which depends on wavelength and eye sensitivity), and the absorption coefficient.

6.3.24 universal indicator

Visual indicator for acid-base titrations composed of a solution of several pH indicators (main components are thymol blue, methyl red, bromothymol blue, and phenolphthalein) that changes colour through a range of pH values, usually from 1 to 14.

• Note: Universal indicators are commercially available in the form of a solution or in paper strips accompanied by a colour matching chart.

Reference: [53]

6.3.25 volumetry

Titration when the titrant is added by volume.

• Note: Titrant is usually added from a calibrated burette.

Source: Adapted from [37].

7.2.1 Uncertainty components

For tared weighing, uncertainty sources are i) repeatability, ii) readability (digital resolution) of the balance scale, and iii) the contribution due to the uncertainty in the calibration function of the scale. The last has two components, the sensitivity of the balance and its linearity. Of these, sensitivity (the slope of the calibration) gives a negligible contribution to uncertainty, because the mass by difference is done on the same balance over a very narrow range. In this discussion, corrections (and their uncertainties) for the buoyancy of air will be also neglected (see 3.2.2), although even when a buoyancy correction is deemed unnecessary, buoyancy may still add a non-negligible uncertainty [17].

Standard uncertainty due to repeatability is taken as the standard deviation of repeated measurements (u_r=s_r) and is established for a given operator as part of their induction or taken from validation data. That the repeatability still applies may be checked from time to time by duplicate weighing and ensuring that the difference in duplicate measurements is within the repeatability limit (r=2√2 s_r), which is the 95 % interval about zero for the difference [59].

Errors due to linearity are given in the calibration certificate as maximum permissible errors for different ranges of weighing, ±a. This guarantees that weighing a mass m many times will not give a mean result outside m±a, but makes no statement about the probability of values within the range. Therefore, a uniform distribution is assumed and the standard uncertainty is u=a/√3.

The readability of the scale of an electronic balance is±half the smallest digit. The standard uncertainty is this value divided by √3.

The components are combined in quadrature:

The standard uncertainty can be multiplied by a coverage factor to obtain an expanded uncertainty with 95 % coverage probability, but as masses (and volumes) are usually used as part of an ongoing calculation, the combined standard uncertainty is often quoted.

7.2.2 Example

Calculated values in these examples are rounded to two significant figures. It is recommended that, when calculating uncertainties, the manipulation of results be done in a spreadsheet that records full precision. Only when reporting results should rounding be applied.

A four-figure tared balance measuring a sample of 0.5 g has a linearity of ±0.15 mg, and the least significant digit is 0.1 mg. Repeated weighing of a standard mass had a standard deviation of 0.18 mg.

The combined standard uncertainty is

In this example, repeatability represents about 80 % of the total uncertainty. The relative combined uncertainty is 0.0002/0.5=0.0004 or 0.04 %.

7.3.1 Examples

In these examples we estimate the uncertainty of delivering an aqueous solution from a Grade A 100 mL volumetric flask. In the first calculation, uncertainty components from temperature changes and from the calibration error of the flask are included. Fill-and-weigh experiments are then used to measure the volume of the flask and to correct the nominal volume. Thirdly, the temperature is measured and the volume corrected back to 20 °C. Each step improves the uncertainty.

EXAMPLE 1

A 100 mL Grade A volumetric flask is certified to within ±0.1 mL. The standard uncertainty of this calibration, assuming a triangular distribution, is 0.1 mL/√6=0.041 mL. Repeatability standard deviation from ten fill-and-weigh measurements is 0.022 mL and is used directly as a standard uncertainty. If the laboratory temperature is known to vary between 16 and 24 °C, the standard uncertainty of the temperature is 4/√3 °C=2.3 °C. This gives a standard uncertainty in volume of 0. 00021×100×4/√3=0.048 mL.

The three contributions are combined to give the combined standard uncertainty u_c(V) of a single measurement of the volume V=100 mL

The relative combined standard uncertainty is 0.067 mL out of 100 mL, or 0.07 %.

EXAMPLE 2

Suppose we now take the results from the fill-and-weigh experiment to correct the volume of the flask. The mean of the ten fill-and-weigh experiments is 100.03 mL and standard deviation, as before, 0.022 mL. To simplify the calculation, we ignore any uncertainty of the weighing. It is now known that the flask delivers 100.03 mL with standard deviation of the mean of 0.022/√10=0.0070 mL. This new standard uncertainty of 0.007 mL replaces the standard uncertainty calculated from the manufacturer’s maximum permissible error (0.04 mL). Now for a single delivery of 100.03 mL, the uncertainty is

By performing a calibration on a particular piece of glassware we have reduced the uncertainty somewhat. However, uncertainty of temperature is now the major component.

EXAMPLE 3

We now measure the temperature of the laboratory while we use the volumetric flask as 22.3 °C with a standard uncertainty of 0.5 °C. The volume of 100.03 mL at 22.3 °C may now be corrected back to 20 °C (if this is required for comparison and other calculations)

with uncertainty from the temperature of 0.00021×100×0.5=0.011 mL. The new combined standard uncertainty for delivery of a single volume is

Having corrected systematic effects, the remaining uncertainty is dominated by repeatability.

The calculated uncertainties for these three examples are shown in Table 7.3-1.

7.4.1 Examples

Potassium hydrogen phthalate (KHP) is a commonly used standard for acid-base titrations (see 4.1.2). An approximately 0.1 mol L⁻¹ solution is obtained by dissolving 2.0 g in water and making up to 100 mL. The NIST certificate informs that the purity is (99.9934±0.0076) %, where 0.0076 % is the expanded uncertainty with a coverage factor [VIM 2.38], k=2.04. The standard uncertainty of the purity (as a fraction) is therefore 0.000076/2.04=0.000037.

Assume the measured mass is 2.051 g. The calculated concentration is 0.100423 mol L⁻¹. In this example the uncertainties of mass and volume measurements are 0.2 mg and 0.067 mL, respectively, and M_KHP is 204.223 with u=0.004. The combined standard uncertainty is 0.000068 mol L⁻¹. Almost all arises from the uncertainty in volume. A spreadsheet calculation is shown in Table 7.4-1. The standard uncertainties are entered in the column ‘u’, converted to relative uncertainties (u/x) by dividing by the value of the quantity (note volume is in litres), squared and summed (Excel function SUMSQ), and then the square root is taken to give the combined relative uncertainty. This is multiplied by the calculated concentration to give the combined standard uncertainty.

Note that in this calculation the repeatability contributions are contained within the uncertainties for mass and volume.

The concentration should therefore be reported as 0.10004 mol L⁻¹ with standard uncertainty u=0.00007 mol L⁻¹.

As a further example, the EURACHEM guide Quantifying Uncertainty in Analytical Measurement [57] gives the example of estimating the uncertainty in the mass concentration of a cadmium standard (see Appendix 1 of [57]). Here 0.1 g of cadmium is weighed with a relative standard uncertainty of 0.0005, some five times greater than the relative uncertainty in the example of potassium hydrogen phthalate. The combined standard uncertainty now has similar contributions from mass and volume and an overall relative uncertainty of 0.0009.

7.5.1 Example

Using the standard solution of potassium hydrogen phthalate (7.4.1), a sodium hydroxide solution was standardised to be 0.10214 mol L⁻¹ with u=0.00010 mol L⁻¹ (see Example A2 in [57]). Four 10 mL-aliquots of a solution of hydrochloric acid of unknown concentration were titrated with the standard sodium hydroxide solution giving end-point volumes of 10.91 mL, 10.86 mL, 10.88 mL, and 10.89 mL, with an average of 10.885 mL and standard deviation 0.0208 mL.

The concentration of the hydrochloric acid solution is

Taking each term and evaluating the uncertainty:

c _NaOH. The standard uncertainty is given as 0.00010 mol L⁻¹.

V _NaOH. The maximum permissible calibration error for a 25 mL burette is ±0.03 mL, assuming a triangular distribution u=0.03/√6=0.0122 mL. If the temperature range in the laboratory is ±4 °C, the standard uncertainty for the mean titration volume is u=0.00021×4/√3×10.885=0.0053 mL (see also 7.3.1). The repeatability standard uncertainty is calculated from the standard deviation of the four replicate titration volumes by assuming a scaled-and-shifted t-distribution [61]

Note that this replaces the assumption of a Gaussian distribution, which then requires a Student-t value at (n–1) degrees of freedom. Therefore u=0.01803 mL. Combining these uncertainties gives

Titration error for a strong acid – strong base is considered negligible.

V _HCl. The maximum permissible calibration error for a 10.00 mL pipette is ±0.02 mL, assuming a triangular distribution u=0.02/√6=0.0082 mL. By a similar calculation to that of the previous section, the uncertainty arising from temperature is u=0.00021×4/√3×10=0.0049 mL. Random effects are included in u(V_NaOH). Therefore,

The combined standard uncertainty is, therefore,

The relative combined standard uncertainty is 0.33 %, showing titration is an exceptionally accurate analytical method. Figure 7.5-1 shows the contributions made by the components of the titration, where repeatability of the titration is shown separately from the other components of u(V_NaOH).

Fig. 7.5-1:

Uncertainty contributions to standardization of HCl by titrating with standardised NaOH.

7.6.1 combined standard measurement uncertainty

combined standard uncertainty

standard measurement uncertainty that is obtained using the individual standard measurement uncertainties associated with the input quantities in a measurement model

• Note: In case of correlations of input quantities in a measurement model, covariances must also be taken into account when calculating the combined standard measurement uncertainty; see also [58] 9-2.3.4.

Source: [VIM 2.31]

7.6.2 coverage factor

number larger than one by which a combined standard measurement uncertainty is multiplied to obtain an expanded measurement uncertainty

• Note: A coverage factor is usually symbolised k (see also [58] 2.3.6).

Source: [VIM 2.35]

7.6.3 expanded measurement uncertainty

expanded uncertainty

product of a combined standard measurement uncertainty and a coverage factor larger than the number one

• Note: The coverage factor depends upon the type of probability distribution of the output quantity in a measurement model and on the selected coverage probability.

Source: [VIM 2.35]

7.6.4 maximum permissible error

extreme value of measurement error, with respect to a known reference quantity value, permitted by specifications or regulations for a given measurement, measuring instrument, or measuring system.

Source: [VIM 4.26]

7.6.5 measurement uncertainty

uncertainty of measurement

uncertainty

non-negative parameter characterising the dispersion of the quantity values being attributed to a measurand, based on the information used

• Note 1: Measurement uncertainty includes contributions arising from systematic effects, such as contributions associated with corrections and the assigned quantity values of measurement standards, as well as the definitional measurement uncertainty. Sometimes estimated systematic effects are not corrected for but, instead, associated measurement uncertainty contributions are incorporated.

• Note 2: The parameter may be, for example, a standard deviation termed standard measurement uncertainty (or a specified multiple of it), or the half-width of an interval, having a stated coverage probability.

• Note 3: Measurement uncertainty comprises, in general, many contributions. Some of these may be evaluated by Type A evaluation of measurement uncertainty from the statistical distribution of the quantity values from series of measurements and can be characterised by standard deviations. The other contributions, which may be evaluated by Type B evaluation of measurement uncertainty, can also be characterised by standard deviations, evaluated from probability density functions based on experience or other information.

• Note 4: In general, for a given set of information, it is understood that the measurement uncertainty is associated with a stated quantity value attributed to the measurand. A modification of this value results in a modification of the associated uncertainty.

Source: [VIM 2.26]

7.6.6 quality

degree to which a set of inherent characteristics of an object fulfils requirements

• Note 1: The term “quality” can be used with adjectives such as poor, good, or excellent.

• Note 2: “Inherent”, as opposed to “assigned”, means existing in the object.

Source: [62], 3.6.2

7.6.7 standard measurement uncertainty

standard uncertainty of measurement

standard uncertainty

measurement uncertainty expressed as a standard deviation

Source: [VIM 2.30]