Chapter 4 - Basic Requirements for Analysis

8/9/2019 Chapter 4 - Basic Requirements for Analysis

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Nor Anashatul Afini..UiTM N.Sembilan1

CHM 256 :

BASIC ANALYTICALCHEMISTRY


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Chapter 4 2

4.1 Treatment of samples prior to chemicalanalysis4.1.1 Techniques of sampling, drying and weighing of

samples.4.1.2 Dissolution of samples inclusive of both dry and

wet methods.4.1.3 Elimination of interferences.

4.2 Standards4.2.1 Properties of primary and secondary standards.4.2.2 Preparation of standard solutions : primary &

secondary4.2.3 Calculations of required amount of

reagents/standards

4.3 Storage and dilution of stock solution.Preparation of serial dilutions.

CHAPTER OUTLINE


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Population

The group of things, items or units under investigation.

Sample

Obtained by collecting information only about somemembers of a "population.

Sampling

Act of collecting sample to produce meaningfulinformation.

Chapter 4 3


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What is the purpose of sampling?

The purpose of SAMPLING is to obtain aREPRESENTATIVE SAMPLES andHOMOGENOUS of the whole sample that canbe taken to the laboratory for chemical analysisand the results obtain will be ACCURATE.

Representative means that content ofanalytical sample reflects content of bulksample.

Homogeneous means that the analyticalsample has the same content throughout theanalysis.

Chapter 4 4


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Classification of Sampling1. Base on method used for sampling

Sample Weight (mg) Sample Volume (L)

Meso >100 >100

Semi-micro 10 100 50 - 100

Micro 1 10 < 50Ultra-micro < 1

2. Based constituents presents in a sampleMajor > 1 %

Minor 0.1 1 %

Trace < 0.1 %

Ultratrace in the range of few parts per million or less.

Chapter 4 5


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A chemical analysis is usually performed ononly small portion of the material collectedto be characterized.A. If the amount of material is very small

and it is not needed for further use, thenthe entire samples may be used for

analysis.B. The material to be sampled must either inthe form of solid, liquid or gas.

C. Sampling method depends on type of

sample solid, liquid or gas.

Chapter 4 6


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SamplingDeciding how to obtain a sample for analysisdepend on,

1. The size of the bulk to be sampled.

2. The physical state of the fraction to beanalyzed. (solid, liquid, gas)

3. The chemistry of the material to beassayed.

*Sampling techniques that that would destroyor alter the identity or quantity of theanalyte are forbidden.

Chapter 4 7


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Obtaining a representativesample is the first step ofan analysis.

Thegross sampleis several smallportions of the sample. Sampletaken at random an assumed tobe as representative.

Gross sample that is reduced to asufficiently small size to behandled is called the laboratory

sample.An aliquot of this sample is takenfor the analysis sample.


9/80Chapter 4 9

Identify the population fromwhich the sample is to be

obtained.

Collect a gross sample thatis truly representative of the

population being sampled.

Reduce the gross sample toa laboratory sample that is

suitable for analysis.

Sampling the bulk material


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Sampling Solids Inhomogeneity of the material, make

sampling of solids more difficult.

The easiest way to sample a material isgrab sample the sample taken at random

and assumed to be representative. For reliable results, it is best to take 1/50

to 1/100 of the total bulk. The larger the

particle size, the larger the gross sampleshould be.

The gross sample must be reduced in size toobtain a laboratory sample.

Chapter 4 10

APR

2008


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Coning and QuarteringThis process is continued until the grosssample is small enough to be transported tothe laboratory.

Chapter 4 11

Method of Sampling SolidsOCT

2006


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There is specific techniques that can be used fortaking the samples. Using an example, explain how tosample either a solid, liquids or gases sample (5m)

Sampling the solid1. Using the method cone and quarter.

2. Divide a pile of material into quarter.

3. Take sample from each quarter of the pile andcrush these sample and form into smaller conicalpile.

4. Flatten the conical pile and cut into smaller equal

quarter.5. Two opposite quarters are chosen at random.

6. Crush the quarter further.

7. The whole steps are repeated until the laboratory

samples are obtained. Chapter 4 12


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The purpose of drying solid samples is toensure that the exact weight is obtainedduring the QUANTITATIVE chemicalanalysis.

If drying not done, the weight of sample isreported as is basis.

How it is done?

Solid samples dried in oven at 105oC 110oC for1-2 hours.Plant and tissue samples dried by heating.

Drying Solid Samples

Chapter 4 13


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Chapter 4 14

Problems in drying of the solid

samples1. Samples might decompose at high

temperature.2. Some samples are sensitive to heat,

therefore drying can be carried out in adessicator.


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Sampling Liquids Liquid samples are homogeneous and are much

easier to sample. If liquid samples are nothomogeneous and have only small quantity, they canbe shaken and sampled immediately.

The gross sample can be relatively small.

If water sample is taken from the river, then thewater samples is collected at the SURFACE,MIDDLE and at the BOTTOM of the river bed.

If the liquid is in a large container, then the liquidshould be stirred first before the samples aretaken at the top, middle and at the bottom ofthe container.

Chapter 4 15


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Sampling LiquidsSampling techniques will depend on the typesof liquid.

1. Large volume of liquids (impossible to mix)

2. Large stationary liquids (lakes, rivers)3. Biological fluids

Chapter 4 16

A specialsampling

bottle or bagis used to

collect liquidsamples.


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1. Grab sampling. An actual sample of air is

taken in a flask, bottle, bag or othersuitable container. Done over a period offew seconds or up to 1-2 minutes.

2. Continuous or integrated sampling. Gases or

vapours are removed from the air over ameasured time-period and concentrated bypassage through a solid or liquid sorbent.

Examples

Rotary vane pump with open flows to 32 L/min,locking flow valve, 10 feet tubing, and silencer forquieter operations. Suitable for use with asbestoscassettes, VersaTrap cassettes, and bioaerosolimpactors

Sampling Gases

Chapter 4 17


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Sampling Gases Tend to be homogeneous.

Large volume of samples is requiredbecause of their low density.

Air analysis:

Use a `Hi-Vol sampler that is

containing filters to collectparticulates.

Liquid displacement method:

The sample must has little solubility

in the liquid and does not reactwith the liquid.

Breath sample:

The subject could blow into

evacuated bag. Chapter 4 18


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Sample Storage There is a time gap between when the

sample is taken and the actual analysis isbeing carried out.

Therefore the sample should not beadulterated by foreign matter or there is a

lost of analyte during storage. During the storage of samples certain side

reactions can occur. This will change theproperties of the samples that will beanalysed.

Therefore, the sample, once collected mustbe handled and stored in a manner so as toprotect it from contamination or alteration.

Chapter 4 19


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Sample Storage

Example in the air pollution studies thecontent of SO2 in air is not stable due tothe following reaction .

2SO2 + O2 2SO3

To avoid the above reaction the sample iscooled to 4oC.

Chapter 4 20


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Sample StorageFor liquids samples, make sure that it is keptin bottles with stoppers.

Acidic liquid samples can be stored in glasscontainer.

Whereas basic liquid samples in plasticcontainer.

Solid samples is easier to keep and have lesschance to be adulterated by foreignmatters. Sometimes it can also get absorbedor adsorbed to the wall of the container.

Chapter 4 21


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Problems encounter during storage of samples

1. The sample can be contaminated by foreign matter.

2. Lost of analyte during storage.

3. Decomposition of sample.

4. Side reactions can occur.

5. The sample should not react with the wall of the

container or get contaminated.6. Lost of analytes during storage of volatile liquid

samples in which the container is not tightly closed.

7. Loss of water from hygroscopic material.

8.Precipitation of metals from water samples.

Chapter 4 22


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Why we do proper and suitablestorage of sample?

To prevent:1. Contamination or alteration that may cause

by container, the atmosphere or the light

2. Loss of the analyte due to exposure toatmosphere or adsorption onto the wall ofthe container

3. Decomposition due to heat, light or

bacteria

Chapter 4 23


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Examples of precaution taken to preserve the sample

Trace amount of metal should not be stored in glass container

since glass can leach small amount of metal

Trace organic compound may react with plastic container.

Preservatives may have to be added to sample which is

unstable. Eg. NaCI is added to blood sample to preserve the

blood.

Samples may need to be stored in refrigerator to avoid

decomposition of sample from bacterial sources.

Chapter 4 24


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Desiccator


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Sample Storage and Preservation

Preparing a laboratory sample Converting the sample to a useful form.

Solids are usually ground to a suitable

particulate size to get a homogeneoussample.

Dry the samples to get rid of absorptionwater.

Chapter 4 26


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Chapter 4 27

Electronic analytical balance.

Modern balances are electronic. They still compareone mass against another since they are calibratedwith a known mass. Common balances are sensitive to

0.1 mg.


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Chapter 4 28

Weighing bottles.

Weighing bottles are used for drying samples.Hygroscopic samples are weighed by difference,keeping the bottle capped except when removing the

sample.


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Chapter 4 29

Weighing dish.

A weighing dish or boat is used for direct weighingof samples.


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Chapter 4 30Volumetric flask.

Volumetric flasks arecalibrated to contain anaccurate volume. See theinside back cover of the

text for tolerances ofClass A volumetricglassware.


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Typical pipets


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Chapter 4 32

Transfer of volumetric pipettes.

Volumetric pipets accurately deliver afixed volume.

A small volume remains in the tip.


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Chapter 4 33

Measuring pipettes.

Measuring pipets are straight-borepipets marked at different volumes.

They are less accurate than volumetricpipets.


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Chapter 4 34

Hamilton microliter syringe.

Syringe pipets precisely deliver microliter volumes.

They are commonly used to introduce samples into agas chromatograph.

Th i i


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Chapter 4 35

Single-channel andmultichannel digitaldisplacement pipetsand microwellplates.

These syringe pipets canreproducibly deliver a selectedvolume.

They come in fixed andvariable volumes. The plastictips are disposable.


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Chapter 4 36

Typical burette.

A 50-mL burette is marked in0.1 mL increments.

You interpolate to 0.01 mL,good to about 0.02 mL.

Two readings are taken forevery volume measurement.


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Chapter 4 37

Meniscus illuminator.

Position the black field just below the meniscus.

Avoid parallax error by reading at eye level.


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Chapter 4 38

Proper technique for titration.

Place the flask on a white background.

Place the burette tip in the neck of the flask whileyour swirl.

U th f tit ti t f f i it t


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Chapter 4 39

Wash bottles:(a) polyethylene, squeeze type; (b) glass, blow type.

Use these for quantitative transfer of precipitatesand solutions, and for washing precipitates.


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Replicate Samples

Most chemical analyses are performed onreplicate samples whose weights or volumeshave been determined by carefulmeasurements with an analytical balance or

with a precise volumetric device.

Obtaining replicate data on samples improves

the quality of the results and provides ameasure of their reliability.

Chapter 4 40


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Defining replicate samples

Replicate samples are always performedunless the quantity of the analyte, expenseor other factors prohibit.

Replicate samples are portion of a materialof approximately the same size that iscarried through an analytical procedure atthe same time and the same way.

Chapter 4 41


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Five general principles of sample preparation

The sample preparation should be done without losing any ofthe analytes.

The sample preparation should include bringing the analytesinto the best chemical form for the method to be used.

Sample preparation should include removing someinterferences.

Sample preparation should be done without adding newinterferences.

Sample preparation may include diluting or concentrating theanalytes to get the best concentration for the method

chosen. Chapter 4 43

S l Di l ti


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Samples DissolutionSample dissolution is digestion or mineralization the

analyte into solution and to get rid of the interferingorganic substances in the samples.

A. Dissolving inorganic materials or sample thatcan be converted to an inorganic derivative

for measurements. Solution that totallydestroy the sample matrix.

B. Dissolving organic materials either by wetdigestion or dry ashing. Solution that are

nondestructive or partially destroy thesample matrix.

Chapter 4 44

A M h d Di l i


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A. Method DissolvingInorganic Materials

1. Inorganic materials that dissolve inacids.Using strong mineral acids.

Example such as Hydrochloric acids,Perchloric acids and nitric acids.2. Inorganic materials that do not

dissolve in acids.

Using the acidic and basic flux in themolten state.Example such as acid-sodium carbonateflux.

Chapter 4 45


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Dissolving the Inorganic

materials (Acidic-basic flux) Sample is mixed with the flux in a sample toflux ratio of about 1 to 10 or 20.

The mixture is heated in an appropriatecrucible until the flux becomes molten. (Theinsoluble materials react with the flux toform a soluble products)

The reaction is considered complete as themelt becomes clear.

The cooled solid is then dissolved in diluteacid or water.

Chapter 4 46

B Method Dissolving


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1. Wet Digestion

A method for decomposition of an organic

material, such as resins or fibers into an ashby treatment with a boiling oxidizing acid ormixture of acid.

The acids oxidize organic matter to carbondioxide, water and other volatile productswhich are driven off leaving behind the salts

or acids of the inorganic constituents.Chapter 4 47

B. Method DissolvingOrganics Materials

B Method Dissolving


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2. Dry Ashing The sample is slowly combusted at a high

temperature (400C -700C) in a mufflefurnace.

Atmospheric oxygen serve as oxidant, that isorganic matter is burned off leaving theinorganic residue, that is soluble in acid.

Example

Hydrochloric acid, nitric acid or aqua regia (3:1)dissolve many inorganic substances.HF acid decompose silicates.Perchloric acid is used to break up organiccomplexes.

Chapter 4 48

B. Method DissolvingOrganics Materials


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TOTAL DISSOLUTIONProcess whereby the entire sample

dissolve in a solvent

PARTIAL DISSOLUTION

Process whereby only the analyte

dissolve in a solvent

Chapter 4 49


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Usually usethe

combination of

acidEg, sulphuric

acid and nitricacid

Performed inkjedahl flask

Boiled of theacids, white

fumes evolve

Wet digestion methods

Chapter 4 50

1 W t Di ti


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1. Wet DigestionPriciple of wet digestion (wet ashing)

Usually use combination of acids to achieve acomplete dissolution.

A small amount (5 mL) of H2SO4 is used withlarger volumes of HNO3 (20 to 30 mL).

Usually performed in a Kjeldahlflask.

HNO3 destroys the bulk of organic matter,but it does not get strong enough to destroythe last traces.

. Chapter 4 51


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It is boiled off during the digestion processuntil only H2SO4 remains and dense, white SO3fumes are evolved and begin to reflux in theflask.

At this point, the solution gets very hot, H2SO4acts on the remaining organic material.

If the organic matter persists, more HNO3may added.

Digestion is continued until the solution clears. All digestion procedures must be performed in a

fume hood.

Chapter 4 52

cont


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Performed at hightemperature

Atmospheric O2serves as oxidant

Organic matter burnt off,leaving inorganic residue

Chapter 4 53

Process of dry

ashing(dry method)

l D h


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2. Simple Dry Ashing Most commonly employed techniques is simple dry

ashing whereby there is no chemical aids involved. A porcelain crucible can be used usually.

Trace analysis of Pb, Zn, Co, Cr, Mo, Sr, Fe used thesimple dry ashing method since there is little loss

by retention and volatilization during analysis.

Example:

lead is volatilized at temperature more than 500c,

especially if chlorine if present (blood and urinesamples). The crucible are preferred for lead forminimal retention losses. If an oxidizing material(Mg (NO3)2 is added to sample, the ashing

efficiency is enhanced. Chapter 4 54


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2. Dry Ashing

If the sample are liquids and wet tissuesa. The sample are dried on a stream bath

or by gentle heat before they areplaced in a muffle furnace.

b. The heat from the furnace should beapplied gradually up to fullTemperature to prevent rapid

combustion and foaming.

Chapter 4 55


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2 D h


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Care must be taken to ensure that non ofthe volatile elements (Hg, Arsenic) fromescaping during ashing.

Dry ashing often used to remove organicsubstances from interfering with theanalyte.

Further reading page 55 (G.D Christian).

Chapter 4 57

2. Dry Ashing


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Chapter 4 58

ADVANTAGESAND

DISADVANTAGES WET DIGESTION

DRY ASHING


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WET DIGESTION

Chapter 4 59

ADVANTAGES DISADVANTAGE

Superior in term ofrapidity

Introduction ofimpurities from thereagent necessaryfor the reaction

Freedom from loss byretention

Low level oftemperature maintained

DRY ASHING


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Chapter 4 60

ADVANTAGES DISADVANTAGE The ability to decompose

large sample sizes. Free from contaminationssince little or no reagents isrequired. The technique is

relatively safe and simple(simplicity) The ability to preparesamples containing volatilecombustion elements such as

sulfur, fluorine and chlorine(the Schniger oxygen flaskcombustion technique is verypopular in this case). The technique lends itselfto mass production.

Losses due to retention to the

ashing container. Losses due to volatilization. Contamination from the ashingcontainer. Contamination from the muffle

furnace. Physical loss of 'low density asheswhen the muffle door is opened (aircurrents). Difficulty in dissolving certain

metal oxides. Formation of toxic gases in poorlyventilated areas. (Note that allcharring should take place in a hoodand the muffle furnace must have ahood canopy for proper ventilation).

Oth T h i


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Microwave In some cases the dissolution of sample can

be done by using microwave oven toacceleratethe dissolution process (at

microwaves T=100 250o

C). The sample is sealed in specially designed

microwave digestion vessel with a mixtureof appropriate acids.

Chapter 4 61


Other Technique

Oth T h i


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Microwave Microwave ovens can be used for rapid

and efficient drying and aciddecomposition of samples.

Advantages of microwave digestionsinclude reduction in times from hours tominutes and low blank levels due toreduced amounts of reagents required.

Chapter 4 62


Other Technique

Oth T h i


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Fusion A weighed sample is mixed with a flux

(sodium peroxide) in a metal (zirconium) orgraphite crucible. The mixture is heated

over a flame, or in furnace and the resultingfused material is leached with either wateror appropriate acid (dilute mineral acid) oralkali.

These techniques are required for sampletypes that are inorganic in nature andunreactive toward acid decomposition.

Chapter 4 63

Other Technique

F i id d t b f


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Fusions are considered to be more of alast resort' by trace analysts because

1. They are expensive and often not available(high purity fluxes).

2. They yield high solids solutions that can

salt out in the nebulizer.3. Large dilutions of the sample are a

necessity.4. They often require expensive equipment.

Chapter 4 64

F si s sid d t b f


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Fusions are considered to be more of alast resort' by trace analysts because

5. Spectral interferences from the fluxand/or crucible construction material mustbe considered.

6. Contamination of the sample with thecrucible construction element andimpurities must be considered

7. They are labor intensive.

Chapter 4 65

Eliminating Interferences


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Eliminating InterferencesInterferences are substances that prevent

direct measurement of the analyte and mustbe removed.

May included separation steps:

precipitation chromatography

distillation

dialysis extraction into an immiscible solvent

Chapter 4 66

ANALYTES - are

components of a

sample that are to

be determined.

Standard Solution


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Standard SolutionDefinition

Standard solutions are solution whoseconcentrations are known to a high degreeof accuracy.

Characteristics:

i) Maintain its concentration over a longperiod of time (months or years) afterpreparation. This eliminates the need for

restandardization.ii) Must be able to undergo rapidly,

stoichiometric, and complete reaction withthe analyte.

Chapter 4 67

St d di ti


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Standardization

A process to determine the concentration ofsolution (impure titrant) by titrating withprimary standard solution or with a solution ofknown concentration.

Mtitrant = mass (g)primary std x 1 x b (mol titrant)

RMM primary std (g/mol) volume (L)titrant a (mol primary std)

Chapter 4 68

An approximate solution of 0.100 M HCl was prepared by


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An approximate solution of 0.100 M HCl was prepared bydiluting the HCl stock solution. It was then standardized bytitrating with 0.1905 g primary standard, Na2CO3. Thetitration reaction is follow:

CO32- + 2H+ H2O + CO2The titration requires 36.10 mL of HCl. Calculate themolarity of HCl. Molar mass for Na2CO3 is 105.99 g/mol.

Answer:2 mol HCl = 1 mol Na2CO3

Convert volume titrant (HCl) in mL to L36.10 mL x 1 l

1000 mLMHCl = 0.1905 g x 1 x 2 mol HCl

105.99 g/mol 0.036 L 1 mol Na2CO3= 0.09985 M

Chapter 4 69

St d d S l ti


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Standard Solution

Standard Solution


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Standard SolutionHow to prepare 1L 0.77M of NaCl?

Calculate amount of NaCl needed to prepare thestandard solution. Weighed accurately mass of NaCl needed in a weighing

boat using an analytical balance and transfer to thebeaker.

Add distilled water and stir using glass rod to dissolvethe solute. Transfer the solution from beaker into the volumetric

flask using filter funnel. Add distilled water into the volumetric flask until the

solution level near the calibration mark. Use a dropper to carefully add the water until the

meniscus is exactly on the calibration mark. Homogenize the solution by shaking the volumetric flask

for 20 seconds.

Commercial Standard


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1. Technical or commercial grade. Not use in

analytical work, but cleaning solution.2. USP (United States Pharmacopoeia) grade.

The specifications are designed to limitimpurities that are physiological hazards

3. Reagent grade. Minimum specifications ofthe Reagent Chemical Committee of theAmerican Chemical Society.

4. Primary standard grade/Referencestandards.

5. Special purpose reagents. Spectroscopic,chromatographic etc.

Commercial Standard

Standard Solution


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Standard SolutionA standard solution can be prepared in

either of two ways:1. A primary standard is carefully weighed, dissolved,

and diluted accurately to a known volume. Itsconcentration can be calculated from this data.

2. A solution is made to an approximate concentrationand then standardized by titrating an accuratelyweighed quantity of a primary standard.

Types of standard solutions:

i) Primary standard

ii) Secondary standard Chapter 4 73

Primary Standard


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Primary StandardDefinition

A compound of highest purity and it is used to determine,

directly or indirectly, the concentration of the standardsolution for a titration.

Ideal primary standards for volumetric titration should have thefollowing characteristics:

i. Highest purity, up to 99.99% (0.01 to 0.02% impurity).

ii. Stability, the substance should stable at room conditions orduring heating and does not react with constituents of theatmosphere.

iii. Free from hydrated water and should be non-hygroscopic.iv. Soluble in titration medium.

v. High formula weight to minimize weighing errors.

vi. Easily available at reasonable cost

Chapter 4 74

P i St d d


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Primary StandardThe number of primary standards available is very limited for example

oxalic acid (H2C2O4.2H2O), sodium carbonate (Na2CO3), calciumcarbonate (CaCO3), sodium chloride and arsenic trioxide.

Others:

1) potassium hydrogen phthalate2) potassium dihydrogen phosphate

3) potassium hydrogen tartarate,

4) Sodium carbonate

5) Sodium oxalate

6) Benzoic acid

7) Potassium dichromate

Chapter 4 75

Wh HCl r N OH c n n t b th prim r st nd rd?

APR 2008


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Why HCl or NaOH can not be the primary standard?

A primary standard should essentially available in pure form ,stable towards light and heat and react in a stoichiometric

proportion. HCl is a gas which is dissolved in water to form the solution.

The concentration expressed is very approximate so its not aprimary standard.

The exact concentration of any hydrochloric acid solution is notknown, unless it is prepared from standard ampoules. Laboratorygrade hydrochloric acid is not sufficiently pure

Ampoule is small sealed vial which is used to contain and preserve asample, usually a solid or liquid. Ampoules are commonly

made of glass, although plastic ampoules do exist.

NaOH cannot be weighed in open air because it is

highly hygroscopic.

Chapter 4 76

Secondary Standard


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Secondary Standard

A less pure substance whose composition is

reliably known. The purity or the concentration of a

secondary standard must be established by

careful stoichiometric analysis, usuallyagainst a primary standard.

Examples: HCl, HNO3, NaOH, KMnO4,

AgNO3, KOH, Ba(OH)2 ,HClO4,Sulfamic Acid(HSO3NH2), Na2S2O3,Ce(HSO4)4 ,H2SO4, andNH3(g)

Chapter 4 77

Stock Solution


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Stock Solution

Is a large volume of common reagent, such as HCl or NaOH

at a standardized concentration. As a concentrate that is a solution to be diluted to some

lower concentration for actual use.

This term is commonly used in analytical chemistry for

titration procedures where it is important that exactconcentrations of solutions are used.

Many laboratory chemicals such as acids are purchased asconcentrated solution (stock solution). Eg : 12 M HCl, 18 M

H2SO4

More dilute solutions are prepared by taking certain quantityof the stock solution and diluting it with water

Chapter 4 78

Dilution


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DilutionDilution

a process of changing the concentration of asolution from high to low.

Involves distributing a fixed amount of solute into alarger volume of solvent. Amount of solute remain the

same (n1 = n2) during the process.Volume changes from (V1 to V2).Addition of solvent do not change amount of solute inthe solution but only change the Molarity of thesolution.

Formula of dilution isM1V1 = M2V2

1 indicate as initial while 2 indicated as new or final.

Ad t f t k l ti


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Advantages of stock solution

Can save a lot of time

Converse material

Reduce needed storage space

Improve accuracy with which we prepare

solutions and reagents

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Chapter 4 - Basic Requirements for Analysis

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