An introduction to Physical Chemistry: Principles and

An introduction to Physical Chemistry: Principles and Practicals

ALaboratory Manual

For Physical Chemistry ExperimentsB. Sc. Chemistry (Hons.) Programme

Physical Chemistry Lab

Prepared by: Dr. Anurag Prakash SundaWebpage: www.apsunda.com

DEPARTMENT OF CHEMISTRYJ. C. BOSE UNIVERSITY OF SCIENCE AND TECHNOLOGY, YMCA

NH-2, Sector-6, Faridabad- 121006 Haryana (INDIA)E-mail: [email protected]

Contents

1 Your Responsibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 General Laboratory Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Laboratory Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.1 Surface tension measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Viscosity measurement using Ostwald viscometer . . . . . . . . . . . . . . . . . . . . . . . . 73.3 SASB volumetric titration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 SASB conductometric titration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.5 WASB conductometric titration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.6 Molecular weight determination using Dumas method . . . . . . . . . . . . . . . . . . . . . 193.7 Solubility & Enthalpy Change for benzoic acid . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Chemical Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Cleaning Glassware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.1 Grease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Safe Use of Chromic Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3 Rinsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.4 Cryogens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6 Volumetric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.1 Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2 Standard Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 NaOH Standardization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7 Record writing proforma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 Books for further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1. YOUR RESPONSIBILITY www.apsunda.com

1 Your Responsibility

The prevention of laboratory accident is a collective responsibility of everyone present inside the laboratory.Lab accidents generally occur due to:

• an indifferent attitude,

• failure to use common sense,

• failure to follow instructions, making mistakes and hurry/haste.

Doing things safely is not merely the right way to work in laboratory-it is the only way.

Never work alone in the laboratory. Confine long hairs and loose clothing. Never Pipet by mouth. Always use Pipetaid or suction bulb. Never perform un-authorized experiment.

Before you help another person, evaluate potential danger to yourself. If you try to help and are injured,you cannot be of much further help to someone else.

Plagiarism: Plagiarism is defined as the submission or presentation of work, in any form, which is not one‘s ownwithout acknowledgement of the source(s). It is an attempt to deceive the reader that the work or ideas presented areyour own, whereas, in fact they are the words/ideas of others. If you obtain information from an outside source, thatsource must be acknowledged accordingly.

Reading Reference:Safety in Academic Chemistry Laboratory, Seventh Edition, Volume I, ACS, 2003.

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www.apsunda.com 2. GENERAL LABORATORY PRACTICE

2 General Laboratory Practice

1. All students must wear safety glasses throughout all practical sessions for eye protection.

2. All students are required to wear a laboratory coat and no student will be permitted to work in the laboratorywithout one.

3. All students must wear closed shoes in the laboratory.

4. Work areas must be kept clean at all times, and free from chemicals/apparatus which are not required.

5. All solids must be discarded into the bins provided in the laboratory. Never throw matches, paper, or anyinsoluble chemicals into the sinks.

6. Waste solvents must be placed into the special waste solvent bottles where provided.

7. Before leaving the laboratory at the end of a practical session, make sure that all electrical equipment areswitched off and all gas and water taps are shut off.

8. Do NOT heat graduated cylinders or bottles, it will result in volume change.

9. Do NOT pour water in acid bottles. If you add water to acid, you form an extremely concentrated solutionof acid initially. So much heat is released which will result in boiling of solution very violently, splashingconcentrated acid out of the container!

10. Balances, spectro-photometers and other expensive equipments must be treated with care and kept clean andtidy at all times.

11. Never hold a container above eye level when pouring a liquid.

12. While carrying out a reaction or boiling a liquid in a test tube, point the mouth of the test tube away fromyourself and from others in the laboratory.

13. Always read material safety data sheet (msds) before using any chemical/reagent.

14. Do not run through laboratories and along corridors. You could easily collide with someone coming the otherway carrying something nasty. Always take particular care when passing through doorways.

Reading Reference:Garland, Nibler and Shoemaker, Experiments in Physical Chemistry, Seventh Edition, McGraw-Hill, 2003.

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3. LABORATORY EXPERIMENT www.apsunda.com

3 Laboratory Experiment

3.1 To determine surface tension of given liquid using Stalagmometer.

Learning Outcome:

Understanding the role of surfactants (surface active agents) or detergents in day to day life and their importance inphysical chemistry: Addition of surface active agents in water such as Cetyltrimethylammonium bromide [CTAB]or Sodium lauryl sulfate [SLS] lead to change in surface tension of water. This experiment aims to explain surfacetension behaviour of aqueous solution of surface active agents at their different concentration.

Preparation:

1. Aqueous Sol.n of 2%, 4%, 6%, 8%w/v Cetyltrimethylammonium bromide [CTAB i.e. CH3(CH2)15N(Br)(CH3)3]

2. Aqueous Sol.n of 2%, 4%, 6%, 8% w/v Sodium lauryl sulfate [SLS or SDS i.e. CH3(CH2)11OSO3Na]

Requirements:

Stalgmometer, a small rubber tube with a screw pinch cork, distilled water.

Figure 1.1: Cohesive forces andadhesive forces.

Theory:

Cohesive forces are the forces exist between molecules of one phase. Adhesiveforces are the forces exist between molecules of two different phases. The cohe-sive forces among liquid molecules are responsible for the phenomenon of surfacetension. In the bulk of the liquid (Point A in Figure 1.1), each molecule is pulledequally in every direction by neighboring liquid molecules, resulting in a net forceof zero. The molecules at the surface (Point C in Figure 1.1) do not have othermolecules on their all the sides and therefore are pulled inwards. It forces liq-uid surfaces to contract to the minimal area and this force is known as ‘SurfaceTension’.

Figure 1.2: Stalagmometer.

Surface tension is the force per unit length (dyne/cm) applied par-allel to the surface so as to counterbalance the net inward pull ofmolecules of interface together OR it is the force per unit length(dyne/cm) on the surface of a liquid which opposes expansion ofthe surface area. Surface tension (γ) of a liquid can be deter-mined by drop weight method using stalagmometer. Stalagmome-ter consists of a dropping tube with a capillary, the end of whichis flattened. This flattened end helps to give a large droppingsurface. This surface is already ground flat and polished. Theother end of the capillary is sealed on a tube containing a bulb.There are two marks etched on the stalagmometer, one above thebulb ‘p’ and another one below the bulb ‘q’. The γ of a liq-uid is related to the weight of a drop of that liquid which fallsfreely from the end of a tube. Due to the surface tension ofliquid, drop of liquid grows at the tip of capillary tube of sta-lagmometer till the forces of liquid surface (i.e. γ) remain bal-anced with gravitational force of liquid drop (mg). Once gravita-tional force of liquid drop (mg) becomes higher then surface ten-sion of liquid drop, drop falls down from the tip of Stalagmome-ter.

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www.apsunda.com 3.1: Surface tension measurement

γ1γ2

=n2n1× d1d2

where, density d = W/V ; hence we can write:

γ1γ2

=n2n1× W 1V 2

V 1W 2

here, γ1 and γ2 are surface tension of given liquid and reference liquid (i.e. water) respectively. The value of γ2 can betaken from literature at particular temperature. For dropping of liquid through Stalagmometer from mark ‘P’ to ‘Q’,the counted number of drops n1 and n2 are of given liquid and reference liquid (i.e. water) respectively. The W 1 andW 2 are weight of given liquid drops and reference liquid drops (i.e. water) respectively collected in volumetric flask.Since, the volume of both the liquids from mark ‘P’ to ‘Q’ will remain same (V 1 = V 2) at particular temperature.Thus, surface tension at the recorded temperature of given liquid can be expressed as:

γ1γ2

=n2n1× W 1

W 2(1.1)

Procedure:

1. Measure the temperature of distil water using thermometer.

2. Clean the stalgmometer and volumetric flask with chromic acid, wash with water and dry it. Measure the weightof empty volumetric flask.

3. Attach a small piece of rubber tube having a screw pinch cock at the upper end of the stalgmometer.

4. Immerse the lower end of the Stalgmometer in distilled water and pull the water 1-2 cm above mark P andadjust the pinch cork so that 10-15 drops of water fall per minute.

5. Allow the water drops to fall and start counting the number of drops (n2) when the meniscus crosses the uppermark ‘P’ and stop counting when the meniscus passes mark ‘Q’.

6. Weigh the volumetric flask and find out weight of water drops (W2) collected in volumetric flask.

7. Repeat the exercise to take minimum three readings for number of drops (n2) and weight of water drops (W2)respectively.

Table 1.1: Number of drops and weight of collected drops in volumetric flask for water and given liquid respectively and theiraverage value from three readings.

Sr.No.

Sample Number of drops Avg. Number of drops Weight of drops Avg. Weight of drops

1Water

-n2

-W22 - -

3 - -1

2 % CTAB-

n1

-W12 - -

3 - -

8. Rinse the stalgmometer with alcohol and dry it.

9. Take the given liquid in stalgmometer and allow the liquid drops to fall and start counting the number of drops(n1) when the meniscus crosses the upper mark ‘P’ and stop counting when the meniscus passes mark ‘Q’.

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3.1: Surface tension measurement www.apsunda.com

10. Weigh the volumetric flask and find out weight of liquid drops (W1) collected in volumetric flask.

11. Repeat the exercise to take minimum three readings for number of drops (n1) and weight of liquid drops (W1)respectively.

Temperature of distil water = −−−−−−−C

Surface tension of distil water at −−−−−−C = −−−−−−−−− dynes/cm

Calculate the surface tension of liquid (dynes/cm) for different concentrations using eqn 1.1 and conclude observa-tions.Note: Make certain there are NO BUBBLES! in Stalgmometer

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www.apsunda.com 3.2: Viscosity measurement using Ostwald viscometer

3.2 To determine the intrinsic viscosity of aqueous solution of a given solid using Ostwald or Ubbe-lohde viscometer.

Learning Outcome:

Demonstration of forces acting in between layers of a liquid and their measurement as viscosity. Study the Character-istic behaviour of a solid in a solvent using different concentration .

Preparation:

1. Aqueous Sol.n of 2%, 4%, 6%, 8% w/v Starch

2. Aqueous Sol.n of 2%, 4%, 6%, 8% w/v Glucose

Requirements:

Ostwald viscometer, rubber tube with screw pinch cock, stand, beaker (50 ml), volumetric flask (50 ml), & distilledwater.

Theory:

The force of friction which one layer of the liquid offers to another layer of the liquid is called viscosity. The SIphysical unit of viscosity is the pascal-second (Pa·s), (i.e., kg·m−1·s−1). This means: if a fluid with a viscosity ofone Pa·s is placed between two plates, and one plate is pushed sideways with a shear stress of one pascal, it moves adistance equal to the thickness of the layer between the plates in one second. The cgs unit for the same is the poise(P), (named after J. L. Marie Poiseuille). It is more commonly expressed, as centipoise (cP). [1 cP = 0.001 Pa·s].Water at 20 C has a viscosity of 1.0020 cP.

Figure 1.3: Ubbelohde viscometer

For measuring the viscosity coefficient, Ostwald viscometer method is usedwhich is based on Poiseuille’s law. According to this law, the rate of flow of liquidthrough a capillary tube having viscosity, η, can be expressed as

η =πr4P

8vl× t (1.2)

where, v = vol. of liquid (in ml)t = flow time (in sec.)r capillary = radius of the capillary (in cm)l = length of the capillary (in cm)P = hydrostatic pressure (in dyne/sq.cm)η = viscosity coefficient (in poise).

Ostwald viscometer is a two neck apparatus consist of one reservoir and mea-suring bulb. A liquid solution can be introduced into the reservoir then suckedthrough the capillary in the measuring bulb. The liquid solution is allowed to travelback through the measuring bulb and the ‘time (t)’ it takes for the liquid to passthrough two calibrated marks ‘A’ to ‘B’ is a measure for viscosity. The Ubbelohdedevice (See Figure 1.3) has a third arm extending from the end of the capillaryand open to the atmosphere. In this way, the pressure head only depends on a fixedheight and no longer on the total volume of the liquid. In order to determine relativeviscosity, first the ‘time (t0)’ taken by the solvent to pass through two calibratedmarks ‘A’ to ‘B’ is measured. Similarly, the ‘time (t)’ taken by the solution to passthrough two calibrated marks ‘A’ to ‘B’ is measured. The ratio of ‘time (t)’ takenby the solution with the ‘time (t0)’ taken by the solvent is known as relative vis-cosity. The specific viscosity and inherent viscosity can be deduced from relative

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3.2: Viscosity measurement using Ostwald viscometer www.apsunda.com

viscosity. For constant P and v we can write η ∝ t

If solvent used is water (η0 ∼ 1 cP), then relative Viscosity (ηrel.) can be expressed such that

ηrel. =t

t0

Specific Viscosity ηsp. = ηrel. − 1

Reduced Viscosity ηred. =ηsp.c

Inherent Viscosity ηinh. =1

cln ηrel.

The extrapolation of specific viscosity or inherent viscosity for concentration tends to zero gives intrinsic viscosityof the solute.

Intrinsic Viscosity

ηint. = limC→0

[1

c(ln ηrel.)

](1.3)

OR,

ηint. = limC→0

[1

c(ηrel. − 1)

](1.4)

Figure 1.4: Variation in Reduced or Inherent Viscosity as a function of concentration.

Procedure:

1. Clean and rinse the viscometer properly with distilled water. Fix the viscometer vertically on the stand.

2. Fill the solution in the reservoir (mark ‘C D’) of viscometer with specific amount (say 20 ml) of solvent i.e.distil water (every time take the same volume).

3. Pull the solvent up in measuring bulb with the help of suction bulb.

4. Measure the time taken by solvent (medium) to pass through two calibrated marks ‘A’ to ‘B’ i.e. ‘t0’.

5. Repeat the exercise to take minimum three readings of ‘t0’.

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www.apsunda.com 3.2: Viscosity measurement using Ostwald viscometer

6. Prepare standard solution of known concentration (molar) in same solvent/medium.

7. Pull the solution up in measuring bulb with the help of suction bulb.

8. Record the time taken by solution to pass through two calibrated marks ‘A’ to ‘B’ i.e. t.

9. Repeat the exercise to take minimum three readings t.

10. Calculate inherent viscosity and reduced viscosity of given solution.

11. Plot a graph between inherent viscosity and concentration. Extrapolate it to zero to find out ‘intrinsic viscosity’.

12. Plot a graph between reduced viscosity and concentration. Extrapolate it to zero to find out ‘intrinsic viscosity’.

Table 1.2: Time taken in viscometer by solvent and given liquid of known concentration respectively and their average valuefrom three readings.

Sr.No.

Sample Time (t) Average Time(tavg)

Reduced Viscosity(ηred.)

Inherent Viscosity(ηinh.)

1Water

-t0 (avg.) - -2 -

3 -1

2 % CTAB-

t(avg.) - -2 -3 -

Temperature of distil water = −−−−−−C

Calculate the standard deviation in ‘intrinsic viscosity’ obtained from ηred. & ηinh. respectively and conclude obser-vations.Note: Make certain there are NO BUBBLES! in Ubbelohde/Ostwald device

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3.3: SASB volumetric titration www.apsunda.com

3.3 To determine the end point of strong acid versus strong base through volumetric titration.

Preparation:

1. 0.1 N Oxalic Acid aq. solution (100 ml) using (H2C2O4 · 2 H2O)

2. 0.1 N HCl aq. solution (100 ml)

3. 0.1 N NaOH aq. solution (100 ml)

4. Phenolphthalein indicator solution in methyl alcohol.

Requirements:

Burette, burette stand, beaker (50 ml), volumetric flask (100 ml), conical flask (50 ml) & distilled water.

Theory:

An Arrhenius acid is a substance that produces H+ ions in aq. solution, and an Arrhenius base is a substance thatproduces OH− ions in aq. solution. A Brønsted acid is a proton donor, and a Brønsted base is a proton acceptor. ABrønsted acid-base reaction is a proton transfer reaction. A strong acid dissociates in water solution almost completelyinto H+ (aq) ions and anions characteristic of the acid. A strong base is completely dissociated into OH− (aq) ionsand cations characteristic of the base. Common strong acids include hydrochloric acid (HCl), nitric acid (HNO3), andsulfuric acid (H2SO4) etc. The hydroxides of the alkali metals and of the alkaline earth metals are strong bases.

Figure 1.5: A set up of acid-basetitration.

Acid-base reactions, or neutralization reactions, are commonly usedto determine the concentrations of acids (or bases) in solutions. Ifthe concentration and volume of one of the reactants in a neutraliza-tion reaction is known, the concentration of the second solution canbe determined if its volume is known. This procedure requires themeasurement of volumes, and is therefore called volumetric analysis.The base solution is added slowly from a burette to a measured vol-ume of the acid solution until all of the acid is neutralized. Thepoint at which neutralization occurs is usually detected by the changein color of an organic dye such as litmus or phenolphthalein, whichhas one color in acidic solution and a different color in basic solu-tion. The organic dye used for this purpose is called an indica-tor. This type of volumetric analysis is known as titration. Fig-ure 1.5 illustrates a setup for an acid-base titration. The concentra-tion of one solution is determined by titration with another solution ofknown concentration, is called standardization. In a standardization pro-cedure, the solution whose concentration is accurately known is calleda standard solution. In these titration’s, H+ ions of the acid com-bine with OH− ions of the alkali to form ionized molecules of wa-ter.

HA + BOH −−→ BA + H2O

HCl + NaOH −−→ NaCl + H2O

The end point represents all the H+ ions of the acid gets neutralized with base. Thus, the normality of unknownsolution can be calculated using the expression given below:

N1V 1 = N2V 2 (1.5)

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www.apsunda.com 3.3: SASB volumetric titration

where, N1 = Normality of unknown solutionV 1 = Volume of unknown solution (in ml)N2 = Normality of standard solutionV 2 = Volume of standard solution (in ml)

The end point in these titration’s is determined by the use of organic dyes which are either weak acids or weakbases. An acid-base indicator is a weak organic acid (HIn) or a weak organic base (InOH), where the letter ‘In’ standsfor a complex organic group. Methyl orange and phenolphthalein are both weak acids. They change their colourswithin a limited range of hydrogen ion concentrations, i.e., pH of the solution. Phenolphthalein is a suitable indicatorin the titrations of strong alkalies (free from carbonate) against strong acids or weak acids. Methyl orange is used asan indicator in the titration’s of strong acids against strong and weak alkalies. Phenolphthalein is pink in base solutionand colourless in acid solution. Thus, when we add two drops Phenolphthalein to the acid solution in the flask, itshows no colour. As the added base is in slight excess, it becomes pink. Thus phenolphthalein signals the end-pointby a colour change from colourless to pink. Similarly methyl orange indicates the end-point by a colour change fromred (in acid) to yellow (in base).Theories of acid-base indicators:

1. Ostwald’s theory:Phenolphthalein is a weak acid and exists as the following equilibrium in solution,

HPh −−−− H + + Ph –

colourless pinkHPh molecules are colourless, while Ph− ions are pink. Thus in acid solution, phenolphthalein is colourless andin basic solution it is pink. With an addition of a base to the solution, H+ ions are removed as H2O by reactingwith OH− ions of the base. This shifts the equilibrium to the right, resulting in the increase of Ph− ions that arepink. Thus in acid solution the unionized HPh molecules predominate and the solution is colourless, while inbasic solution Ph ions are in excess and the solution is pink.

The actual colour shade of the indicator depends on the ratio of concentrations of Ph− ions and HPh present insolution. From the equilibrium constant expression, we can write

Kind. =[H+][Ph−]

[HPh](1.6)

[H+] = Kind.[HPh]

[Ph−](1.7)

2. Quinonoid theory:The cause of colour is due to change of an indicator in acid-base solutions. The unionised molecule and itscorresponding anion show a change in tautomeric transformation of the respective indicator. Phenolphthaleinexists in two tautomeric forms:(i) the benzenoid form which is colorless and present in acid solution; and(ii) the quinonoid form which is pink and present in basic solution.

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3.3: SASB volumetric titration www.apsunda.com

Procedure:

1. Clean and rinse the burette properly with distilled water. Fix the burette vertically on the stand.

2. Fill the NaOH solution in the burette and take 20 ml standard oxalic acid solution in conical flask.

3. Add two drop of phenolphthalein indicator in acid solution. Swirl the solution gently.

4. For titration, add NaOH solution drop-wise from burette in acid solution and keep swirling the mixture gently.

5. Stop addition of NaOH solution when acidic solution turns colorless to pink. Note down the volume consumedof NaOH in burette.

6. Repeat the exercise to take minimum three readings or till for average volume of base, there should be 2consistent readings observed.

Table 1.3: Volumetric titration of NaOH solution against standard oxalic acid solution using phenolphthalein indicator.

Sr. Volume of acid Volume of NaOH in Volume of NaOH Avg. volume ofNo. (in ml) burette (in ml) consumed in burette (ml) NaOH (in ml)

Initial Final

1 20.0 0.0 - --2 20.0 0.0 - -

3 20.0 0.0 - -

7. Calculate normality of NaOH solution using eqn 1.5.

8. Fill the NaOH solution in the burette and take 20 ml of unknown HCl acid solution in conical flask.


10. For titration, add NaOH solution drop-wise from burette in acid solution and keep swirling the mixture.

11. Stop adding NaOH solution when acidic solution turns colorless to pink. Note down the volume consumed ofNaOH in burette.

12. Repeat the exercise to take minimum three readings.

13. Calculate normality of HCl solution using eqn 1.5.

Table 1.4: Volumetric titration of standardized NaOH soln against unknown HCl soln in presence of phenolphthalein.

Sr. Volume of HCl Volume of NaOH in Volume of NaOH Avg. volume ofNo. (in ml) burette (in ml) consumed in burette (ml) NaOH (in ml)

Initial Final

1 20.0 0.0 - --2 20.0 0.0 - -

3 20.0 0.0 - -

Normality of standard oxalic acid = −−−−−− N

Normality of standard NaOH solution = −−−−−− N

Normality of Unknown HCl solution = −−−−−− N

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www.apsunda.com 3.4: SASB conductometric titration

3.4 To determine the equivalence point of strong acid and strong base (SASB) through conducto-metric titration using conductivity meter.

Preparation:

1. Aq. 0.1 N & 0.01 N KCl solution (50 ml each)

2. Aq. 0.2 N oxalic acid solution (100 ml)

3. Aq. 0.2 N HCL solution (100 ml)

4. Aq. 0.2 N NaOH solution (100 ml)

Requirements:

Platinum electrode, Conductivity meter, Burette, burette stand, beaker (50 ml & 100 ml), volumetric flask (100 ml),conical flask (50 ml) & distilled water.

Theory:

The principle of conductometric titration is based on variation in ionic conductivity due to replacement of one of theions by the other and invariably these two ions differ in the ionic conductivity. The equivalence point can be obtainedgraphically by plotting the variation in conductance as a function of the volume of titrant added. The ‘equivalencepoint’, or stoichiometric point, of a acid-base chemical reaction is the point at which chemically equivalent quantitiesof base and acid are available. In other words, the moles of acid are equivalent to the moles of base, which does notnecessarily imply a 1:1 molar ratio of acid:base. Whereas, the ‘end point’ in volumetric titration refers to the point atwhich the indicator changes its colour.

The conductance (C) is the reciprocal of electrical resistance (R).

C =1

R

Figure 1.6: A set up of conductometrictitration.

Where, resistance R = ρl

a& it is expressed as Ohm (Ω).

l

ais cell

constant.In case of strong acid with a strong base, for e.g. conductometric titration

HCl with NaOH: Before NaOH is added, the conductance is high due to thepresence of highly mobile H+ ions. When the base is added, the conductancefalls due to the replacement of H+ ions by the added cation Na+ and H+

ions react with OH− ions to form water. This decrease in the conductancecontinues till the equivalence point.

HCl + NaOH −−→ NaCl + H2O

At the equivalence point, the solution contains only NaCl. After theequivalence point, the conductance increases due to the presence of OH−

ions which results in increase in conductivity with further addition of NaOH.

Procedure:

1. Clean and rinse the burette properly with distilled water or soap solution. Fix the burette vertically on the stand.

2. Fill the burette with NaOH solution and take 20 ml oxalic acid solution (as primary standard) in a conical flask.


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3.4: SASB conductometric titration www.apsunda.com

Figure 1.7: Variation in conductance of HCl solution with addition of NaOH solution.

4. For standardization of NaOH, add NaOH solution drop-wise from burette in acid solution and keep swirling themixture.





Initial Final

1 20.0 0.0 - --2 20.0 0.0 - -

3 20.0 0.0 - -


8. Fill the NaOH solution in the burette.

9. Determine temperature of KCl solution and calibrate conductivity meter using aq. 0.1 N & 0.01 N KCl solution.

Table 1.6: The observed conductance of HCl solution with addition of NaOH.

Sr. No. Volume of NaOH added (in ml) C (in Ω−1)

1 0.0 -2 1.0 -3 2.0 -

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www.apsunda.com 3.4: SASB conductometric titration

10. Take 50 ml of HCl solution and determine the conductance of neat HCl solution.

11. Now add NaOH solution drop-wise using burette and note down the conductance of HCl-NaOH mixture atdefinite volume interval of NaOH.

Plot a graph for conductance of HCl solution with addition of NaOH. Draw a straight line passing through maximumpoints for both decreasing and rising curve. Extrapolate both the curves such that decreasing and rising curve inter-sects. The point where, both the lines intersect, corresponding volume of NaOH is equivalence point for the givenHCl-NaOH conductometric titration.

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3.5: WASB conductometric titration www.apsunda.com

3.5 To determine the equivalence point of weak acid and strong base (WASB) through conducto-metric titration using conductivity meter.

Preparation:

1. Aq. 0.1 N & 0.01 N KCl solution (50 ml each)


3. Aq. 0.2 N Acetic Acid solution (100 ml)


Requirements:

Platinum electrode, Conductivity meter, Burette, burette stand, beaker (50 ml & 100 ml), volumetric flask (100 ml),conical flask (50 ml) & distilled water.

Theory:

The principle of conductometric titration is based on variation in ionic conductivity due to replacement of one of theions by the other and invariably these two ions differ in the ionic conductivity. The equivalence point can be obtainedgraphically by plotting the variation in conductance as a function of the volume of titrant added. The equivalencepoint, or stoichiometric point, of a acid-base chemical reaction is the point at which chemically equivalent quantitiesof base and acid are available. In other words, the moles of acid are equivalent to the moles of base, which does notnecessarily imply a 1:1 molar ratio of acid:base.

The conductance (C) is the reciprocal of electrical resistance (R).

C =1

R

Figure 1.8: A set up of conductometrictitration.

Where, resistance R = ρl

a& it is expressed as Ohm (Ω).

l

ais cell

constant.In case of Weak Acid with a Strong Base, for e.g. conductometric titra-

tion of CH3COOH with NaOH. Since CH3COOH is a weak acid and it re-mains partially dissociated in neat solution. For neat CH3COOH acid solu-tion, the conductance is due to presence of partially dissociated H+ ions ofacetic acid. Hence, the initial conductance is very low for acetic acid solu-tion. On addition of NaOH, the dissociated CH3COO – ions combines withNa+ ions and give rise to CH3COONa.

CH3COO – H+ + Na+OH – −−→ CH3COO – Na+ + H2O

In the beginning, at very low concentration of NaOH, the dissociation ofboth ‘Acetic acid and CH3COONa’ is suppressed due to common ion effect.The conductance which is solely due presence of H+ ions which cancel outwith OH− ions to form water and lead to decrease in concentration.

CH3COOH −−→ CH3COO – + H+

CH3COONa −−→ CH3COO – + Na+

Common Ion

On certain addition of NaOH volume, a limiting concentration of CH3COONa produced at which the ionizationof CH3COONa starts dominating and results in conversion of CH3COONa to CH3COO – and Na+ ions. Simulta-neously, acetic acid gets dissociated. The conductance starts increasing due to continuous increase of feebly mov-able CH3COO – and Na+ ions. At the equivalence point, acetic acid dissociates completely and gets converted to

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www.apsunda.com 3.5: WASB conductometric titration

Figure 1.9: Variation in conductance of CH3COOH solution with addition of NaOH solution.

CH3COO – and Na+ ions (i.e. replacement of H+ ions with OH− ions to form water). After the the equivalence point,the conductance of acetic acid-NaOH mixture shows a sharp increase due to excess of OH− ions. The conductancecontinuously increases afterwards with further addition of NaOH.

Procedure:









Initial Final

1 20.0 0.0 - --2 20.0 0.0 - -

3 20.0 0.0 - -


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3.5: WASB conductometric titration www.apsunda.com

8. Fill the NaOH solution in a burette.

9. Measure the temperature of KCl solution using thermometer and calibrate conductivity meter using aq. 0.1 N& 0.01 N KCl solution.

10. Take 50 ml of CH3COOH solution and determine the conductance of neat CH3COOH solution.

11. Now add NaOH solution drop-wise using burette and note down the conductance of CH3COOH-NaOH mixtureat definite volume interval of NaOH.

Table 1.8: The observed conductance of CH3COOH solution with addition of NaOH.

Sr. No. Volume of NaOH added (in ml) C (in mS)

1 0.0 -2 1.0 -3 2.0 -

Plot a graph for conductance of CH3COOH solution with addition of NaOH. Draw a straight line passing throughmaximum point for both rising curve. The point where, both the lines intersect, corresponding volume of NaOH isequivalence point for the given CH3COOH-NaOH conductometric titration.

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www.apsunda.com 3.6: Molecular weight determination using Dumas method

3.6 To determine molecular weight of a given volatile organic liquid by the ‘Dumas’ method using‘Ideal Gas Law’.

Learning Outcome:

The purpose of this experiment is to get a practical understanding of a method for analyzing macroscopic properties(i.e. molar mass) of a volatile organic liquid using the Ideal Gas Law.

Requirements:

Beaker (50 ml & 100 ml), water bath, copper wire, aluminium foil, rubber band, conical flask (50 ml) & distilledwater.

Theory:

Many gases at pressures near 1 atm and at normal temperatures are found to exhibit nearly ideal behaviour. For thesegases, we can express their pressure/volume/temperature behaviour using the ideal gas law:

PV = nRT (1.8)

Figure 1.10: A set up for the Molar Massexperiment.

where P is the pressure, V is the volume, T is the absolute temperature,n is the number of moles of gas molecules and R is the ideal gas constant(8.314 J/mol K).

In order to determine molecular weight of the an volatile sample, volumeand mass of an organic compound is measured at the boiling temperature(T) of water and ambient pressure. This allows us to determine the numberof moles of the compound and hence, it’s molar mass. In this approach,the sample is added to a small conical flask, the flask is heated and as thesample evaporates, the air present in the flask is swept out of the container.It followed by cooling of flask, and the mass of liquid which condensesmust be equal to the mass of vapor. Heating the flask with its contents ina water bath, one must assume that all the air is displaced by the vaporizedcompound from the flask. It can also be assumed that the molecular structureis identical in both liquid & gas phase (no dissociation or association ofmolecules). Thus, the volume of the flask is equal to the volume of the puregas at the particular temperature. Upon cooling, weigh the liquid remainingin the flask and get its mass. One assumes in this step that the mass of the airin the flask is identical to what it was before we heated the flask. This allows us to calculate the density (d = m/V )of the gas at T.

The rearrangement of the ideal gas law in terms of density and the molar mass (n = m/M ), the molar mass ofthe unknown compound can be expressed such that:

PV = nRT (1.9)

PV =m

MRT (1.10)

M =m

V

(RT

P

)(1.11)

M = d×(RT

P

)(1.12)

Procedure

1. Weigh a clean, dry conical flask together with a piece of foil and a thin copper wire.

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3.6: Molecular weight determination using Dumas method www.apsunda.com

2. Half fill a pipette with an organic compound and transfer this into the c conical flask.

3. Place the aluminum foil on n the mouth of the flask and carefully press it into place to form a reasonably tightclosure over the top of the flask. Take care not to tear or pierce the foil.

4. Place the wire round the top of the neck and twist the ends together so that the foil is held tightly in place. Placea second aluminium foil over the first one and hold with a rubber band.

5. Place the flask in the water bath. Heat slowly the water bath so that the water is brought to boiling in 10 to 15min.

6. Set the temperature of water bath at boiling point of the water and keep the conical flask in water bath until notrace of the liquid organic compound is left in the flask.

7. Measure the temperature of the boiling water followed by quick removal of the flask from the water.

8. Dry the outside surface of the flask with a clean tissue paper and keep the flask on a clean dry surface to cool.Wait approximately 5-10 minutes (or more) for the flask to cool to room temperature. A small amount of liquidshould condense inside the flask as it cools.

9. Remove the rubber band and outer Al-foil. Remove any moisture from the inner foil with a tissue paper.

10. Weigh the covered flask and contents on the weighing balance. The difference in the mass of the flask beforeand after heating is the mass of the condense vapour.

11. Measure the atmospheric pressure and determine the volume of the flask by filling it to the lip with tap water andpouring this without loss into a graduated cylinder. Use a glass pipette if you feel uncomfortable transferringthe initial volume of water from the flask to the graduated cylinder.

12. Repeat the procedure two more times.

Note down the weight of the volatile organic compound and its volume during the experiment. Put these values alongwith the parameters such as temperature and pressure in the ideal gas equation and report its molar mass.

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www.apsunda.com 3.7: Solubility & Enthalpy Change for benzoic acid

3.7 To determine solubility of given organic solid in water and heat of solution at particular tem-perature range.

Learning Outcome:

The purpose of this experiment is to have a hands-on practice for knowledge of thermodynamics involved in dissolu-tion process of solid compound and live demonstration of thermodynamics changes of a chemical system captured interms of heat of solution.

Preparation:



3. Saturated benzoic acid solution at (30, 35, 40 and 45 C)

Requirements:

Beaker (50 ml & 100 ml), conical flask water-bath, thermometer, burette and pipette & distilled water.

Theory:

Solubility is the property of a solid compound (solute) to get dissolve in liquid solvent to form a homogeneous solutionof the solute in the solvent. The extent of the solubility of a solid in a specific solvent is measured as the saturationconcentration where adding more solute does not increase the concentration of the solution. The dissolution of asolid into a liquid is usually accompanied with a heat effect (heat is either evolved or absorbed). The heat evolved orabsorbed can be determined when 1 mole of the solid is dissolved in a saturated solution. To obtain the solubility atdifferent temperatures using the Van’t Hoff relation, it is possible to determine the enthalpy change or heat of solution(∆H). Van’t Hoff equation in terms of solubility can be expressed as:

logS = − ∆H

2.303RT(1.13)

Figure 1.11: A set up for the heat of solu-tion experiment.

where, S is the solubility (mole/kg) at different temperatures (T in kelvin),∆H is the heat of solution (J/mol), and R = 8.314 J mol degree. The solubil-ity is expressed as grams of solute per 1000 g of solvent. The calculated heatof solution is approximately the average heat of solution over the tempera-ture range studied, and corresponds to the heat of solution at the saturationconcentration.

For example, the heat of solution of aqueous organic solid such asBenzoic acid can be determined by dissolving it in water at 5 differ-ent temperatures (30, 35, 40 and 45 C). In a saturated aqueous so-lution, benzoic acid has a molar solubility with the following equilib-rium:

C6H5COOH(aq.) C6H5COO –(aq.) + H+

The molar solubility can be determined titrimetrically againsta standardized strong base solution such as standard NaOH solu-tion.

Solubility = Normality × Equivalent weight (1.14)

Plot a graph of logS vs 1/T for obtained molar solubility at various temperature using eqn 1.13. Heat of solution ∆H

or enthalpy change can be calculated easily from the slope such that

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3.7: Solubility & Enthalpy Change for benzoic acid www.apsunda.com

Figure 1.12: Variation in solubility of C6H5COOH in aqueous solution as a function of temperature.

∆H = - 2.303 × R × Slope

Where, R = 1.987 cal/mole.

Procedure









Initial Final

1 20.0 0.0 - --2 20.0 0.0 - -

3 20.0 0.0 - -


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www.apsunda.com 3.7: Solubility & Enthalpy Change for benzoic acid

8. Fill the NaOH solution in the burette.

9. Prepare the saturated solution of benzoic acid by taking 100 ml of distilled water in a beaker.

10. Place the beaker inside water-bath at temperature T, add increasing amount of benzoic acid with constant stirringuntil a small amount of solid remains undissolved at desired temperature.

11. Pipette out 10 ml of saturate solution in a conical flask from the beaker and titrate this solution against standard0.05 N NaOH solution using phenolphthalein indicator.

Table 1.10: Volumetric titration of benzoic acid solution against standard NaOH solution using phenolphthalein indicator atvarious temperature.

T (C) Volume of acid Volume of NaOH in Volume of NaOH Avg. volume of(in ml) burette (in ml) consumed in burette (ml) NaOH (in ml)

Initial Final

2510.0 0.0 - -

-10.0 0.0 - -10.0 0.0 - -

3010.0 0.0 - -

-10.0 0.0 - -10.0 0.0 - -

12. Determine the Normality of benzoic acid solution for each temperature and calculate solubility of benzoic acidusing eqn 1.14.

Table 1.11: Variation in solubility of C6H5COOH in aqueous solution at various temperatures.

Sr. No. T (in kelvin) 1/T (in × 10−3 K−1) Normality of benzoicacid

Solubility of benzoicacid (moles/kg)

1 303.15 3.298 - -2 308.15 3.245 - -3 313.15 3.193 - -4 318.15 3.143 - -

Further reading:Khouri, S.J. “Titrimetric Study of the Solubility and Dissociation of Benzoic Acid in Water: Effect of Ionic Strengthand Temperature,” Am. J. Analytical Chemistry, 2015, 6, 429-436.

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4. CHEMICAL PREPARATIONS www.apsunda.com

4 Chemical Preparations

Concentration terms

(A) Molarity & Normality:

Molarity (M) is defined as the number of moles of solute per liter of solution.

Molarity (M) =Mass of solute (g)

MolecularWeight of solute (g)× 1000

V olume of solution (ml)(1.15)

Normality (N) =Mass of solute (g)

EquivalentWeight of solute (g)× 1000

V olume of solution (ml)(1.16)

(B) w/v % concentration:

To calculate w/v % concentration:

w/v (%) =Mass of solute (g)

V olume of solution (ml)× 100 (1.17)

Preparation of Acid solution (For e.g. 0.1 N Acetic Acid)

For Preparation of Acetic Acid solution, we need to take Glacial Acetic Acid. In order to prepare 0.1 Normal Aceticacid from Glacial Acetic Acid, we need to know Normality of Glacial Acetic Acid. We need to know the w/v % andspecific gravity (g/ml) written on Glacial Acetic Acid bottle to calculate Normality of Glacial Acetic Acid.

Normality (N) =w

v× Mass of solute (g)

EquivalentWeight of solute (g)× 1000

V olume of solution (ml+)(1.18)

=w

v× specific gravity (g/ml)

EquivalentWeight of solute (g)× 1000 (1.19)

For example if w/v % and specific gravity (g/ml) written on 1 Ltr. Glacial Acetic Acid bottle are 99 % and 1.018(g/ml) respectively then, Normality of the Glacial Acetic Acid:

Normality (N) =99

100× 1.018 g/ml

60.05 g× 1000 (1.20)

= 16.78 (1.21)

To prepare 100 ml of aq. 0.1 N Acetic Acid solution, we have to take standard volumetric flask of 100 ml. We need tofill it first with 20 ml distil water (Always add acid in water!) followed by addition of Glacial acetic acid as calculatedbelow:

N1V 1 = N2V 2 (1.22)

where, N1 = Normality of desired solution i.e. 0.1 NV 1 = Volume of desired solution solution (in ml) for e.g. 100 mlN2 = Normality of Glacial Acetic Acid solution i.e. 16.78 NV 2 = Volume of Glacial Acetic Acid to be taken (in ml)

0.1N × 100 = 16.78N × V 2 (1.23)

V 2 = 0.5959ml (1.24)

Take 0.5959 ml of Glacial Acetic Acid in volumetric flask and swirl it gently. Add distil water further to make uptothe mark of 100 ml.

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www.apsunda.com 5. CLEANING GLASSWARE

5 Cleaning Glassware

Good laboratory technique demands clean glassware. The safest criterion of cleanliness is uniform wetting of thesurface by distilled water. This is especially important in glassware used for measuring the volume of liquids. Greaseand other contaminating materials will prevent the glass from becoming uniformly wetted. This in turn will alterthe volume of residue adhering to the walls of the glass container and thus affect the volume of liquid delivered.Furthermore, in pipets and burets, the meniscus can be distorted and the correct adjustments cannot be made.

5.1 Grease

Grease is best removed by boiling in a weak solution of sodium carbonate. Acetone or any other fat solvent can alsobe used. Strong alkalies (NaOH/KOH) should not be used. Silicon grease is most easily removed by soaking thestopcock plug or barrel for 2 hours in warm decahydronaphthalene.

Special types of precipitates may require removal with nitric acid, aqua regia or fuming sulfuric acid. These arevery corrosive substances and should be used only when required.

1. Remove stoppers and stopcocks when they are not in use. Otherwise they may ‘freeze’ in place.

2. You can degrease ground glass joints by wiping them with a lint-free towel soaked with ether or acetone.

5.2 Safe Use of Chromic Acid

If glassware becomes unduly clouded or dirty or contains coagulated organic matter, it must be cleansed with chromicacid cleaning solution. The dichromate should be handled with extreme care because it is a powerful corrosive andcarcinogen.

5.3 Rinsing

It is imperative that all soaps, detergents and other cleaning fluids be removed from glassware before use. This isespecially important with the detergents, slight traces of which will interfere with serologic and cultural reactions.

After cleaning, rinse the glassware with running tap water. When test tubes, graduates, flasks and similar contain-ers are rinsed with tap water, allow the water to run into and over them for a short time, then partly fill each piecewith water, thoroughly shake and empty at least six times. Pipets and burets are best rinsed by attaching a piece ofrubber tubing to the faucet and then attaching the delivery end of the pipets or burets to a hose, allowing the water torun through them. If the tap water is very hard, it is best to run it through a deionizer before using.

Reading Reference: Aldrich-Suggestions for Cleaning Glassware

5.4 Cryogens

Liquid Nitrogen (LN2), Liquid Argon, Liquid Helium and Solid CO2 (dry ice) are examples of cryogens. Cryogenicchemicals present a safety hazard due to their extreme coldness. Users should be familiar with this hazard and useappropriate cryogen gloves as well as designated personal protective equipment against the freezing effects. A usershould not allow to contact any cryogen with his/her body under any circumstances. Severe injury can result fromsuch contact. All cryogens listed above can displace the oxygen in the air as they evaporate. Therefore, you mustonly use cryogens in well-ventilated rooms and after having performed an analysis of the amount of air that could bedisplaced by the cryogen proposed for use.

Things to remember:

• Follow the Acid-Into-Water rule: AAA- Always Add Acid.

• Do not disturb persons working with chemicals.

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5.4: Cryogens www.apsunda.com

• Respect your colleagues and follow guidelines!!!

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www.apsunda.com 6. VOLUMETRIC ANALYSIS

6 Volumetric Analysis

Principle: Volumetric analysis (titration) is used for determining the content of a specific component in a sample. Theessence of volumetric determination is the chemical reaction between a standard solution (of known concentration),which is usually added, and a given volume of a substance to be determined in a titration flask. When a chemicalreaction proceeds quantitatively, just the equimolar amount of substance reacts and the equivalence point is reached.We identify this point by:- using the indicator (subjective method)- using devices (objective method) eg. potentiometry, conductimetry etc.

6.1 Indicators

Indicators are substances by which we can find out the equivalence point. According to the usage, we divide indicatorsinto these groups:

• Acid-base indicators that react by changing colour depending on the concentration of H+.

• Oxidation-reduction indicators are substances having different colors depending on their different oxidationnumber.

• The precipitation indicators are used to identify the equivalence point by formation of an insoluble substance,mostly of different color than the product of the titration.

• Chelatometric indicators are used to indicate titration based on the formation of chelates with the reagent andindicator

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6.2: Standard Solutions www.apsunda.com

Indicators Colour change (Acid-Base) pH−rangeMethyl orange Red-orange 3.1-4.4Methyl red Red-yellow 4.4-6.0Litmus Red-blue 5.0-8.0Bromothymol blu Yellow-blue 6.0-7.6Phenolphthalein Colourless-pink 8.3-10.0

6.2 Standard Solutions

An ideal standard solution for a titrimetric method will

• be sufficiently stable so that it is necessary to determine its concentration only once;

• react rapidly with analyte so that the time required between additions of agent is minimized.

• react more or less completely with analyte so that satisfactory end points are realized;

• undergo a selective reaction with the analyte that can be described by a balanced equation.

6.3 NaOH Standardization

It is usually impossible to obtain NaOH of sufficient purity to use it as a primary standard. - Why? - Sodium Hy-droxide is hygroscopic (picks up water from the air) - A solution of a approximate molarity will be prepared andstandardized against a primary standard of known purity.

What is a primary standard?- A sample that is of high purity, remain unchanged in air during massing and remain stable during storage,- Reasonable solubility in the titration medium, - Have a high molar mass to reduce massing errors,- React with the solution to be standardized in a direct, well-defined reaction.

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www.apsunda.com 7. RECORD WRITING PROFORMA

7 Record writing proforma

1. Aim or Objective:-If experiment is performed in a group (more then one student), mention roll no. of all the students in the group.

2. Chemicals: (Amount with CAS number of chemical)For e.g.:(i) Sodium Hydroxide (NaOH) : 0.4 g. [CAS:−−−−−−−−−−−−−−−−](ii) Hydrochloric Acid (HCl) : 5.0 ml [CAS:−−−−−−−−−−−−−−−−]

3. Apparatus required:(for glassware, mention volume if applicable)For e.g.:(i) Beaker : 100 ml.(ii) Volumetric flask : 100 ml.(iii) Conical flask : 50 ml.

4. Equipment required:Name of equipment: Make & Model number of equipment

5. Theory:

6. Procedure:

7. Calculations and observations:For e.g.:Table 1: Volumetric titration of hydrochloric acid solution against standard NaOH solution in presence of phe-nolphthalein indicator.

Sr. No. Volume of HCl Sol. (ml) Volume of NaOH Sol. (ml) Final∗ Volume of NaOH Sol.(ml)

Initial Final

1 20.0 0.0 -2 20.0 0.0 - -3 20.0 0.0 -

∗If two subsequent reading are not consistent, then take extra reading. Put consistent value only in Final Vol.

8. Results:

9. Discussion and Error:Calculate standard deviation, if applicable. Discus the results and do complete error analysis.

10. Conclusions:

11. Precautions (if any):

Note:- Each observation table and graph should have caption (title) and numbered accordingly,- Graph paper should not be trimmed from any side for pasting,- Paste the graph in such a way that graph (front side) appears on right page of the record in portrait orientation.

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8. BOOKS FOR FURTHER READING www.apsunda.com

8 Books for further reading

1. Fundamentals of Analytical Chemistry (9th Edition)Douglas A. Skoog and Donald M. West (CENGAGE LEARNING publication, 2014)

2. Aqueous Acid-Base Equilibria and TitrationsR. deLevie (Oxford University Press, 1999)

3. Potentiometry and Potentiometric TitrationsE. P. Serjeant (Wiley, 1984)

4. Electrochemical MethodsA. J. Bard and L. R. Faulkner (Wiley, 2001)

5. Kinetic methods in Analytical ChemistryD Perez-Bendito and M. Silva (Wiley, 1988)

6. Ionic Equilibrium: Solubility and pH calculationsJ. N. Butler (Wiley, 1998)

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Documents

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