BRO Density Determination Manual e

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Manual of Weighing Applications

Part 1

Density


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BK - 07.07.99- VORWOR_E.DOC

Forew ord

In many common areas of application, a weighing instrument or the weight it measures is just a

means to an end: The value that is ac tually sought is ca lculated from the weig ht or mass that wa sdetermined by the weighing instrument. That is why this Manual of Weighing Applications describes

the most widely used weighing applications in a series of separate booklets, with each booklet

representing a complete and independent manual.

Each of these booklets begins with a description of the general and theoretical basis of the

application concerned which in many cases cannot be done without using formulas and

equations taken from the fields of physics and ma thematics. The equations used are explained in

the text, and the intermediate steps necessary for arriving at the results given here are also

included. Some readers may get the impression, at first gla nce, that the manual is just a co llection

of (many) equations, but we do hope that the most important points are familiar to all readers, even

in this context.

Each separate booklet also contains a chapter giving detailed examples of applications, as well as

an index where you can look up keywords to find the information you need.

At the back of each booklet, just before the index, we have included a list of questions on the

subject so that readers can check whether they have understood what they have read and can put

it into practice.

This manual was written to provide sartorius employees and associates, as well as any interested

customers, with a comprehensive compilation of information on the most widely used applications,

both to introduce the subject and to increase existing knowledge in the field, as well as to providea reference source. W e also hope that read ers who make active use of this manual will ga in

confidence in finding solutions when using these applications and gain a feeling for the

possibilities that these weighing applications open up, so they can begin to create custom solutions

where needed.

Contributions from users both in the laboratory and in industry will help ensure that this manual

"lives" and g row s w ith use. W e are especially interested in receiving your app lication reports,

which we would like to include in future editions of the Manual of Weighing Applications, to make

information about new and interesting applications of weighing technology available to all our

readers.

Marketing, Weighing Technology

February 1999


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SymbolsSymbols

IndicesIndicesaa Ai r

f lf l Fluid (flowing system)

ss Solid

(a)(a) Determined in air

(fl)(fl) Determined in a fluid

BB Buoyancy

bb Bulk

tt Total

SymbolSymbol UnitUnit UnitUnitmm Mass mm kgkg

VV Volume VV mm33

AA Area AA mm

FF Force FF N = kgN = kgm/ sm/ s22

W = mW = m gg Weight (force) W = mW = mgg N = kgN = kgm/ sm/ s22

gg Gravitationalacceleration

gg m/ sm/ s22

p = F/ Ap = F/ A Pressure p = F/ Ap = F/ A Pa=N/ mPa=N/ m

rhorhoDensityDensity

rhorho

kg/ mkg/ m33 ; g / cm; g/ cm33

TT Temperature in

Kelvin

TT KK

tt Temperature in

Celsius

tt CC

t=T-273,15 K

gammagamma

Specific gravity (old!) gammagamma

kp/dmkp/dm33

alphaalpha

Linear expansion

coefficient of (of solid s)

alphaalpha

1/K = K1/K = K-1-1

gammagamma

Expansion coefficient (of

liquids or gases)

gammagamma

1/K = K1/K = K-1-1

phiphi

Relative humidity phiphi

%%

pipi

Porosity pipi

Volume-%Volume-%


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Contents:

C O N TEN TS:CO N TEN TS:. .... ..... .... .... .... ..... .... .. . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . .. . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . .. . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . .. .. . . . . . . . . . . . . . . . . . . 11

THE G EN ERAL PRIN C IPLES O F DEN SITY DETERM IN ATIO NTHE G EN ERAL PRIN C IPLES O F DEN SITY DETERM IN ATIO N .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . 33

EXAM PLES OF APPLICA TIO N S FOR DEN SITY DETERM IN ATIO N ......................................................................... 3DEN SITY: A DEFIN ITIO N .................................................................................................................3UN ITS FO R M EASURIN G DEN SITY ........................................................................................................ 4DEPEN DEN CE OF DEN SITY O N TEM PERATURE.......................................................................................... 4THE ARC HIM EDEAN PRIN CIPLE ........................................................................................................... 7

G RAVIM ETRIC M ETHO DS O F DEN SITY DETERM IN ATIO NG RAVIM ETRIC M ETHO DS O F DEN SITY DETERM IN ATIO N .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . .. . . .. . . . . . . . . . . .. . . . . . . . . 1 01 0

DEN SITY DETERM IN ATIO N BASED O N THE ARC HIM EDEAN PRIN C IPLE.............................................................10Buoyancy M ethod.................................................................................................................1 0Displacement M ethod ............................................................................................................ 12Determining the Density of Air..................................................................................................13

DEN SITY DETERM IN ATIO N USIN G PYCN O M ETERS ...................................................................................1 5W eighing a Defined Volume ("W eight per Liter ").. ......... ......... ......... .......... ......... ......... ......... ...... 1 5Pycnometer Method...............................................................................................................16

O THER M ETHO DS O F DEN SITY DETERM IN ATIO NO THER M ETHO DS O F DEN SITY DETERM IN ATIO N . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . .. . . .. . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . 181 8

O SCILLATIO N M ETHO D .................................................................................................................1 8SUSPEN SIO N M ETHO D .................................................................................................................1 8HYDRO M ETERS ........................................................................................................................... 20

PRAC TIC AL APPLIC ATIO N SPRAC TICAL APPLICATIO N S.. ... ... ... ... ... ... ... ... ... .... . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . .. . . .. . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . .. .. . . . . . . . . . .. . . . . . . . . . . .. . . . . . . 212 1

DETERM IN IN G THE DEN SITY O F SO LIDS...............................................................................................2 1

Characteristic Features of Sample Ma terial .......... ......... ......... ......... .......... ......... ......... ......... ...... 2 1Choosing a Density Determination M ethod .......... ......... ......... ......... .......... ......... ......... ......... ...... 2 1Performing Density Determination using the Displacement M ethod .... .... .... .... .... .... .... .... .... .... .... .... .. 2 1

DETERM IN IN G THE DEN SITY O F POROUS SO LIDS....................................................................................2 2Characteristic Features of Sample Ma terial .......... ......... ......... ......... .......... ......... ......... ......... ...... 2 2Choosing a Density Determination M ethod .......... ......... ......... ......... .......... ......... ......... ......... ...... 2 3Performing Density Determination using the Buoyancy M ethod (in Accordance w ith European StandardEN 99 3-1)........................................................................................................................... 23

DETERM IN IN G THE DEN SITY O F PO W DERS AND G RAN ULES .......................................................................25Characteristic Features of Sample Ma terial .......... ......... ......... ......... .......... ......... ......... ......... ...... 2 5Choosing a Density Determination M ethod .......... ......... ......... ......... .......... ......... ......... ......... ...... 2 5

Performing Density Determination using the Pycnometer M ethod (in Accordance w ith German andEuropean Standard DIN EN 72 5-7) ......................................................................................... 26DETERM IN IN G THE DEN SITY O F HO M O G E N O U S LIQUIDS..........................................................................27

Characteristics of Sample Material...........................................................................................27Choosing a Density Determination M ethod .......... ......... ......... ......... .......... ......... ......... ......... ...... 2 7Performing Density Determination using the Buoyancy M ethod .... .... .... .... .... .... .... .... .... .... .... .... .... ... 2 7

DETERM IN IN G THE DEN SITY O F DISPERSIO N S........................................................................................ 28Characteristics of Sample Material...........................................................................................28Choosing a Density Determination M ethod .......... ......... ......... ......... .......... ......... ......... ......... ...... 2 9Performing Density Determination using the Displacement M ethod .... .... .... .... .... .... .... .... .... .... .... .... .. 2 9

ERRO RS IN AN D PRECISIO N O F DEN SITY DETERM IN ATIO NERRO RS IN AN D PREC ISIO N O F DEN SITY DETERM IN ATIO N .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . .. . . . . . 3 13 1

A IRBUOYANCY C O RRECTION ......................................................................................................... 31Displacement M ethod ............................................................................................................ 31Buoyancy M ethod.................................................................................................................3 2


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Pycnometer Method...............................................................................................................32A IRBUOYANCY C O RREC TIO N FO R THE PAN HANGER ASSEMBLY ................................................................32

Buoyancy M ethod.................................................................................................................3 2Displacement M ethod ............................................................................................................ 34

PREVEN TION O F SYSTEM ATIC ERRO RS.................................................................................................3 5

Hydrostatic M ethod...............................................................................................................3 5Pycnometer Method...............................................................................................................36ERRO R CALC ULATIO N ....................................................................................................................3 6

Buoyancy M ethod.................................................................................................................3 7Displacement M ethod ............................................................................................................ 45Pycnometer Method...............................................................................................................46

C O M PARISO N O F DIFFEREN T M ETHO DS O F DEN SITY DETERM IN ATIO NC O M PARISO N O F DIFFEREN T M ETHO DS O F DEN SITY DETERM IN ATIO N ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . 505 0

APPEN DIXAPPEN DIX ...... ..... .... .... .... ..... ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . .. . . .. . . . . . . . . . .. . . . . . . . . 5 45 4

TEM PERATURE DEPEN DEN CY O F DEN SITY.............................................................................................5 4HYDRO STATIC DEN SITY DETERM IN ATIO N ELIMIN ATION O F THE VO LUM ES IN THE EQ UATIO N FO R R.......................55

A IRDEN SITY DETERM IN ATIO N .......................................................................................................... 56A IRBUOYANCY C O RRECTION ......................................................................................................... 57

Q UESTIO N S ABO UT DEN SITYQ UESTIO N S ABO UT DEN SITY ..... ... ... ... ... ... ... ... ... .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . . .. . . . 585 8

TIPS FO R AN SW ERIN G THE Q UESTIO N STIPS FO R AN SW ERIN G THE Q UESTIO N S. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . 606 0

REG ISTERREGISTER................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . .. . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . .. . . .. . . . . . . . . . . .. . . . . . . . . 6 26 2


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The General Principles of Density Determinat ion

Examples of Applications for Density Determination

Density is used in many areas of application to designate certain properties of materials orproducts. In conjunction with other information, the density of a material can provide someindication of possib le causes for alterations in prod uct characteristics. Density determination isamong the most often used gravimetric p rocedures in laboratories.

Density can indicate a change in the composition of a material, or a defect in a product, such asa crack or a bubble in cast parts (known as voids), for instance in sanitary ceramics or in foundriesin the iron and steel industries.

In aluminum foundries, the melt quality is monitored by taking two samples: one under ai r pressureand one under, for example, 8 0 mbar pressure. O nce they have set and cooled , the density of the

samples is determined. The ratio of bo th density values provides information on the purity of themelt.

Determining the density of plastics used in engineering can help to monitor the proportion ofcrystalline phase, because the higher geometric order of crystals makes them denser than the non-crystalline portion. The density of glass is determined by both the chemical composition of thesample and the cooling rate of the melted mass.

W i th porous materials, the density is affected by the quantity of pores, which also determinescertain other qualities of the material; for instance, the frost resistance of roof tiles, or the insulatingproperties of wall materials such as clay and lime malm bricks or porous concrete.

O ne of the factors measured to determine the quality of w ine is know n as the must weight, w hich ismeasured in deg rees chsle this is also a density value, beca use the density is proportional tothe concentration of a substance in the solvent (e.g ., sugar, salt or a lcohol in an aqueous solution).

The density of products also plays an important role in the average weight control of prepackagedproducts, in those cases where a package is filled by weight but must carry a label indicating thecontents in volume.

Density: A Definition

Density (

) is the proportion o f mass (m) to volume in an amount of material. The terms used hereare defined in the G erman Industrial Standa rd (DIN ) 13 0 6 .

=m

VEquation 1

Different fields of industry and technology have also developed various special terms relating todensity:

Normal density the density of gases under normal physical conditions (0C and 1013 hPa) Tap density the density of a powdery material under undefined conditions; for example, in

shipp ing (DIN 3 0 9 0 0 ) or the mass quotient of the uncompressed dry granules in a designatedmeasuring container divided by the volume of the container (DIN EN 1 0 9 7 -3 )


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Apparent density the density of a powder when filled in accordance with the relevantdesignated testing p rocedure; this is important for determining the fill q uantity of pressing molds

Bulk density the mass/ volume ratio that includes the cavities in a p orous material Solid density the mass/ volume ratio of a porous materia l; i.e., excluding the pore volume True density Term still used for "solid density"

Relative density the proportion of a density value to a reference density 0 taken from areference material; relative density is a ratio with the dimension 1

The value forspecific gravityis still sometimes given as well, although it is rarely used today .

= =W

V

m g

VEquation 2

In contrast to the density, this value indicates the weight force in relation to the volume; i.e., thedifference between density and specific gravity is that the calculation of specific gravity includesthe gravitational acceleration (g).

Units for Measuring density

In the International System of Units, the unit for density is kg/ m3; the unit used most often is g/ cm3 this correspo nds to the results in g/ ml. Results are also sometimes given in kg/ dm3 .

1 kg/ m3 = 0.001 g/ cm3 or:

1 g/ cm3 = 1 kg/ dm3 = 1000 kg/ m3

Dependence of Density on Temperature

The density of all solid, liquid and gaseous materials depends on temperature.

Aside from tempera ture, the density of gaseous materia ls also dep ends on pressure. G ases arecompressible at "normal" pressure; this means that air density changes when the air pressurechanges.

The normal density is the density of a gas (or combination of gases) under normal physicalconditions: temperature T = 0 C, pressure p = 1 0 1 .3 2 5 kPa.

A general rule is: The higher the temperature, the lower the density. M aterials expand w henheated; in other words: their volume increases. Therefore the density of materials will decrease as

their volume increases. This is more noticea ble in liquids than in solids, and especially in gases.

The change in density over a certain temperature interval can be calculated using the heatexpansion coefficient; this will yield the change in volume of a material in relation to thetemperature (see Appendix, page xx).

The following diagrams show the density of various substances calculated in relation to thetemperature the x axes show density in intervals of 0 .0 6 g/ cm3 (except in the case of air).

As can be seen from these diagrams, temperature affects some substances more strongly thanothers. For density determination, this means that depending on the required accuracy of

measurement, of course the test temperature must be set very precisely and kept constant.

In hydrostatic density determination methods, for example, it is usually better to use water thanethanol as a liquid for buoyancy; when the temperature increases, for example, from 20C to


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2 1 C, the density of the water only decreases by 0 .0 0 0 2 1 g/ cm3 , where the density of ethanoldecreases by 0.0 00 85 g/ cm3 more than 4 times as much. This means that the temperature hasto be controlled more precisely, or a greater error must be assumed in the results of the densitydetermination using ethanol.

Figure 1: Temperature dependency of density for water and ethanol (above) and for steel and

aluminum (below)

0.9700

0.9800

0.9900

1.0000

1.0100

1.0200

1.0300

10 20 30 40 50

Tempera ture / CTempera ture / C

Density/g/cm

Density/g/cm

Water (cal c. from density (20C) and expansion coeff icient)

Water (PTB-table)

0.7500

0.7600

0.7700

0.7800

0.7900

0.8000

0.8100

10 20 30 40 50

Temperature / CTempera ture / C

Density/g/cm

Density/g/cm

Ethanol

7.8700

7.8800

7.8900

7.9000

7.9100

7.9200

7.9300

10 20 30 40 50


Density/g/cm

Density/g/cm

Steel

2.6700

2.6800

2.6900

2.7000

2.7100

2.7200

2.7300

10 20 30 40 50


Denity/g/cm

Denity/g/cm

Aluminum


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Figure 2: Temperature dependency of glass, polyethylene and air

2.1700

2.1800

2.1900

2.2000

2.2100

2.2200

2.2300

10 20 30 40 50


Density/g/cm

Density/g/cm

Quartz Glass

2.4500

2.4600

2.4700

2.4800

2.4900

2.5000

2.5100

10 20 30 40 50

Temperature / CTemperature / C

Density/g/cm

Density/g/cm

AR-glass (plummet of the Density Determination Kit)

0.9000

0.9100

0.9200

0.9300

0.9400

0.9500

0.9600

10 20 30 40 50


Density/g/cm

Density/g/cm

Polyethylene

0.00110

0.00115

0.00120

0.00125

0.00130

15 20 25

Temperature / CTemperature / C

Density/g/cm

Density/g/cm

Ai r


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The Archimedean Principle

In accordance with the definition of density as = m/ V, in order to determine the density ofmatter, the mass and the volume of the sample must be known.

The determination of masscan be performeddirectly using a w eighing instrument.

The determination of volume generally cannot be performed directly. Excep tions to this ruleinclude

cases where the accuracy is not required to be very high, and measurements performed on geometric bodies, such as cubes, cuboids or cylinders, the volume

of which can easily be determined from dimensions such as length, height and diameter. The volume of a liquid can be measured in a graduated cylinder or in a pipette; the volume of

solids can be determined by immersing the sample in a cylinder filled with water and thenmeasuring the rise in the water level.

Because of the difficulty of determining volume with precision, especially when the sample has ahighly irregular shape, a "detour" is often taken when determining the density, by making use of theArchimedean Principle, which describes the relation between forces (or masses), volumes anddensities of solid samples immersed in liquid:

From everyday experience, everyone is familiar with the effect that an object or body appears tobe lighter than in air just like your own body in a swimming pool.

Figure 3: The force exerted by a body on a spring scale in air (left) and in water (right)

Both the cause of this phenomenon and the correlation between the values determined in itsmeasurement are explained in detail in the following.

A body immersed in water is subjected to stress from all sides simultaneously due to hydrostaticpressure. The horizontal stress is in equilibrium, which is to say that the forces cancel each otherout.

The vertical pressureon the immersed body increases as the depth of the body under the surfaceincreases. The pressure at a certain d epth in liquid exerted by the liquid ab ove that point is ca lledweight pressure. The weight pressure can be ca lculated from the density of the liquid , the height ofthe liquid and the gravitational a cceleration: p = flgh.

The same pressure is exerted on area A at depth h:

F p A g A hfl= = Equation 3


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fl = 1g/ cm3

h1h2

h3

A

A

Figure 4: Gradient of pressure in liquid

F1

F2

fl

x

h

x + h

Figure 5: Buoyancy effects

The weight pressure on the surface of an immersed body with area A causes a force ofF1 = Axflg to be exerted on the upp er surface of the body, and F2 = A(x+h)flg on the lowersurface. The resulting force on the bod y can be ca lculated from the difference between these twoforces:

[ ] [ ]

F F - F

= A (x+h) g - A x g

= A (x+h- x) g

= A h g

res 2 1

fl fl

fl

fl

=

Equation 4

The product of area and height of the bod y is equal to the volume of the body. This volume is inturn equal to the volume of water that is displaced by the immersed body.

(A h) V Vs fl = = Equation 5

Thus the resulting force is

F V g Fres fl fl B= = . Equation 6

This force is known as buoyancy force or simply buoyancy, and it directly includes the value for

volume to b e determined.

with fl = 1 g/cm3 and g 10 m/s2

h p A F

1 1 cm 0.01 N/cm2

1 cm2

0.01 N

2 10 cm 0.1 N/ cm2

1 cm2

0.1 N

3 50 cm 0.5 N/ cm2

1 cm2

0.5 N


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O bserving the ratio o f forces exerted on the immersed b ody and the w ater displaced by the bod y,it can be seen that the forces exerted include both the weight W s, a downward force, andbuoyancy FB, an upw ard force. The resulting force can be calculated from the difference betweenthese two forces: Fres = W s - FB . The buoya ncy FB exerted on the body is equal to the weight W flof the liquid displaced by the body.

WS

FBFB = fl Vfl g

Wfl = mfl g = FB

Figure 6: On the Archimedean Principle: The figure on the left represents the body immersed in liquid;on the right, the liquid element.

The result is

W m g = V gfl fl fl fl= Equation 7

If the body and the liquid element are at equilibrium, the buoyancy FB must, by module, be equalto the weight W fl; thus

F = WB fl . Equation 8

The buoyancy is the result of the level of hydrostatic pressure in a liquid. Buoyancy is inverse to theweight of a body immersed in liquid. This explains why a body seems to be lighter in water than inair. Depending on the ratio of body weight to buoyancy, the immersed body may sink, float or besuspended.

If the buoyancy is less than the weight (FB < W s), the body will sink. In this case, the density of thebody is greater than that of the liquid (s > fl). The wid ely used method o f determining densityaccording to the buoyancy method is usually used under these conditions.

If the buoyancy is equal to the weight (FB = W s), the body remains completely immersed and is

suspended in the liquid. Because both the volume and ma ss of the bo dy are equal to the volumeand mass of the displaced water, it follows that the body and the liquid have the same density.There are a number of density determination procedures that make use of this condition (see Page18).

If the buoyancy is greater than the weight (FB > W s), the body floats; i.e., it rises to the surface ofthe liquid and remains only pa rtia lly immersed. In fact, i t dips so far into the liquid until the weightof that volume of liquid that is displaced is equal to the weight of the body. In this case, thevolume of the displaced liquid is less than the volume of the body (V fl < Vs ) and the density of theliquid is greater than the density of the body fl > s. These are the conditions for densitydetermination using a hydrometer (see page 20).


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Gravimetric Methods of Density Determination

Density Determination Based on the Archimedean Principle

The relationships between themass, the volume and the density of solid bodies immersed in liquidas described by Archimedes form a basis for the determination of the density of substances. Thedifficulty in this method of density determination lies in the precise determination of the volume ofthe sample.

W hen a bod y is completely immersed in liquid, the mode o f procedure demands that the volume ofthe body is equal to the volume of the displaced liquid . Thus w e can derive the following generalequation between the density and mass of a liquid and of a solid , in which the volumes are notexplicitly named (see App endix, page 5 4 for the derivation):

s fl

s

fl

m

m= Equation 9

Accordingly, the unknown density of a solid substance can be determined from the known densityof the liquid for buoyancy and two mass values:

fl sfl

s

m

m=

or

flfl

s

m

V=

Equation 10

Recip rocally, the density of liquids can be determined from one mass value and the known densityof the immersed body.

Simple and precise methods of mass determination can eliminate the need to measure volume.

Hydrostatic b alances and M ohr ba lances are still used in some ca ses for measuring density; theMohr balance, a beam balance has been widely replaced by the use of density sets in conjunctionw ith laboratory balances.

There are two basically different procedures for hydrostatic weighing methods. The actualmeaning of the values displayed on the weighing instrument depends on the mode of procedureused. The buoyancy method(see Figure 7 and Figure 8) entails measuring the weight of the body,which is decreased by buoyancy, while the displacement method (see Figure 9) calls for the directmeasurement of the weight or mass of the displaced fluid.

O ther methods that are b ased on the Archimedean princip le include density determination usinghydrometers (see pag e 2 0 ) as well as various suspension methods (see pag e 1 8 ).

Buoyancy Method

The buoyancy method is often used to determine the density of bodies and liquids. The apparentweight of a body in a liquid, i.e., the weight as reduced by the buoyancy force is measured. Thisvalue is used in combination with the weight in air to calculate the density.


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Weighing pan

Figure 7: Basic procedure for the buoyancy method with below-balance weighing

Liquid in a beaker on a metal platform;no contact with the weighing pan

Weighing pan

Figure 8: Basic procedure for the buoyancy method with a frame for hanging the plummet and abridge for holding the container for liquid

In the procedures illustrated in Figure 7 and Figure 8, the values displayed on the weight readoutindicate the mass of the immersed body as reduced by buoyancy (see also Figure 3).

This means that, in light of the equation s = fl(ms/ mfl), the mass of the body weighed in air isknown: ms = m(a). The mass of the liquid mfl is not directly known, but is yielded by the differencebetween the weights of the body in air (m(a)) and in liquid (m(fl)):

mfl = m(a) - m(fl).

This changes Equation 9 for determining the density of the bodyinto:

s fl(a)

(a) (fl)

m

m m=

. Equation 11

To determine the density of a liquid, mfl is again calculated from the values measured for mass ofthe body in air and in liquid m fl = m(a) - m(fl) and the result used in Equation10. The buoyancymethod maintains the relationship for determination of the density of the liquid:

fl s(a) (fl)

(a)

(a) (fl)

s

m mm

m mV

= = Equation 12


15/65

12

Vs is the known volume of the plummet used to determine the liquid density. Thus the density of asubstance can be determined in two weighing operations.

Displacement Method

The displacement method is another way that the Archimedean Principle is used in determining thedensity of bodies and liquids.

The procedure for the displacement method entails determining the mass of the liquid displaced bythe body. A container of liquid is placed directly on the weighing pan while the body isimmersed. In most cases, the bod y is suspend ed from a hanger assembly.

W hen the bod y is immersed in the liquid, it displaces a volume of liquid Vfl with density fl andmass mfl. The buoyancy force exerted on the body is FB = flVflg = mflg . Because the weig ht ofthe body W s = msg is carried by the hanger assembly and the balance is not loaded, the balancereadout directly indicates the mass of the liquid mfl assuming the weight of the container wastared beforehand.

Figure 9: Basic mode of procedure for the displacement method

This means that, in the case of the displacement method, Equations 9 and 10 (see page 10) canbe di rectly applied for density determination. For the density of a solid:

s fls

fl

m

m=

Equation 13

while for the density of a liquid:

fl sfl

s

fl

s

m

m

m

V= =

Equation 14

If you use a plummet with a known volume Vs, the unknown density fl of a liquid can becalculated from just one measurement.


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13

Determining the Density of Air

To convert a weight value to the true mass value, you must know the value for the air density. Thisvalue can vary over the course of a day by an average of 0 .0 5 mg / cm3 in relation to thenormal density of 1 .2 mg/ cm3 . This is why the air density must be determined at the time the

value is required, with a relative uncertainty factor of < 5

1 0-4

.The air density a depends on the temperature T, the pressure p and the relative humidity of the air. There are various ap proxima tion formulas used to d etermine the density of ai r as dependent onair pressure, temperature and humidity, and even some which consider of the CO 2 -content of theai r (see also "The Funda mentals of W eighing Technology," pp. 4 7 -4 8 .)

It is also possible to determine the density of air(with a 1 % margin of error) with high-resolutionweighing instruments. This is done using 2 calibrated weights that are each made of differentmaterials, with different densities (for example, aluminum and steel).

This determination method is also based on the Archimedean Principle. Because air is made up o fmatter, a b ody in air is subject to buoyancy just as it is in liquid . The same regularities ap ply asthose described in the chapter entitled "The Archimedean Principle", on page 7.

O bserving first in a vacuum an a luminum cylinder w ith a density Al 2 .7 g/ cm3 (Figure 10,

left), we see that it is in equilibrium with a standardized weight of the same mass(N = 8 .0 0 0 g/ cm

3).

GN = mN.. g G Al = mAl

.. g GN = mN.. g

GAl = mAl.. g

FBN = a.. VN

.. gFBAl = a

..

VAl

..

g

Figure 10: The effect of buoyancy on weighing in a vacuum (left) and in air (right)

O bserving the same circumstances in air (Figure 10Figure 10 , right), rather than in a vacuum, we see thatthe aluminum cylinder and the standardized weight are no longer in equilibrium. This is due to thedifference in buoyancy forcescaused by the different material densities and volumes.

To determine what mass mN holds the aluminum cylinder (mAL) in equilibrium in air that has adensity a, all effective forces are observed at equilibrium:

W F W F

m g V g m g V g

N BN Al BAl

N

weight

a N

buoancy

Al

weight

a Al

buoancy

=

= 123 1 24 34 123 1 24 34

Equation 15

Conversion with VN mN

N= and VAl mAl

Al= then yields


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14

m m1

1Al N

a

N

a

Al

=

Equation 16

mN is the weight value W . The weig ht value is in general equal to the read out on the weighing

instrument. The w eig ht value W m 11

Al Al

a

Al

a

N

=

is not constant; rather, it is dependent on the air

density on the w eather, so to speak. This relationship app lies in a similar manner for the steel

cylinder in the weight set for the determination of air density: W m1

1St St

a

St

a

N

=

. From these 2

equations, a relation to the determination of the air density can be derived (see Appendix, page57):

a

Al St St Al

m W m W

m W m W

Al St

Al

St Al

St

=

Equation 17

W St and W Al are the weight values currently measured.

mSt and mAl are calculated according to the following formula, using the conventional mass and thedensities of the certified weights:

m M1

1St St

1.2kg/m

8000kg/m

1.2kg/m

3

3

3

St

=

or m M1

1Al Al

1.2kg/m

8000kg/m

1.2kg/m

3

3

3

Al

=

Equation 18

The conventional mass M of a w eig ht is not the mass of the weig ht itself, b ut rather is equal to themass of the reference weight (standard mass) which under certain defined conditions1 is inequilibrium with the weight being measured.

The air density a can be calculated from the conventional mass values given in the weight set forthe determination of air density for the weights (designated the characteristic values of the weights),the material densitiesof the weights and the current weight values.

A number of Sartorius weighing instruments have the formulas for calculating the air density,

including the values ST = 8 .0 0 0 g/ cm3 and Al = 2 .7 0 0 g/ cm

3 , integrated in their software.The current air density value can be saved and is then used to convert weight values to the actual

masses of the samples weighed, using the formula derived at the beginning of this chapter:

m W1

1v

a

N

a

x

=

.

1

Temperature T = 20 C Density of the standard mass at 20 C: = 8 0 0 0 kg / m3

Air density a = 1 .2 kg / m3


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15

Density Determination using Pycnometers

A pycnometer is a glass or metal container with a precisely determined volume, used fordetermining both the density of liquids and dispersion by simply weighing the defined volume (see

also next chapter), but especially for determination of the density of powders and granules.Pycnometers can also be used in determining the density of the solid phase in a porous solid, butthe sample must first be crushed or ground to the point where all pores are opened.

The pycnometers used in different areas of application have different shapes and standards.During measurement, it is important to make sure that all weighing operations are performed at aconstant temperature and that there is no air trapped either in the liquid or between the sampleparticles.

Figure 11: Different glass pycnometers for density determination in laboratories: The pycnometersmade according to Gay-Lussac, DIN 12 797 (c) and to Hubbard, DIN 12 806 (f) are used fordetermining the density of solids; the volume indicated applies to complete filling after the stopper isinserted. The pycnometers made according to Bingham, DIN 12 807 (b) and to Sprengel, DIN12 800 (d) have a line marking the fill level for the defined volume; the Reischauer, DIN 12 801 (a)and Lipkin, DIN 12 798 (e) pycnometers are marked with scales for checking the fill level.

Weighing a Defined Volume ("Weight per Liter ")

An especially simple gravimetric method for determining the density of flowing substances (liquids,powders, disperse systems) is to weigh a sample with a defined volume. In this case, the sample isplaced in a container that has a defined volume, and the mass of the sample (after taring) isdetermined by weighing. The density can easily determined acco rding to = m/ V.

Different standardized containers are available for this purpose in different branches of industry; forexample, a spherical 1 l-container for determining the density of cast material (slips) in the ceramicindustry. In the lime industry, the tap density of unhydrated lime g ranules is determined using astandardized procedure, in which both the container for the sample and the procedure for filling

the container are precisely defined. US and British standards call for the use of cylindrica l stainlesssteel containers, called specific gravity cups, with various volumes and error margins.

Fill level

Fill level


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16

Pycnometer Method

The pycnometer method is a very precise procedure for determining the density of powders,granules and d ispersions that have poor flow ab ility characteristics. The pycnometer method ismore lab or-intensive a nd far more time-consuming than the b uoyancy and displacement methods.

This method also entails the difficulty of precise volume determination of a powder sample V s fordensity determination of the solid s. The need for explicit determination of the volumeof powdersor granules can generally be avoided by performing 3 weighing operations and using anauxiliary liquid with a known density.

ss

s

m

V= Equation 19

Vgesm1flfl

Vfl ,

m2fl

,fl

Vs, ms, s

Figure 12: Pycnometer with contents

The volume of the solid Vs can only be determined indirectly:

V V Vs ges fl= Equation 20

The procedure is as follows: First the pycnometer is completely filled with liquid, and the mass of the liquid in the

pycnometer determined. O nce this value has been determined, the volume of thepycnometer Vges is known.

Vm

ges1fl

fl

=

Equation 21

N ext (after the pycnometer is emptied, cleaned, d ried and brought to the requiredtemperature) the pycnometer is filled to about 2/ 3 w ith sample material; this yields the mass

of the powder ms. The next step is to fill the pycnometer the rest of the way with liquid and weigh it again,

which gives the combined mass of the sample with the liquidm(fl+s).

The mass of the liquid m2 fl can be calculated from this data

m m m2fl (fl s) s= + Equation 22

which also yields the volume Vfl of the liquid in the pycnometer filled with water and liquid

V

m m -mfl

2fl

fl

(fl+s) s

fl= = . Equation 23


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17

The volume of the powdered sample Vs, the value actually sought, is yielded by the differencebetween the total vo lume Vges and the volume of the liquid Vfl .

V V Vs ges fl=

V m m -ms1fl

fl

(fl+s) s

fl

=

Equation 24

Using the volume Vs in the orig inal equation s = ms/ Vs results in the following conversion2

s fls

1fl (fl s) s

m

m - m m=

++or

s fl2

1 2 3

m

m m m=

+ Equation 25

with the masses m1 , m2 and m3 in the order of the procedural steps:

m1 mass of the liquid in the pycnometer filled completely w ith liquid

m2 mass of the sample materialm3 mass of the sample and liquid together in the pycnometer

Thus this procedure represents another method for determining density "via detours," i.e. using aseries of mass determination measurements.

2

ss

s

s

m m ms

m -(m m )fl s

1fl (fl s) s

m

V

m m m

m -(m m )1flfl

(f l s) s

fl

1fl (fl s) s

fl

= =

= =

+ + +


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18

Other Methods of Density Determination

There are also other methods of density determination that are based on the ArchimedeanPrinciple. The density of air ca n be determined using tw o solid b od ies of different densities (e.g .,

2 weights made of different metals).Density can also be determined by rad ioa ctive absorption by the material being tested. Theabsorption of the radiation will depend on the mass absorption coefficient, thickness of layers anddensity of the material. O nce the mass ab sorption co efficient and thickness as well as the physicalinterrelationships are known, the density of the substance can be calculated.

O n magnetic samples, the magnetic fo rces can also be utilized in density determination o f solid s orliquids.

Oscillation Method

The oscilla tion method is w idely used to determine the density of homogenous liquids. Thisprocedure is not suitable for use with suspensions or emulsions which, because they are made upof different phases, could separate.

The sample being tested is placed in a measuring chamber (usually a U-shaped glass tube) andmechanica lly vibrated. Calculation of the density uses the physical interrelationship between thefrequency of the oscillation and the mass of the oscillation channel (the U-shaped tube with thesample). The equipment must be ca libra ted using liquids that have a known density and a viscositysimilar to that of the sample material.

Suspension MethodThe suspension method makes use of the Archimedean Principle in the special case of suspensionin which the densities of the liquid and of the suspended solid are equal.

The density of the solid body can be determined by setting the density of the test liquid so that thesample bo dy reaches a state of suspension. The density setting o f the test liquid ca n also beachieved by mixing two liquids of different densities; the density of the sample is then determinedfrom the proportions of the liquids mixed, or with the oscillation method (see page 18) or thedisplacement method.

Density Gradient Column

W ith a density gradient column, two liquids of di fferent densities are layered in a g lass tube so thatover time, diffusion results in a vertical density gradient (a continuous change of the densitythroughout the height of the co lumn). Small solids of various densities are then suspended atvarious heights, w ith each height indicating a pa rticular density . Colored g lass bea ds of know ndensities are available for calibration.

In addition to the density of small bodies (such as fibers, powder particles, small pieces of metal orplastic foil) this method can also be used to determine the density of drops of liquidof course theliquid tested should be insoluble in the test liquid.


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19

Schlieren Method

If you fill a capillary tube with liquid and hold it horizontally immersed in another liquid, the liquidw ill only flow horizontally from the tube if the densities of the tw o liq uids are equal. If the density ofthe liquid flowing from the tube is lower or higher than that of the liquid in which the tube is

immersed, schlieren (streaks) will form, flowing upward in the former case and downward in thelatter.


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20

Hydrometers

Hydrometers, also known as spindles, are simple measuring instruments for determining the densityof liquid s or dispersions. They are forms of plummets that float on the surface and then sink to a

certain level, depending on the density of the liquid. The density of the liquid can be d eterminedfrom the depth the plummet sinks (from the volume of the displaced liquid) by comparing the heightof the liquid in the container to the sca le marked on the hydrometer. For certain ap plications, thereare also hydrometers that show the concentration of a given substance in an aqueous solution; forinstance; sugar (in a saccharimeter), alcohol (in an alcoholometer), battery acid or anti-freeze.

Figure 13: Special hydrometer with integrated thermometer in accordance with DIN 10 290 fordetermining the density of milk and skimmed milk the density is dependent on the fat content of themilk.

Thermometer scale

Thermometerscale

Hydrometer

scale

Thermometercapillary

W eightingdevice

Thermometer filling

6.25Hydrometer scale

Thermohydrometer for use wi th milk and skimmed milkThermohydrometer for use w ith milk and skimmed milk


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21

Practical Applications

Determining the Density of Solids

Characteristic Features of Sample Material

Solid bod ies retain their volume and shape under atmospheric pressure. Examp les of solids forw hich it can be useful to know the density include metals, glass and pla stics. These solids may bemade up of only one or many phases; one phase may also be embedded in another (for instance,in fiberglass-reinforced plastic) or the different phases may be interlocking, such as the many smallcrystals in a homogenous metallic material.

An important factor in choosing a suitable sample for density determination is the question ofwhether the density is required as a characteristic of a material or whether density determination isperformed to check for defects in a material. The choice of procedure for density determination

will depend on this factor as well.

Choosing a Density Determination Method

The best procedures for density determination on solids are the buoyancy and displacementmethods, both of which are ba sed on the Archimedea n Princip le. Prerequisite for these methods isthe use of a liquid for buoyancy that does not react with the sample material, but wets itthoroughly.

The suspension method, for example, is w idely used in the gla ss processing industry. G lasssamples are placed in an orga nic liquid in which they float at room temperature. Because the

density of the liquid is 100 times more temperature-dependent than that of glass, the glass can bebrought to the point where it is suspended within the liquid by slowly increasing the temperature inthe test system; in this way, the density of the glass can be determined.

Performing Density Determination using the Displacement Method

Equipment Required

W eighing instrument Thermometer Stand with holding device for samples

Beaker with liquid for buoyancy that has a known density distilled water for all materials thatdo not react with water

Preparation of the Sample, Testing Procedure and Evaluation

The beaker is placed on the pan of the balance and the sample-holding device is immersed in theliquid , to the same dep th that it w ill later be immersed w ith the sample on it. The weighinginstrument is tared.

The sample is placed next to the bea ker on the weighing pan. The mass of the sample in ai r m(a)is determined.

The sample is placed in the holding d evice on the stand and immersed in the liquid . The weightreadout shows the mass of the displaced liquid mfl.


25/65

22

The density of the sample is calculated according to = fl(a)

fl

m

m.

Determining the Density of Porous Solids


There are a number of terms used in connection with the density of porous materials, such as soliddensity, true density, bulk density, apparent porosity, open porosity and closed porosity.

Porous solids consist of one or more solid phases and pores. Pores are cavities filled with air (orother ga s). These openings are found ei ther between ind ividual crystals in the solid materia l, o r asga s bubb les in glass phases; i .e. , solidi fied in a non-crystalline form. Thus there are basica lly tw oforms of pores: open and closed. Among op en pores, in turn, there are also di fferent types; forinstance, there are pores through which liquid may flow , and saturab le po res. W ith these

designations, the type of soaking medium and other conditions must be given (e.g., water at atemperature of 22C and pressure of 2500 Pa).

The term "po re" is used for op enings or gap s from 1 nm to 1 mm. O penings larger than 1 mm arereferred to as cracks or void s; those under 1 nm are defects in the crystal la ttice. Pores are animportant element in the micro structure of many materials; the quantity, type, shape, orientation,size and size distribution of pores significantly affect many important characteristics of a material;for example, the frost-resistance of roof tiles, or insulating properties of lime malm bricks or porousconcrete. Such characteristics as mechanical solidi ty and co rrosion resistance a re also a ffected b ypores in a material.

Figure 14: Microstructure of a porcelain plate. Magnification: approximately 80x. Left: porcelainwith irregular pores between phases; right: glaze layer melted during firing, with closed sphericalpores (bubbles); extreme right: synthetic resin as embedding medium for the polishing preparation

The density determination procedure will depend on whether the value sought is the density of thesolid matter only or the density of the material including pores; after all, it may be important toknow the porosity of the material.

The density of the solid material (not the solid body), which used to be termed the "true density" isnow simply referred to as "density": t = m / Vsolid . The po res are not included in this

measurement."Bulk density" is the quotient of mass and total volume of a sample: b = m / Vb . The bulk densityis an average of the density of the solid and the gas found in the pores.


26/65

23

"O pen p orosity" is the ratio o f the volume of op en po res to the total volume of the porous body, inpercent: a = Va / Vb .

"Closed porosity" is the ratio of the volume of closed pores to the total volume of the porous body,in percent: f = Vf / Vb .

The apparent porosity is the ratio of the volume of all pores to the total volume of the material, inpercent: t = Vt / Vb . The app arent po rosity is the sum of open and closed p orosity: t = a + f.


Both the buoyancy and the displacement methods are suitable for density determination on poroussolids. The solid density can also be determined using the pycnometer method, by first grinding thesample until the grain size is roughly equal to the pore size.

To determine the bulk density of porous material, the sample can also be covered in a wax or latexcoating or layer to prevent liquid from entering open pores (see for example the German IndustryStandard (DIN ) 2 7 3 8 ). Density determination can then be performed using the buoya ncy method.

Performing Density Determination using the Buoyancy Method (in Accordance with EuropeanStandard EN 993-1)

Figure 15: Mode of procedure for density determination using the buoyancy method and the DensityDetermination Set from sartorius

Equipment Required

Drying oven; temperature: 1 1 0 5 C W eighing instrument: margin of error: 0.0 1 g Frame to be p laced over the weighing pa n: included w ith the Density Determination Set

Vacuum generator w ith ad justable p ressure a nd p ressure g auge Thermometer with an error margin of 1 C Liquid for saturation distilled water for all materials that do not react with water


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24

Dessicator.

Preparation of the Sample

Shape and size (total volume between 50 cm3 and 20 0 cm3) of the sample, as well as the numberof samples to be tested, are defined in the Standard.

Testing Procedure and Evaluation

The sample is first dried in the drying oven until it reaches a constant mass and then cooled to roomtemperate in the dessicator. The mass of the sample is then determined in air using the weig hinginstrument. m1

The sample is then evacuated under precisely defined conditions and saturated (in the vacuum) untilthe op en pores as stated in the test specifica tions are filled w ith the saturation liquid . Theapparent mass of the saturated sample is then determined using a hydrostatic balance (or using theDensity Determination Set). The samp le must be completely immersed in a b eaker filled w ith thesaturation fluid for buoyancy. m2

The temperature of the saturation liquid must be determined.

Then themass of the saturated sample is determined by weighing in air. Liquid that remains on thesurface of the sample must be removed with a da mp sponge before weighing . The weig hingoperation must be performed quickly, to avoid loss of mass due to evaporation. m3

The density of the saturation liquid fl must be measured or taken from a table of density values at

defined temperatures.

The bulk density b in g / cm3 is calculated as follows:

b1

3 2

f l

m

m m=

Equation 26

The open porosity a in volume percent is calculated as follows:

a3 1

3 2

m m

m m100=

Equation 27

The apparent porosity t is calculated as follows:

t

t b

t

100=

Equation 28

The apparent porosity is the sum of open and closed porosity (t = a + f); thus it follows that forthe closed porosity f:

f t a= Equation 29

m1 M ass of the dried samplem2 Apparent mass of the saturated sample weighed in liquidm3 M ass of the saturated sample w eighed in ai r


28/65

25

rt Density of the solid , determined accord ing to EN 9 9 3 -2 (or calculated from thecomposition)

rfl Density of the fluid for buoyancyrb Bulk density of the sample

O ne of the numeric values often g iven for ceramics for the open po rosity, in a dd ition to the values listedabove, is the water absorption. The wa ter ab sorption in percent is yielded b y the difference in massbetween the saturated sample and the dried sample, relative to the mass of the dried sample. The resultingfigure can be used in dividing ceramics into "dense" and "porous" grades.

Additional information about the type and size distribution of the pores can b e ga ined using a mercurypo rosimeter: The porous samp les are put under pressure w ith mercury, w hereby the pressure is increa sed atcertain stages so that, depending on the pore diameters, a certain number of the pores are filled withmercury. This can yield information on the prop ortion and d iameter of open po res that are accessible fromthe outside.

Another method for determining number, shape and size of pores is image analysis, the quantitative

statistical evaluation of polished sections under a microscope, similar to the image shown in Figure 14 .

Determining the Density of Powders and Granules


The term powder refers to "a heap of particles, usually with dimensions smaller than 1 mm."

Granules are larger particles than those that make up a powder. The term "granules" has differentmeanings in different areas of application:

M aterial made up of "secondary" particles, w hich in turn are mad e up of agg lomeratedparticles of a fine powder; or

M aterial that w as heated to the melting p oint and then cooled very quickly, causing i t to takeon a teardrop shape; for example, intermediate products in the plastics or porcelain enamelindustries.


The pycnometer method is the only method that can be used with powders or granules.

Performing Density Determination using the Pycnometer Method (in Accordance with Germanand European Standard DIN EN 725-7)

Figure 16: Pycnometer with integrated thermometer


29/65

26

Equipment Required

Distilled water and another liquid, such as ethanol Pycnometer with thermometer and sidearm with polished glass stoppers

W ater bath Vacuum pump W eighing instrument; error margin: 0 .0 0 0 1 g

Testing Procedure and Evaluation

The pycnometer must be carefully cleaned and dried; it is then filled with distilled water, evacuatedunder precisely defined conditions, and brought to within 0.1 K of the required temperature in aw ater bath. Then the pycnometer is filled . The volume of the pycnometer is ca lculated from themass of the water at the test temperature: Vpycnometer = mwater / water .

The pycnometer is then dried, filled with ethanol and weighed, following the same procedure asthat describ ed abo ve. The density of the ethanol at the test temperature can be ca lculated from themass of the ethanol and the volume of the pycnometer:

ethanol = methanol / Vpycnometer .

methanol = m1.

O nce the pycnometer has been cleaned and dried aga in3, it is loaded with about 10 g (at adensity between 2.5 and 4 g/ cm3 ) of the powder, which has been dried at a temperature 10 Kbelow the decomposition point of the pow der. m2

Enough ethanol is now added to the pycnometer to wet the powder; the pycnometer is thenevacuated and shook to release as many ai r bubbles as possible. M ore ethanol is ad ded to thepycnometer; after this has been heated to the test temperature, the pycnometer is completely filled.The total mass of powder and ethanol is determined. m3

The data is evaluated using the formula = + ethanol

2

1 2 3

m

m m m, the density of the sample

material is given w ith a precision of 0 .0 0 1 g/ cm3 .

Determining the Density of Homogenous Liquids

Characteristics of Sample Material

Homogenous liquids are relatively simple systems; unlike dispersions, they are always single-phasesystems. W hen substances are mixed, such as w ater wi th alco hol or sugar wi th w ater, this isreferred to as the solution of one substances in the other. A genuine solution is clear, the particlesof the dissolved substance are present as molecules or ions in the solution.

3 The latest weighing instruments from sartorius come equipped with density determinationsoftware that eliminates with time-consuming steps for drying in the drying oven and cooling in thedessicator; the program has a second tare memory which can be used to tare the weight of waterremaining in the pycnometer. This considerab ly simplifies wo rk in the laboratory.


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27

The density of a solution depends on the concentration of the dissolved substance; in other words,when the interrelationships are known, the density value can be used to derive the concentration ofthe solution.

At 20 C most fluids have a density betw een 60 0 kg/ m3 a nd 2 0 0 0 kg / m3 o r 0 .6 g/ cm3 to

2 .0 g/ cm3

. The density of fluids is far more tempera ture-depend ent than that of solids. This meansthat the temperature must alwaysbe monitored carefully and, if necessary, the sample heated orcoo led to the required temperature.


There are several methods that can be used to determine the density of liquids, including thehydrometer, pycnometer, oscillator, buoyancy and displacement methods. The choice of methoddepends, among other things, on the degree of precision required and the amount of samplematerial avai lable.

Performing Density Determination using the Buoyancy Method

Figure 17: Determining the density of a liquid using the buoyancy method

Equipment Required

W eighing instrument

Density Determination Set Plummet with known volume (10 cm3 in the sartorius Density Determination Set; see Figure 17) Thermometer In some cases: w ater bath for ad justing the tempera ture of the samp le

Preparation of the Sample, Test Procedure and Evaluation

Position an empty beaker on the bridge and hang the plummet from the frame provided in theDensity Determination Set.

Tare the weighing instrument with the plummet.Fill the beaker with the liquid to be tested up to a level 10 mm higher than the plummet.


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28

The negative value shown on the weight readout corresponds to the buoyancy of the plummet inthe liquid.

The density of the liquid is calculated by dividing the measured value by the volume of the plummet

=m

V

fl

TK

.

Determining the Density of Dispersions

Characteristics of Sample Material

Disperse systems or dispersions are combinations of two or more phases, each of which is insolublein the other(s). O ne phase, ca lled the dispersion medium, is alw ays contiguous, w hile the otherphase or phases are present in the medium in the form of finely distributed isolated particles.

In a colloid dispersion, the particles are generally between 1 m and 1 nm in size. If the particles

are larger than > 1 m, this is referred to as a coarse dispersion; if particles are smaller than< 1 nm, it is a molecular dispersion.

There are many examples of dispersions, because"dispersion" is the generic term for all systems,independent of the state of the phases. Different types of dispersions include:

Suspensions M ixtures of solid particles in a liquidExamples: "Dispersion" paint, ceramic slips, abrasive liquid c leanser, too thpa ste,ink, etc.

Emulsions M ixtures of two liquids that are mutually insoluble, w here one is present in theform of finely distributed minuscule drops in the otherExamples: Cremes, lo tions, mayonnaise, milk, the classic o il-and-vinegar salad

dressing, etc. Foams M ixtures of ga s bubbles in a liquid (or a solid ) M ist M ixtures of small drops of liquid in a g as phase Smoke M ixtures of solid pa rticles in a gas phase

The term "stability" in reference to dispersion is somewhat problematic, because these are actuallyunstab le systems. This can be seen in their tendency to separate. The terms "stable suspension"and "stable emulsion" are often used to refer to systems that remain constant over a certain periodof time.


M any o f the same methods used on liq uids or solids can a lso b e used for determining the densityof dispersions. The best choice for a given sample materia l w ill depend on the consistency of thesample.

The oscillation method is not well-suited for use here, for a number of reasons. The many phaseboundaries are a disadvantage; the viscosity has an effect on the measurement, and the vibrationduring measurement can promote separation of the phases, which means the values obtained willnot be representative of the overall averag e.

Hydrometers can be used, but it must be ensured that the suspension or emulsion does not show

signs of separating.


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The pycnometer can also be used on dispersions, just as with density determination on liquids orpow ders. Different containers are used in different branches of industry; the containers may befilled to the rim or up to a marking w ith the sample material and w eighed. The potentia lseparation o f the sample phases during measurement is not a p roblem w ith this proced ure.

The buoya ncy or displa cement method ca n be used for many suspensions and emulsions. In thiscase, too, care must be taken to avoid separation of the sample phases; the flow behavior of thesuspension must also allow the plummet to sink quickly.

Performing Density Determination using the Displacement Method

Figure 18: Density determination using the displacement method, with a gamma sphere as a plummet,affixed to a holder mounted next to the weighing instrument

Equipment Required

W eighing Instrument

Plummet with known volume Holder Thermometer In some cases: water bath for adjusting the temperature of the sample

Preparation of the Sample, Test Procedure and Evaluation

Bring the sample to the required temperature and p lace i t in a b eaker. Place the beaker on thew eig hing p an and tare the w eig hing instrument. Immerse the plummet in the test substance up tothe marking.

The weight reado ut show s the mass of the displaced liquid directly (see Pag e 1 2 ). The density of

the sample is determined by dividing the measured value by the volume of the plummet =m

Vfl

TK

.


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Errors in and Precision of Density Determinat ion

In the two previous chapters (Fehler! Verweisquelle konnte nicht gefunden werden.Fehler! Verweisquelle konnte nicht gefunden werden. and Fehler!Fehler!Verweisquelle konnte nicht gefunden werden.Verweisquelle konnte nicht gefunden werden. ) the fundamentals of these two hydrostatic density

determination methods were explained and the formulas for calculation of the density werederived.

If a high degree of precision is required, the existing cond itions must be allow ed for. Technicallyspeaking, the weighing instrument does not show the mass of the samplesthe value given in theequationsbut rather the weight value for the samp les in air. For more precise results, use theweight values obtained after air buoyancy correction.

W hen using the buoyancy method, the immersion level of the pan hanger assembly is affectedwhen the sample is immersed, which producesadditional buoyancy. This must also be consideredin more p recise ca lculations.

In general, density determination procedures require very careful work; it is especially important tomake sure the temperature remains constant during testing.

Bubbles entering the liquid when the sample is immersed can also affect results; bubbles adheringto the test piece will distort the measurement results.

Air Buoyancy Correction

For high-precision density determination, it is important to note that the weighing instrument doesnot directly determine the mass of the sample, but its weight value. This value is dependent on theair density, which in turn depends on the pressure and temperature and must be corrected by thevalue for air buoyancy.

Betw een the mass of a solid b ody m and its w eight value in air W ; that is, under consideration ofthe air buoyancy on the sample, the following relationship is generally valid (G = density of thestandard):

m W1

1v

a

G

a=

. Equation 30

Displacement Method

If you include this equation in the calculation of the density when the displacement method is used,the equation for determining the density of the solid body follows with allowance for the airbuoyancy(see Appendix, page 58, for derivation).

s fls

fl

fl as

fl

a

m

m( - )

W

W= = + Equation 31

This is the manner in which density is calculated by the software that comes with sartoriusweighing instruments.


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Buoyancy Method

If you include the equation m W1

1v

a

G

a=

for m(a) and m(fl) in the equation for density

determination using the buoyancy method s fl (a)(a) (fl)

m

m m= , mathematical conversion results in

s fl a(a)

(a) (fl)

a( )W

W W=

+ . Equation 32

the formula for calculating the density of solid bodies with allowance for air buoyancy.

This is the manner in which the air buoyancy is accounted for in the software that comes withsartorius weighing instruments (for details, see the operating instructions for the instrument in

question). M oreover, another correction factor is included in the calculation w hich takes intoaccount the additional buoyancy caused by the immersion of the wires on the pan hangerassembly (see below).

Pycnometer Method

The formula for calculating density with the air buoyancy correctionusing the pycnometer methodis:

( ) s fl a2

1 2 3

a

W

W W W=

+ + . Equation 33

This is the formula used by the software that comes with sartoriusweighing instruments.

Air Buoyancy Correction for the Pan Hanger Assembly

Buoyancy Method

For high-precision measurement, another consideration besides the air buoyancy is the additionalbuoyancy of the pan hanger wires, caused by the fact that the height of the liquid is increasedwhenthe sample is immersedwhich means the wires are deeper under the surface than without thesample. (The buoyancy of the pan hanger assembly is not included in the density calculation if theweighing instrument is tared with the pan hanger assembly.)

h

d

D

Figure 19: Diagram illustrating the calculation of the buoyancy caused by increased height of liquidwhen sample is immersed

W ith the procedure for using the buoyancy method w ith the Density Determination Set, the w ires ofthe pan hanger assembly are deeper under the surface of the water when the sample is immersed


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beca use the volume of the sample causes more liquid to be displaced. Because more of the w ire isimmersed, it causes more buoyancy; the additional buoyancy can be calculated and corrected forin the results.

The amount of volume by which the liquid increases in a container with diameter D corresponds to

the volume Vfl in Figure 1

V D4

hfl= . Equation 34

The volume Vs of the sample is

Vm m

s

(a) (fl)

fl

=

Equation 35

These two volumes are identical, Vfl = Vs. Using the above equations and solving for h, the

increase in height of the liquid, yields

h(m m ) 4

D

(a) (fl)

fl

=

Equation 36

The buoyancy force FBD exerted on two wiresof diameter d at the fluid level h is

F V g (2 h) gBD fl D fld4= = Equation 37

and with the value for h included yields

F 2 d4

(m m )4D

gBD fl(a) (fl)

fl

=

F2 d (m m ) 4 g

4 D

2d

D(m m ) g

BD

fl2

(a ) (fl)

fl2

2

2 (a ) (fl)

=

=

Equation 38

This means that the buoyancy caused by the wires is proportional to the relation between thediameters of wires and beaker.

In addition to the buoyancy of the sample which is to be determined the measured value alsoincludes part of the "wire buoyancy;" thus the wire buoyancy must be subtracted from the measuredvalue to yield the buoyancy of the sample alone FBS(corr):

[ ]F (corr) (m m ) (2 dD m m ) gBS (a) (fl) (a) (fl)=

F (corr) (1 2 dD

) (m m ) gBS

correction factor

(a) (fl)= 1 24 34

Equation 39

W hen the diameters of the w ires and the beaker are know n, the factor by which the valuemeasured for buoya ncy must be multiplied can be ca lculated. In the Density Determination Set


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from sartorius, the wire diameter is d = 0.7 mm, the beaker diameter D = 76 mm, and the panhang er assembly has two w ires. Thus the correction factor is:

Corr 1 2 dD

1 2 0.776

0.99983= = = .

The smaller the diameter d of the wires, the larger the diameter D of the beaker and the fewerwires on the pan hanger assembly, the nearer the correction factor will be to 1; i.e., the correctionis neglig ible. These cond itions are easy to crea te when performing density determination using thebelow -scale or b elow-ba lance w eighing method.

Returning to the formula for density determination using the buoya ncy method: For the buoyancy

correction for the wires, the equation s fl(a)

(a) (fl)

m

m m=

yields

s fl(a)

(a) (fl)

m

[m m ] Corr

=

or, when the air buoyancy is also accounted for,

s fl a(a)

(a) (fl)

a( )W

(W W ) Corr=

+ Equation 40

This is the formula used for density determination in the software that comes with sartoriusweighing instruments. The weighing instruments can work with the preset values for a standard airdensity of a = 0 .0 0 1 2 g/ cm

3 and a correction factor of 0.99983 for the sartorius DensityDetermination Set. Alternatively, user-defined correction factors can be entered.

Displacement Method

Errors caused by the additional buoyancy when the pan hanger assembly for the sample or lines orwires of the pan hanger can be eliminated at the outset by immersing the pan hanger assembly justas deep in the liquid when it is weighed empty or tared as it will be later with the sample on it.

In addition, the test conditions can be arranged to make the correction factor 1; i.e., by usinglarge container diameter, small pan hanger assembly diameter, only one pan assembly if possible.The equation for calculation of the solid body density using the displacement method withallowance for air buoyancy and the correction factor for the sample hanger is:

s fl a(a)

(fl)

a( )W

W Corr=

+ Equation 41

This is the formula used by the software that comes with sartorius weighing instruments. The defaultsettings in this software include a numerical value of 1.0 for the correction factor andra = 0 .0 0 1 2 g/ cm3 for the standard air density. User-defined values can also be entered.

W hen the displacement method for density determination on liquids is used, including on paintsand varnishes, the plummet is usually made of metal (gamma sphere) and is tapered in one section

(see Figure 20); the nominal volume of the plummet is calculated to the middle of the taperedpo rtion. There are different sizes of plummets ava ilable for use with di fferent degrees of surface


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tension in the liquid being tested. The influence of the bulge of liquid around the stem of theplummet is also accounted for.

Figure 20: Plummet (in accordance with DIN 53 217 Part 3) for determining the density of paints,varnishes, and similar coating materials using the displacement method.

Prevention of Systematic Errors

Hydrostatic Method

To limit the errors in density determination with different hydrostatic procedures, the followingshould be observed:

The temperature must be kept constant throughout the entire experimental proced ure. W ithwater as buoyancy medium, for example, a temperature variation of 0.1C changes the density

b y 0 .0 0 0 0 2 to 0 .0 0 0 0 3 g / cm3 ; w ith alcohol, by 0 .0 0 0 1 g/ cm3 . The measuring instrument should be loaded exactly in the center, to maximally limit off-center

loading errors. W hen w eighing in air using the sartorius Density Determination Set with thesample on top of the frame, eccentric positioning of the sample due to the shape of the framecauses force to be conducted to two external points on the base of the frame resulting in agreater torque than with the use of "normal" weighing pans with the same deviation from thecenter.

After submersion in the liquid there must be no air bubbles on the sample or on the pan hangerassembly. These cause an add itional buoya ncy and falsify the measured weight. To preventthis, one can wet the sample in a separate beaker or in an ultrasound bath.

Errors due to adhesion of liquid to the wire of the pan hanger assembly can be prevented bytaring the w eighing instrument w ith submersed pa n hanger assembly before measuring.

Area for labelingLabel a ccording to section 31

d = 3mm w ith a 1 00 -ml plummetd = 1mm with a 10-ml plummet

100 ml or 10 ml up to the middle of thetapering section.

10 to 12 mm


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Air buoyancy causes an error of density of 0 .0 0 1 2 g/ cm3 (corresponding w ith air densityunder normal conditions); therefore in calculating the density, the equation should take airbuoyancy into account (see p. 31, "Air Buoyancy Correction").

After submersing the sample in the container, the level of the liquid rises so that the wires of thepan hanger assembly cause additional buoyancy. Depending on the diameter of the beaker

used and the number of wires of the assembly, this additional buoyancy can be corrected (seep. 3 2 Air Buoyancy Correction for the Pan Hanger Assembly.)

Pycnometer Method

To limit the errors in determining density with the pycnometer method, the following should beobserved:

The temperature must be kept constantthroughout the entire experimental procedure; thetemperature of the samples must be carefully controlled. W ith wa ter as the liquid med ium forexample, a temperature change of 0.1C changes the density by 0.00002 to0 . 00 0 0 3 g/ cm3 ; w ith alcohol, by 0 . 00 0 1 g/ cm3 .

There should be no air bubbles in the liquid medium or on the sample. The air buoyancy causes an density error of 0 .0 0 1 2 g/ cm3 (corresponding to air density

under normal conditions); therefore in calculating the density, the equation should take airbuoyancy into account (see p. 33, "Air Buoyancy Correction").

W hen proper ca re is exercised using the pycnometer, this method can be used for hig hly accuratedetermination of the density of materials.

Error Calculation

W ith careful work and w ith the prevention o f the above-mentioned systematic errors, the errors ofdensity determination ca n be ca lculated accord ing to the rules of error reprod uction. The densityerror is based mainly on measuring results in mass determination.

Generally, for the determination of the total errors F of a dimension that is calculated from severalmeasuring values:

W ith the sum (and the difference) the absolute single errors add up quadratically:

F F F ...12

2

2= + + Equation 42

W ith products (and quotients) the relative single errors ad d up quadratica lly. (The relative error isthe absolute error in relation to the measured value.):

FF

F

F

F

F...1

1

2

2

2

2

=

+

+ Equation 43

Buoyancy Method

For the density determination of solid bodies using the buoyancy method the following relationship

is applied s fl(a)

(a) (fl)

m

m m=

or s fl a(a)

(a) (fl)

a( )W

(W W ) Korr

=

+


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Because the correction factor for the pan hanger assembly and the density of air have nonoticeable effect on the error of density, they do not need to be regarded in error calculation.

If one uses the basic rules of error calculation, the absolute error of the denominator [m(a)-m(fl)] iscalculated next:

[ ] m m m m(a) (fl) (a)2

(fl)

2 = + Equation 44

followed by the total relative error of density /

=

+

+

fl

fl

2

(a)

(a)

2

(a) (fl)

(fl)

2m

m

(m m )

mEquation 45

The total error of weighing in air (m(a)) is the sum of reproducibility and a linearity error of onedigit regardless of the type of weighing instrument, because differential weighing is performed.

The maximum error of weighing in liquids (m(a) - m(fl)) is on average assumed to be 10 times asgreat as with weighing in air this assumption is based on many measurements in densitydetermination using the buoyancy method.

For the error of liquid density determination, the value of 0 .0 0 0 0 3 g/ cm3 for water and 0 .0 00 09g/ cm3 for ethanol are applied; that is the value of a thermometer reading error of 0.1C whichcorresponds to a temperature variation during measurement of 0.1C.

The following figures (see Figure 2 1 : Solid d ensity determination using the buoyancy method therelative error of density is dependent on the sample size and the sample density ) show the relative

error of solid density determination with the buoyancy method in water or ethanol, depending onthe sample size and the density of the sample examples, using sartoriusweighing instruments withdifferent readab ilities and w eighing cap acities.

It is clear that the error in density determination is dependent on the density of the sample to asignificant degree: the lower the density of the sample, the greater the error of the end result4.

A further figure (see figure 27) shows the error in density determination of liquids using thebuoyancy method for liquid d ensities betw een 0.5 and 2 .2 g/ cm3 with the use of the glassplummet included with the sartorius Density Determination Set. The plummet has a volume of(10 + 0 .01 ) cm3 and a density of 2.4 8 g/ cm3 with a tolerance of 0.5 mg in relation to the

buoyancy of w ater. Here too the error of density is dependent on the density value of theinvestigated sample.

4 For the ca lculation of

m m 1(fl) (a)

fl

s

=

is plugged in for m(fl);

for fl the density of water 1 .0 g/ cm3 is used, for the density of ethanol 0. 7 8 9 g/ cm3


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Figure 21: Solid density determination using the buoyancy method the relative error of density isdependent on the sample size and the sample density

Readability: 1 mg

0.0%

0.1%

0.2%

0.3%

0.4%

0.5%

0.6%

0.7%

0.8%

0 10 20 30 40 50 60

Sample Size / g

RelativeErrorofDensity/

Sample Density = 2 g/cm3



Water= 1 g/cm3


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Figure 22: Solid density determination using the buoyancy method the relative error of density isdependent on the sample size and the sample density

Readability: 0,1 mg

0.0%

0.1%

0.2%

0.3%

0 5 10 15 20 25 30Sample Size / g

RelativeErrorofDensity/

/




Water= 1 g/cm3

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BRO Density Determination Manual e