Density Determination Using Various Methods to Measure Volume

Experiment 1

Experiment 1

Goal: To accurately determine densities of objects

Method: Determine volume of objects (3 methods) Calculate densities from mass and volume

Densities

Substance Density

(g/mL)

wood 0.35

water 1.00

quartz 2.65

diamond 3.51

Al 2.7

brass 8.4

Au 19.3

Os 22.4

Increasing density

=Increasing “heaviness”

Density from lab data

V

m

volume

mass Analytical balance Three methods

Volume measurements:

–by geometry

–by water displacement

–by pycnometry

Uncertainties/Error

Mass: balance ±0.0001 g

Geometry: calipers ±1 of last digit cm

Displacement: grad. cylinder ±0.1 mL probably

Pycnometry: pynometer from average

Propagate: all but mass fractions

Procedure

1. Four cylinders

brass aluminum plugged hollow

Record:

2. Number #

3. Type type

4. Mass (analytical balance) m#

1) Volume by geometry

1. Lengths and diameters l, d

2. Calculate volume and error V, V

length, l

ldπ

ld

πV 22

42

diameter, d

1) Al example – density using volume by geometry

Mass 17.4640 gLength 5.08 cmDiameter 1.29 cm

32 632085

2

291

464017cm

g

cm.cm.

π

g.Vm .d

3

3

050

2632 291010

085010

46401700010

cmg

cm.cm.

cm.cm.

g.g.

cmg

ddlm

.

.

ddlm

Using vernier calipers

Object falls between 3.3 and 3.4 cm

Line on auxiliary scale matches at 8

Section 3B in manual

Length: 3.38 ± 0.01 cm

2) Volume by water displacement

1. Record initial volume of water Vi

2. Add metal cylinder

3. Record final volume Vf

4. Calculate volume and error V, V

initialfinalcylinder VVV

2) Al example using volume by water displacement

mass 17.4640 g

Vwater+cylinder 66.55 mL 0.05mL

Vwater 60.00 mL 0.05mL

Vcylinder 6.55 mL 0.1mL

mLg

mL.g.

Vm .672556

464017

mLg

mL.mL.

g.g.

mLg

Vm ..Vm 040672 6610

46401700010

Water displacement

46.5 mL

Meniscus: liquid’s curved surface

66.0 mL

66.0 mL– 45.5 mL 20.5 mL(object’s volume)

3) Volume by pycnometry

Pycnometry: pertaining to specific gravity

V = mH2O/H2O

1. Make pycnometer

2. Calibrate

Calibration example

Calibration – multiple trials (how well you fill to the mark)

trial masswater+pycnometer

1 92.7829 g

2 92.7825 g

3 92.7826 g

4 92.7831 g

Average: 92.7828 g

σmass: 0.0003g (using big equation)

Pycnometry continued

3. Add H2O to mark

record initial mass mi

4. Add metal cylinder

5. Remove H2O until at mark

6. Record final mass mf

7. Calculate volume and error V, V

waterwaterdisplaced waterdisplaced cyl ρ

m VV1

Pycnometry continued

llArea 22

d42

d

llArea 2

2

d42

d

waterdisplacedBobjectA massmassmassmass

A B

m

V

3) Al example using volume by pycnometry

massA 92.7828 g (calibration average) 0.0003g

massB 103.7227 g 0.0003g

masscylinder 17.4640 g 0.0001g

g.)g.g.g.(

g.g.g.g.massmassmassmass

water displacedmass

BobjectAwater displaced

00070000300001000030

52416782892464017782892

mL.mL.VV

mL.g.

mLg . VV

g.g.

mcylmcylV

waterdisplaced cyl

mm

cyl000705386

538699780

52416

5241600070

mLg

mL.mL.

g.g.

mLg

Vm

mLg

mL.g.

Vm

..

. Vm 000306712

6712

538600070

46401700010

5386464017

Record temperature, T

Density of Water vs. Temperature

0.9950

0.9960

0.9970

0.9980

0.9990

1.0000

15 20 25 30

Temperature (oC)

Den

sity

(g/m

L)

4) Volume of void in hollow cylinder

voidmetalcyl VVV

pycnometry

m metalV =

Example void calculation – brass cylinder

If Vcylinder = 4.970 mL (by pycnometry)

& mcylinder =34.5964 g (analytical balance)

& ρbrass = 8.387g/mL (pycnometry of solid cylinder)

mL.

g.mL.

VVV

g.mL

metalcylvoid

8450

5964349704 38781

No error propagation required for this

5) Mass Fractions in Mixed Cylinder

brassAlcyl mmm

Let: mAl = X.mcyl

mbrass = (1-X)mcyl

cylcylcyl mXmXm 1

Mixed cylinder

brassAlcylinder VVV

brassAl brassAl

cyl

mmV mAl = X.mcyl

mbrass = (1-X)mcyl

m V =

brassAl

XX

cylcyl

cyl

m1mV

by

mcyl

brassAl

XX

11

cyl

Collect terms with X

brassAlbrass

XX

11

cyl

Solve for X

brassAlbrass

X

1111

cyl to isolate X

Mixed cylinder

X = Al fraction

1-X = brass fraction

brassAl

brassX

11

11

cyl

No error propagation required for this

Pycnometry: volume Vmixed

Balance: mass mmixed

Mixed cylinder – example

X = Al fraction (here: 13.97%)

1-X = brass fraction (here: 86.03%)

860301

13970

38781

67121

38781

45761

11

11

.X

.

..

..X

mLg

mLg

mLg

mLg

brassAl

brasscyl

No error propagation required for this

Use pycnometry data to find:

density of pure brass 8.387 g/mL

density of pure Al (solid cylinders) 2.671 g/mL

density of mixed cylinder (example) 6.457 g/mL

ReportAbstractResults/ Samples calculations including:

Mass and volume by each method Volume of void Mass fraction Error analysis (parts 1 – 4)

Volume of objects (3 methods) yesVolume of inner void noMass fractions for mixed cylinder no

Discussion/review questions

Equipment cylinders: brass, aluminum, mixed, hollow Vernier calipers 50 mL Erlenmeyer flask lab marker Pasteur pipet 100 mL graduated cylinder 400 mL beaker thermometer (analytical balance)

Caution

NO

YES