density determination using various methods to measure volume experiment 1
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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
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