# Experiment or accuracy of the results for each

Experiment R: Relative Densities

Abstract

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This report discusses how to find relative densities of
unknown materials using three different methods and includes explanations of
the results achieved, as well as comparisons to known data. It was found that
each method was better fitted for the physical state of the exercised material.
Improvements that may benefit the safety or accuracy of the results for each
method used is discussed further into the report. The purpose of this report
was to compare the three methods to discover which was the most accurate or
reliable in determining the density of each sample.

Introduction

The purpose of this report was to compare the methods of
measuring density for accuracy and reliability. This can later be used to
determine which method is best, given the physical state of the material. This
has been based on the assumption that there is no provided access to better
equipment than the ones used in this experiment, which may significantly
increase accuracy or reliability of data. These experimental procedures can
also be used to find out what a liquid sample or a solid sample is, this can be
regarded as the motivation for this report.

Theory

Throughout this subsection, there will be references to
method numbers 1 to 3, method one is called ‘Hydrostatic Balance’, method 2 is
called ‘Relative Density Bottle’ and method 3 is called ‘Hare’s Apparatus’.
Although similar in name, there is a distinct difference between relative
density and density. The difference is the following: ‘the relative density is
the ratio of the density of the substance to the density of water’ 2.
Therefore, relative density has no unit of measurement, but density does. As
the relative density requires the density of the substance, the density of the
sample must be measured. This is done using one of the three methods mentioned
in this report. These methods tend to use the concept of Archimedes’ Principle
3, which is when a sample or material (in this case) is completely submerged
in water, it tends to feel lighter 3. This ‘weight lost’ equates to the
‘buoyant force’ on the object which is the upward force the water is applying
to a floating object for example 3. This principle can also be described as a
body in liquid experiences an upward force which equates to the weight of the
liquid that this body had displaced 3. Using this principle to find the
density of an object we can then use the following formulae to work out the
Relative density:

(Equation
1)

(Equation
2)

However, there is an error associated with using Archimedes’
Principle. When a body is underwater, air bubbles occur at the surface of the
sample 4 which adds extra weight and therefore effects the result of relative
density. Relative density will appear lower than its most accurate value. Once
results have been recorded, they will be compared to Kaye and Laby’s Book of
Physical Constants 1 to decide on which method is the most reliable or
accurate. This book is used as reference during the experiment which was put
together by Physicists Kaye and Laby 1. During the procedure there will be
errors occurring, the formula for calculating this error for these procedures
are found below:

(Equation
3)

Where the denominators of each bracket shown above are the
mean value of each set of data and the numerators are the errors associated
with them, such as using a ruler with an error of 0.05cm. Equation 3 will have
to be rearranged to find the value for .

(Equation
4)

Where:  = Weight of Water and Bottle,  = Weight of Bottle,  = Weight of Bottle, Shot and Water and  = Weight of Bottle and Shot. For the second
method, a Relative Density Bottle is needed. This Bottle allows the same volume
of water to be put in the bottle each time as the lid has a hole which is
pushed down inside the bottle, pushing any unwanted water out. This means that
the bottle must be slightly overfilled to get the same volume each time. The
third method includes the use of a glass tube called Hare’s apparatus, shaped
like the letter y 4. Method 3 uses the equations below:

(Equation
5)

Where:  = Density of Water,  = Density of Ethanol,  = Height of Ethanol on Hare’s apparatus and  = Height of Water on Hare’s apparatus. In theory,
methods: ‘Hydrostatic balance’ and ‘Relative Density Bottle’ would be the most
suitable for measuring the density of material’s in the solid physical state
whereas method ‘Hare’s Apparatus’ would be most suitable for testing the
density of Liquids. It is also worth noting that, to work out the mean of a set
of data, the following equation was used:

(Equation
6)

Where  is the mean, the numerator is the set of
values and n is the number of values. To work out the standard error on the
mean, combine the following equations with equation 6 and equation 3:

(Equation
7)

(Equation
8)

(Equation
9)

(Equation
10)

Throughout this report, the value of 1 g/cm³ will be used as the density of water,
this value has been taken from Kaye and Laby’s Book of Physical Constants 1.

(Equation
11)

Where  is the relative density of x in salt solution,
is the density of salt and  is the density of x.

Experimental Methods

Firstly, the Hydrostatic Balance requires the following
equipment: an electronic crane scale (for example the one used during this
experiment is the Kern FOB 500-1S Balance (0.5kg)) 5, string, metal samples
that are preferably around 50g-200g, a beaker and a raised platform with about
a 5cm in diameter hole in the middle. To begin the experiment, place the
electronic balance on the raised platform so that the hook on the underside of
the balance is coming out of the hole. With the hook attached, re-calibrate the
device so that the digital display shows 0g, this makes sure that the weight of
the hook is not included in the recorded value of the sample. Then the string
must be used to attach the metal sample to the hook but be sure to allow enough
string so that the metal can be submerged in water in the beaker below. Ensure
that the sample has stopped spinning and record the value that the balance
shows. Fill a beaker with water high enough so that the metal sample can be
completely submerged and place it under the metal sample and into the water
below. Check for any air bubbles clinging onto the surface of the sample, if
so, try rubbing the bubbles off with your fingers. If the sample is not
touching the beaker and it has stopped spinning, record that value as well. Now
repeat all the above with the rest of the metal samples you have – each 2 more
times for reliability – take a value for the mean for each set of data and use
equation 2 to find the relative density. Rearrange equation 1 to find the
density of the material and use equations 3, 6, 7, 8 and 9 to find the error
associated with that value, then compare the recorded value with the value in Kaye
and Laby’s Book of Physical Constants 1 to find out what the material is. The
equipment should be set up as Figure 1 below:

Figure 1: A sketch of the ‘Hydrostatic Balance’ apparatus with the
beaker included, used to measure the weight of a metal sample when submerged
and when not submerged in water.

For the second experimental procedure ‘Relative Density
Bottle’, the following equipment is needed: A Relative Density Bottle, Salt
(NaCl), Shot Pellets of Lead and the Kern FOB 500-1S Balance (0.5kg) 5.
Before starting the procedure, the density of salt solution must be calculated.
Ensure that the Relative Density Bottle is rotated on its side while filling it
to get rid of any air bubbles which would mean less liquid would get into the
bottle than usual. To calculate the density of the salt solution, the apparatus
from the first method must be reused. Add about 10g of Sodium Chloride to the
beaker filled with water, then submerge one of the metal samples into the
solution. The system should already be re-calibrated to the hook from the last
time it was used in this experiment Make sure that all the salt has dissolved,
that there are no air bubbles on the surface of the sample and that the metal
sample is attached and not touching the beaker before weighing. Record your
data. Now using data concluded from the past method use equation 11 to work out
the density of the salt solution, repeat two more times. Then use equations 7,
8 and 9 to work out the combined error. Now to begin the procedure, fill the
relative density bottle with salt solution, add the stopper to the relative
density bottle and wipe off any excess liquid. Weigh it then record your value.
Now empty the bottle and fill with distilled water, weigh, then record this
value down also. Get 50g of the Shot Pellets of Lead and put it into the
relative density bottle filled with distilled water make sure to attach the
stopper and wipe away any excess liquid each time. Now weigh the relative
density bottle with the shot pellets and distilled water and record this value.
Repeat the above 2 more times to ensure your data is reliable. Use equations 4
and 1 to find out the density and relative density for the shot pellets, then
use equation 3, 6, 7, 8, 9 and 10 to work out the combined error on your value
for the density and relative density of the shot pellets. Finally, for this
method, compare your data to Kaye and Laby’s Book of Physical Constants 1 to
find out whether this method is reliable and/or accurate. Set up the equipment
for this procedure as illustrated below in Figure 2:

Figure 2: A sketch of the ‘Relative Density Bottle’ apparatus, used
to measure the weight of water that is displaced due to the addition of Lead
shot pellets.

The third and final method ‘Hare’s apparatus’, you will need
Hare’s Apparatus, two beakers, a boss, a clamp and a ruler. Get two beakers and
fill them both, one with ethanol and one with distilled water. Attach Hare’s
apparatus onto a boss and clamp low enough so that the two ends of the y-shaped
tube can be lowered into the liquid. Suck on the only end that isn’t submerged
into either liquid until the liquid stops at a reasonable height within the
Hare’s Apparatus. Place your thumb over that tube so that the liquid does not
run down the tube back into the beaker and record the heights:  and . Note that  and  is measured from the surface of the liquid in
each beaker, to where it reaches in the Hare’s Apparatus. Record the values for
and  using the ruler and repeat 2 more times for
reliability, then rearrange equation 5 to find your density of ethanol. After
this is done, use equation 1 to find the relative density. To work out the
combined errors, use propagation error equations 3, 6, 7, 8 and 10. The
equipment should be set up like the following in Figure 3 below:

Figure 3: A sketch of the ‘Hare’s Apparatus’ set up, used to
measure the density of Ethanol.

Results

Here are the results that were concluded from when the
experimental procedure was carried out:

Figure 4: A Graph showing the recorded relative density of the
unknown metal samples labelled 1 to 4 using method 1 ‘Hydrostatic Balance’

In reference to Figure 4, when comparing the relative
densities of objects 1 to 4 with Kaye and Laby’s Book of Physical constants 1,
it was concluded that object number 1 was Lead, object number 2 was Copper,
object number 3 was Silver and object number 4 was Titanium.

Sample

Relative Density

(10.540.2)

Ethanol (Method 3)

(1.30.2)

Figure 5: A table
showing the recorded relative density of Ethanol and Lead. The relative density
for Lead was acquired by using the ‘Relative Density Bottle’ method whereas the
Ethanol value was found using the ‘Hare’s Apparatus’ method.

The errors included in the results were calculated using
equations 3, 6, 7, 8, 9 and 10 under the ‘Theory’ subsection.

Discussion

The value for the relative density of ethanol was slightly
off when compared with Kaye and Laby’s Book of Physical Constants 1. The
value referred to in Kaye and Laby’s Physical Book of Constants 1 did not lie
within the range of the error calculated within this report either. This could
have been because of the parallax error on the ruler from the surface of the
liquid to the top of where the liquid had stopped in Hare’s apparatus. It was
difficult to tell the height of the ethanol or water as the ruler was held free
hand against the apparatus during the procedure. This left the data for method
3 unreliable when compared to universal data 1. Kaye and Laby’s 1 value for
the density Lead also did not lie within the range that was calculated however,
it wasn’t far off from it. Kaye and Laby 1 reported 11.43 g/cm³ for Lead and
0.79 g/cm³ for Ethanol. Method 1 is the only procedure where all values
referred to was within the range of Kaye and Laby’s 1 data. Therefore, for
this experiment Method 1 ‘Hydrostatic Balance’ is the most reliable.

Conclusion

The most reliable and accurate method out of the three to
find Density and Relative Density of a material is method 1 ‘Hydrostatic
Balance’. However, this method can only be used when testing the density of a
solid, not a liquid or gas phase material.

References

1 Kaye and Laby, Book of Physical and Chemical Constants
and Some Mathematical Functions “Densities and Relative Densities of known
materials” (1911)

2 https://en.wikipedia.org/wiki/Relative_density
(2018)

3 https://en.wikipedia.org/wiki/Archimedes%27_principle
(2018)

4 Christian McQueen, Physics Labs Foundations of Physics
Laboratory Book (2017)

5 https://scales-measuring.com/bench-scales/369-kern-fob-500-1s-stainless-steel-bench-scale.html