Experiment R: Relative Densities

Abstract

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) 6

(Equation

2) 6

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) 6

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) 6

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) 6

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) 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) 6

(Equation

8) 6

(Equation

9) 6

(Equation

10) 6

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) 6

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

Lead (Method 2)

(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

6 Foundations of Physics Laboratory Handbook 2017/18,

University of Bristol (2017)