2. We would like to sort some numbers using liquids; i.e.:
1. Prepare for each given number a uniquely-colored solution having
its density the specified value;
2. Work with a comparison-based sorting algorithm to mix the solu-
tions in pairs in a transparent vesel to observe the ordering.
1
3. Everybody knows that when you combine water and oil they don’t
mix, and oil rises above the water, as opposed to sinking or staying
in place. Why?
2
4. Why won’t water and oil mix?
As like dissolves like (old chemistry saying), only polar substances
dissolve in polar solvents; likewise, non-polar solutes only dissolve in
non-polar solvents.
3
5. Water is known to be a polar molecule
There is an unequal sharing of electrons between the hydrogen atoms
and the electronegative oxygen atom. This results in a slightly positive
charge on the hydrogen and a slightly negative charge on the oxygen.
Therefore, water is a polar solvent.
4
6. Oil is known to be a non-polar molecule
The chains of carbon atoms bond to hydrogen atoms making them
non-polar, or hydrophobic (water-fearing).
5
7. Why will oil float on top of water?
The density of the substances doesn’t have anything to do with the
miscibility properties of oil and water; it only explains why oil will layer
atop of water.
6
8. The density is defined as:
m
ρ= (1)
V
Examples:
1. Pure water at 4◦ Celsius has a density of 0.99g/ml (grams per
millilitre);
2. Olive oil at 20◦ Celsius has a density of 0.8 − 0.92g/ml.
7
9. If we consider the volume to be constant and mix a couple of liquids
of different densities, their densities will be directly proportional to
their masses. Thus, gravity will order them accordingly.
8
10. There are two approaches to solve the comparison problem. When
we mix two liquids of different densities, we can use:
(A) A polar liquid and another non-polar liquid such that these won’t
mix, and the order in which these will be added in the transparent
container will not matter;
9
11. (B) Either two polar or two non-polar liquids of different coloring; the
order in which these will be added in the transparent container
matters, such that:
• If the colors mix, then the second liquid (that was added) has
a higher density than the first;
• If the colors don’t mix, then the second liquid has a lower
density than the first one.
10
12. Food coloring
A food coloring is any substance that is added to food or drink to
change its color.
11
13. The experiment
The density of water is dependent on the dissolved salt content, as
well as the temperature of the water.
12
14. Water changes its density in respect to its temperature, but not on
a linear scale, and not even continuously in one direction.
Temperature (◦C) Density (kg/m3)
+100 958.4
+80 971.8
+60 983.2
+40 992.2
+30 995.6502
+25 997.0479
+22 997.7735
+20 998.2071
+15 999.1026
+10 999.7026
+4 999.9720
0 999.8395
13
15. Density is weight divided by volume. The density of fresh water is 1
gram (mass) per cubic centimeter (volume). In other words, if you
had a cube with the dimensions: 1cm x 1cm x 1cm; and filled it with
pure water, that cube of water would weigh 1 gram. This density
is expressed as 1g/cm3. If you dissolve salt into the water, the salt
will increase the fluid’s mass, while its volume will remain the same.
Thus, the liquid’s density will increase.
14
16. Determine the density of tap water
1. Measure the mass of the empty graduated cylinder; record its
weight;
2. Fill the cylinder with water to the 100ml line; this is the volume;
3. Measure the mass of the cylinder with water;
4. Subtract the mass of the cylinder from the mass of the filled
cylinder and divide the mass of the water by its volume; this will
yield the density of the top water.
15
17. Determine the density of tap water with salt
1. Use an eyedropper to remove 2g(2ml) of water from the cylinder;
2. While the cylinder is on a scale, add 2g of salt;
3. Read the new water level inside the cylinder; this is the new vol-
ume;
4. Divide the mass of the water inside the cylinder by its new volume;
this is the density of the salt water.
16
18. Formulas
mtap−water
ρtap−water = (2)
Vtap−water
where V = 100ml.
mtap−water + msalt
ρliquid = (3)
Vtap−water
under the assumption that the volume is constant after the salt dis-
solves, such that
msalt = ρliquid ∗ Vtap−water − mtap−water . (4)
17
26. First experiment
Mix two solutions of different densities (i.e. ρ1 = 0.984g/ml and
ρ2 = 1.048g/ml).
25
27. Red container (ρ1 = 0.984g/ml) and Blue container (ρ2 = 1.048g/ml)
26
28. Red container has 250ml of tap water and 3g of red coloring, while
the Blue container has 250ml of tap water plus 21g of salt and 3g of
blue coloring.
27
29. Overview of the prepared solutions
Container Weight Water Salt Food Coloring Density
188 − 189g
1 250ml 0g 3g of Red 0.984g/ml
20 − 21g
2 192g 250ml 3g of Blue 1.048g/ml
28
33. Again as expected, the liquids mix because the Blue liquid is heavier
than the Red liquid and goes to the bottom of the container.
32
34. Second experiment
Mix three solutions of different densities (i.e. ρ1 = 0.984g/ml, ρ2 =
1.048g/ml, and ρ3 = 1.148g/ml).
33
35. Red container (ρ1 = 0.984g/ml), Blue container (ρ2 = 1.048g/ml),
and Orange container (ρ3 = 1.148g/ml)
34
36. Red container has 250ml of tap water and 3g of red coloring, the
Blue container has 250ml of tap water plus 21g of salt and 3g of blue
coloring, while the Orange container has 250ml of tap water plus 42g
of salt and 3g of yellow coloring.
35
37. Overview of the prepared solutions
Container Weight Water Salt Food Coloring Density
188 − 189g
1 250ml 0g 3g of Red 0.984g/ml
20 − 21g
2 192g 250ml 3g of Blue 1.048g/ml
3 193g 250ml 42g 3g of Orange 1.148g/ml
36
38. 1. Orange, Blue, Red
2. Blue, Orange, Red
3. Blue, Red, Orange
4. Orange, Red, Blue
5. Red, Blue, Orange
6. Red, Orange, Blue
37
39. 1. Poured in the following order: Orange, Blue, Red
38
40. 2. Poured in the following order: Blue, Orange, Red
39
41. 3. Poured in the following order: Blue, Red, Orange
40
42. 4. Poured in the following order: Orange, Red, Blue
41
43. 5. Poured in the following order: Red, Blue, Orange
42
44. 6. Poured in the following order: Red, Orange, Blue
43
45. As expected, the solutions do not mix only when poured in the Or-
ange, Blue, and Red order since the numbers validate the 1.148 >
1.048 > 0.948 ordering.
44
46. Limits
• On the interval of the accepted values;
• On the number of decimals;
• On the distance (e.g. equality) between the numbers.
45
47. The interval of the accepted values
• Hydrogen has the lowest density of all elements, measured at
0.00008988g/ml;
• Hassium has the highest density of all elements, estimated at
41g/ml.
46
48. The interval of the accepted values (2)
• Ether has the lowest density of all liquids, measured at 0.07272g/ml;
• Iodine has the highest density of all liquids, measured at 4.92728g/ml.
47
49. The interval of the accepted values (3)
Even if we don’t have an exact match for our number (i.e. a liquid
of that exact density), we can prepare such a liquid by dissolving a
solute in the closest (i.e. lowest) density solvent. Thus, we can sort
an interval – we can sort an infinity of numbers.
48
50. The precision of the values
Ideally, we can compare numbers of any precision as long as we have
an ideal scale to measure the weights and calculate the densities.
Practically, we’ll probably be able to work only with numbers (i.e.
densities) as precise as to the 5th decimal (it greatly depends on the
scale).
49
51. Conclusion
By associating (manufacturing) a coloured liquid of a specified density
to each number to be sorted, we can use this unconventional liquid
sort method to order the given numbers.
50
