Measurement of Relative Density of a Substance

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Relative Density : Definition, Examples, Diagrams

The relative density of a substance is the ratio of its density to that of water. That is :
Relative density of a substance = Density of the substance Density of water
We know that, Density \(=\frac{\text { Mass }}{\text { Volume }}\) so by writing \(\frac{\text { Mass }}{\text { Volume }}\) in place of the density in the above relation,
we get:
Relative density of a substance \(=\frac{\text { Mass of the substance }}{\text { Volume of the substance }} \times \frac{\text { Volume of water }}{\text { Mass of water }}\)

Now, if we take ‘equal volumes of the substance and of water’, then the two volume factors of the above relation cancel out, and we are left with :
Relative density of a substance \(=\frac{\text { Mass of the substance }}{\text { Mass of an equal volume of water }}\)

This relation gives us the following definition of relative density. The relative density of a substance is the ratio of the mass of any volume of the substance to the mass of an equal volume of water. In other words, the relative density of a substance is the mass of the substance relative to the mass of an equal volume of water. As the relative density is a ratio of two similar quantities (masses), it has no units. Thus, relative density is a pure number. The relative density values of some of the common substances are given below :

Relative Densities of Some Common Substances

The relative density of a substance expresses the heaviness (or density) of the substance in comparison to water. For example, the relative density of iron is 7.8. Now, by saying that the relative density of iron is 7.8 we mean/that iron is 7.8 times as heavy as an equal volume of water. Thus, the relative density of a substance is a number which tells us how many times the substance is heavier than an equal volume of water. The relative density of water is 1. Now, if the relative density of a substance is more than 1, then it will be heavier than water and hence it will sink in water. On the other hand, if the relative density of a substance is less than 1, then it will be lighter than water and hence float in water.

Relative density is very important in the accurate determination of density. Actually, the relative density of a substance is found accurately by using Archimedes’ principle. And this relative density is then used to calculate the density of the substance. We will now solve some numerical problems based on relative density.

Example Problem 1.
The relative density of silver is 10.8. If the density of water be 1.0 × 10³ kg m^-3, calculate the density of silver in SI units.
Solution.
We know that:
Relative Density of a substance \(=\frac{\text { Density of the substance }}{\text { Density of water }}\)
Here, Relative density of silver = 10.8
Density of silver = ? (To be calculated)
And, Density of water = 1.0 × 10³ kg m^-3
Now, putting these values of relative density of silver and density of water in the above formula, we get :
10.8 = \(=\frac{\text { Density of silver }}{1.0 \times 10^3}\)
So, Density of silver
So, Density of silver = 10.8 × 1.0 × 10³ kg m^-3
= 10.8 × 10³ kg m^-3
Thus, the density of silver in SI units is 10.8 × 10³ kg m^-3. This can also be written as 10800 kg m^-3.
It is obvious from the above calculations that the density of a substance can be obtained by multiplying its ‘relative density’ by the ‘density of water’. Please note that sometimes the density of water is not given in the numerical problems. So, we should remember the density of water ourselves.

Example Problem 2.
The volume of a solid of mass 500 g is 350 cm³
(a) What will be the density of this solid ?
(b) What will be the mass of water displaced by this solid ?
(c) What will be the relative density of the solid ?
(d) Will it float or sink in water ?
Solution:
(a) Density of solid \(=\frac{\text { Mass of solid }}{\text { Volume of solid }}\)
= \(\frac{500 \mathrm{~g}}{350 \mathrm{~cm}^3}\)
= 1.42 g/cm³
Thus, the density of the given solid is 1.42 g/cm³.

(b) The solid will displace water equal to its own volume. Since the volume of solid is 350 cm³ so it will displace 350 cm³ of water. Now, volume of water displaced is 350 cm³ and the density values for water in the formula :
So, Mass of water = 1 g/cm³ × 350 cm³
= 350 g
Thus, the mass of water displaced is 350 grams.

(c) Relative density of solid \(=\frac{\text { Density of solid }}{\text { Density of water }}\)
= \(\frac{1.42 \mathrm{~g} / \mathrm{cm}^3}{1 \mathrm{~g} / \mathrm{cm}^3}\)
= 1.42
Thus, the relative density of the solid is 1.42.

(d) Since the relative density of this solid (1.42) is greater than the relative density of water (which is 1), therefore, this solid is heavier than water and hence it will sink in water