Relative Density

1.0Density of a Substance

Density is the measure of the concentration of matter of an object and it is measured as the mass per unit volume of a matter/substance.

The density of a substance is defined as its mass per unit volume,

$\rho =\frac{m}{v}$

where m is the mass of a sample and v is its volume. 

Density is a characteristic property of a solid or a liquid. This means for a given solid (or liquid), its density remains constant whatever be its size or mass.

Density of a gas is not a constant, it is variable. It depends on temperature, pressure, and volume.

S.I. unit of density : kg/m³ or kg m^–3. 

Its C.G.S. unit is g/cm³ or g cm^–3.

The density of water is 1000 kg m^–3 or 1 g cm^–3 (4°C).

Solids and liquids tend to be almost incompressible, meaning that their density changes very little with changes in pressure. Thus, the densities for solids and liquids are approximately independent of pressure.

Gases are compressible and can have densities over a wide range of values. Thus, there is not a standard density for a gas, as there is for solids and liquids. The densities for gases are usually the values of the density at STP. For deviations of temperature and pressure from these values, the density of the gas will vary significantly.
[STP-Standard Temperature and Pressure]

(All values are at standard atmospheric temperature and pressure (STP), defined as 0°C (273 K) and 1 atm (1.013 × 10⁵ Pa). To convert kilograms per cubic meter (kg/m³) to grams per cubic centimeter (g/cm³), multiply it by 10^–3)

What Does Density Depend on?

The densities of some solids, liquids and gases are listed in table-1. The table shows that the density of gold, for example, is more than 19 times greater than the density of water. Also, the density of some solids and liquids, such as mercury, can be more than 10,000 times greater than the density of some gases, such as helium. 

Density of a material depends on following factors:

1) Mass of particles : The density of a material depends on the mass of the particles, such as atoms or molecules, that make up the material. The more mass these particles have, the greater the density of the material. For example, the mass of a gold atom is more than seven times the mass of an aluminium atom. As a result, the density of gold is much greater than the density of aluminium.

(2) Distance between particles: The density of a material also depends on the distance between the particles in the material. The greater the distance between the atoms or molecules, the smaller the density. Table 4.1 shows that in gases, particles are much farther apart than in solids or liquids. As a result, the density of a gas is usually much less than the density of a solid or a liquid.

2.0Relative density (R.D.) 

The relative density of a substance is the ratio of its density to the density of water. It is also called ‘specific gravity’.

$R.D=\frac{{\rho }_s}{{\rho }_w}$

Where, ρ_s = density of substance ; ρ_w = density of water

Relative density has no unit (unitless quantity) as it is a ratio of physical quantities having same units. 

3.0Variation of Pressure with Depth

The fluid pressure depends only on the height of the column of fluid above the surface where you measure the pressure. It does not depends on the area of the surface in contact or the shape of the liquid column. The greater the height of the column of fluid above a surface, the greater is the pressure exerted by the fluid on the surface.

If P₁ is the pressure on the surface of liquid (see figure) and P₂ is the pressure at a point within the liquid at a depth h, then, their pressure difference (P₂ – P₁) is given by,

$\text{Pressure}=\frac{\text{Force}}{\text{Area}}$

$=\frac{mg}{\text{Area}}=\frac{\text{volume}\times \text{density}\times g}{\text{area}}$

$=\frac{\text{Area}\times \text{height}\times \text{density}\times g}{\text{Area}}$

$=\text{height}\times \text{density}\times g$

$\rho g=h$

Pressure depends only on the height of the fluid above a surface, not on the shape of the vessel. Pressure at the bottom of each section of the vessel is same (called hydrostatic paradox).

Solved Examples

1. Suppose that a student finds that 24 mL of a certain liquid weighs 36 g. What is the density of this liquid in SI unit ?

Solution

Given, mass of liquid, m = 36 g = 36 × 10^–3 kg ; density of liquid, ρ = ? ;

volume of liquid, V = 24 mL = 24 × 10^–3 L = 24 × 10^–3 × 10^–3 m³ = 24 × 10^–6 m³ 

{Always remember,1L = 10^–3 m³ = 10³ cm³. Also, 1 mL =1 cm³  (also called 1 c.c)}

Density, $\rho =\frac{\text{Mass}}{\text{Volume}}=\frac{m}{V}=\frac{36\times 10^{-3}}{24\times 10^{-6}}=1500{\text{kg m}}^{-3}$

2. Mercury has a density of 13.6 g/cm³. What volume of mercury must be taken to obtain 225 g of the metal ?

Solution

Given, density of mercury, = 13.6 g/cm³ ; mass of mercury, m = 225 g; 

volume of mercury, V = ?

Now, density,

$\rho =\frac{\text{Mass}}{\text{Volume}}=\frac{m}{V}$

$V=\frac{m}{\rho }=\frac{225}{13.6}=16.54{\text{cm}}^3$

3. The two thigh bones (femurs), each of cross-sectional area 10 cm² support the upper part of a human body of mass 40 kg. Estimate the total pressure sustained by the both femurs. (Given g = 10 m s^–2) 

Solution

Given, mass of body, m = 40 kg ; g = 10 m s^–2; 

area, A = 2 × area of each thigh bone

= 2 × 10 cm² = 20 × 10^–4 m² (weight of the body is supported by two thigh bones) 

Force, F = weight of the body = mg = 40 × 10 = 400 N

Pressure, $P=\frac{Force}{Area}=\frac{F}{A}=\frac{400}{20\times 10^{-4}}$ = 2 × 10⁵ Pa

4. The four tyres of an automobile are inflated to a pressure of 2.0 × 10⁵ Pa. Each tyre has an area of 0.024 m² in contact with the ground. Determine the weight of the automobile.

Solution

Given, pressure, P = 2.0 × 10⁵ Pa ; area of tyre, A = 0.024 m² ; weight of automobile, W = ?

Since, automobile is supported by four tyres, load or force (F) on each tyre is one fourth of the weight of the automobile, i.e., F = W/4

Pressure, or 

$P=\frac{F}{A}$

$P=\frac{W/4}{A}=\frac{W}{4A}$

W = 4AP = 4 × 0.024 × 2 × 10⁵ 

= 1.92 × 10⁴ N

4.0Also Read