DENSITY AND RELATIVE DENSITY

The concept of density. 

Figure  below Shows two identical flasks one filled with water to 250cm³ mark and the other filled with kerosene to the same 250cm³ mark, when measured in electronic balance the flask filled with water is found to be heavier than that filled with kerosene why? The answer is in finding the mass per unit volume of kerosene and water in respective flasks.

The mass of the flask filled with water is 330g, and the mass of flask filled with kerosene 280g, the empty flasks were measured and found to be 80g therefore mass of water only is 250g and kerosene only is 200g.

Mass per unit volume of water is 250g/  250cm³ this is 1g/cm³.

Mass per unit volume of kerosene is 200g/ 250cm³this is 0.8g/cm³.

The results 1g/cm³ and 0.8g/cm³ are the densities of water and kerosene respectively.

Therefore the density of a substance is the mass per unit volume of a given substance.

The SI unit of density is kilogram per meter cubic (kg/m³) also gram per centimeter cubic (g/cm³). The symbol for density is rho (ρ) ρ=mass/volume.

Example 1.

A block of ice with volume 5.5m³ has a mass of 5060kg find the density of ice.

Solution

Volume of block=5.5m³

Mass of block=5060kg

          Density=mass /volume

                       =5060/5.5m³.

                       =920kg/m³.

The density of ice is 920kg/m³.

Example 2.

A silver cylindrical rod has a length of 0.5m and radius of 0.4m,find the density of the rod if its mass is 2640kg.

Solution

Mass of cylinder=2640kg

Volume of cylinder= πr²h

                             =3.14 x  0.4 ²  x 0.5

                              =0.2512m³

Density=mass/volume

            =10509kg/m³.

Example3.

A stone has a mass of 112.5g.when the stone totally immersed in water contained in measuring cylinder displaced water from 50cm³ to 95cm³.find the density of the stone.

Solution

Mass of the stone=112.5g

Volume of stone=95cm³-50cm³=45cm³

Density=mass/volume

            =2.5g/cm³.

Example 4.

Beaker contain 262.5cm³ of a certain liquid  weigh 410g,if the mass of an empty dry beaker is 200g,find the density  of the liquid.

Solution

Mass of liquid=410g-200g=210g

Volume of liquid=262.5cm³.

Density=mass/volume

            =0.8g/cm³.

DENSITY BOTTLE

The density bottle (pycnometer) consists of ground glass stopper with a fine hole through it.

The function of the fine hole in a stopper is that, when the bottle is filled and the stopper is inserted, the excess liquid rises through the hole and runs down outside the bottle, by this way the bottle will always contain the same volume of whatever the liquid is filled in provided the temperature remains constant.

density bottle

the bottle is used to measure density and relative density, relative density is comparison of one density to another, thus a density of a given volume of a substance to a density of equal volume of referenced substance, for example a ratio of a density of a  given volume of substance to a density of an equal volume of water, this is referred to a relative density of a given substance or Specific gravity of a given substance. The term specific gravity is used when the reference substance is water.

Measurement of density of liquid by relative density bottle

ü  The mass of bottle is found when dry and empty

ü  The bottle is then filled with the liquid density is to be determined

ü  The stopper is then inserted causing the liquid to overflow

ü  The bottle is dried up by using blotting paper

ü  The mass of the liquid and the bottle is found

ü  Density is found from the collected data

                              Mass of empty bottle=m₁g

                              Volume of liquid in the bottle=V

                              mass of bottle and the liquid=m₂g

                              mass of liquid only=(m₂-m₁)g

density= mass /volume

density= .(m₂-m₁)g/V

The volume of the bottle is known, usually 25ml, 30ml or 50ml 

Example 1

A 30ml density bottle was filled with kerosene and found to weigh 86g.if the mass empty dry bottle was 62g, find the density of kerosene.

Solution

Mass of empty bottle=62g

Mass of bottle and kerosene=86g

Mass of kerosene only=86g-62g=24g

Density= mass /volume

            =24g/30ml

            =0.8g/cm³.

Determination of densities of granules and sand

To find the density of sand or granules such as lead shots a density bottle is used as follows

1.      Find the mass of empty dry density bottle m_o

2.      Put some granules and find the mass  m₁ =( mass of empty bottle + mass of granules)

3.      Pour water in the bottle until it is full and find mass m_{2 =} ( mass of bottle + mass of granules + mass of water on top of granules)

4.      Find the mass of bottle filled with water only  m₃  =( mass of bottle + mass of  water)

The mass of sand  = (m₁-m₀) g

Mass of water above the sand    = (m₂-m₁) g

Mass of water filling the bottle  = (m₃-m₀) g   

Since density of water is 1g/cm³

Volume of sand  =   [(m₁+m₃)-(m_o+m₂)]/1g/cm³

                         = [(m₁+m₃)-(m_o+m₂)] cm³

Density=mass/volume

Example 1

Given the data below find the density of granules

Mass of empty dry density bottle =18g

Mass of density bottle and granules=131g

Mass of density bottle and granules together with water on top =171g

Mass of density bottle full of water=68g

RELATIVE DENSITY

Relative density of a substance is the ratio of the density of substance to the density of water.

Or

 Relative density of a substance is the ratio of mass of any volume of substance to the mass of an equal volume of water.

Example 1

A globe of steel has a mass of 12g and a volume of 15.2cm³,find its relative density.

To measure relative density of liquid by density bottle

ü  Find mass of empty bottle –m₀

ü  Find mass of bottle and liquid-m₁

ü  Empty the bottle and rinse it with water

ü  Fill the bottle with water and find mass m₂

Mass of liquid= (m₁-m₀)g

Mass of equal volume of water= (m₂-m₀)g

Since comparison of density is done with water (referenced substance) the other name of the ratio is specific gravity of a given substance. Because the density of water is 1g/cm³.Relative density has no units it is simply a number or ratio.

Example

The mass of density bottle is 19g when dry and empty, 45g when filled with water and 40g when full of liquid x. calculate the density of the liquid x.

Determinations of relative density by eureka can method

ü  Find the mass m₀ g of solid

ü  Fill the eureka can  and let water overflow until last drop

ü  Place under the spout of overflow can a clean dry beaker of mass m₁ g.

ü  Lower the solid slowly with thin thread until it is totally immersed

ü  Obtain the mass of water that overflow from the eureka can and the beaker itself m₂ g.

The volume of water overflows into a beaker is equal to the volume of solid

Mass of solid = m₀ g

Mass of beaker and water=m₂ g

Mass of beaker=m ₁g.

Mass of water only = (m₂-m₁)g

EXAMPLE

A certain piece of metal has a mass of 282.5 g,if when the block was totally immersed in overflow can displaced water in a beaker of mass 20 g.if the mass of water and the beaker was 45 g,find the relative density of the metal.