Measurement of Volume and Density

Volume is a unit of three-dimensional measure of space that comprises a length, a width, and a height. It is measured in units of cubic centimeters in metric. The volume of a substance can be measured in volumetric glassware, such as the volumetric flask and the graduated cylinder.

Density is simply defined as mass per unit volume of a substance. It is symbolized by rho (ρ) and its SI units are kg/m3.

Density = mass / volume.

It is measure of the amount of matter contained in a given volume.

Density indicates how much of a substance occupies a specific volume at a defined temperature and pressure. The density of a substance can be used to define the substance.

Different substances have different densities, so density is often used as a method to identify a material. Comparing the densities of two materials can also predict how substances will interact. Water is used as the common standard to substances, and it has a density of 1000 kg/m3 at Standard Temperature and Pressure (called STP).

The Variable Density of Water

Water itself is a complicated and unique molecule. Even if the pressure is consistent, water’s density will change based on the temperature. Recall that the three basic forms of matter are solid, liquid and gas (ignore plasma for the time being).

As a rule of thumb, almost all materials are more dense in their solid or crystalline form than in their liquid form; place the solid form of almost any material on the surface of its liquid form, and it will sink.

Water, on the other hand, does something very special: ice (the solid form of water) floats on liquid water.

Look carefully at the relationship between water’s temperature and its density. Beginning at 100 °C, the density of water steadily increases, as far as 4 °C. At that point, the density trend reverses. At 0 °C, water freezes to ice and floats.

This table lists the densities of water at different temperatures and constant pressure.

Temp (ºC)	Density (kg/m³)
100	958.4
40	992.2
25	997.0479
10	999.7026
4	999.9720
0	999.8395
−10	998.117
The values below 0ºC refer to super cooled water

The implications of this simple fact are enormous: when a lake freezes, ice crusts at the surface and insulates the liquid below from freezing, while at the same time allowing the colder water (with a temp of approx. 4 °C and a high density) to sink to the bottom. If ice did not float, it would sink to the bottom, allowing more ice to form and sink, until the lake froze solid! Scuba divers and swimmers often encounter these water temperature gradients, and they might even encounter a water layer at the very bottom of a lake with a temperature of approximately 4 °C. That’s just about as cold as the lake will get at the bottom; as soon as the water gets colder, the liquid water becomes less dense and rises.

Layers of water in a winter lake: During the winter months of seasonal climates, the warmest water in most lakes and rivers is only 4°C. This 4°C water has the highest density and sinks to the bottom of the lake. As the water becomes colder (<4°C), it becomes less dense and rises to form ice on the surface of the lake. As a result, liquid water always exists in lakes and rivers during the winter months. This unique property of water enables animals and plants to survive under the frozen lake or winter, ensuring that all freshwater life does not go extinct each winter.

Examples

A block of glass of mass 187.5 g is 5.0 cm long, 2.0 cm thick and 7.5 cm high. Calculate the density of the glass in kgm^-3.

Density = mass / volume
= (187.5 /1000) /(2.0 × 7.5 × 5.0 /1,000,000)
= 2,500 kgm^-3.

The density of concentrated sulphuric acid is 1.8 g/cm³. Calculate the volume of 3.1 kg of the acid.

Solution

Volume = mass / density
= 3,100 / 1.8
= 1,722 cm3 or 0.001722 m³.

The mass of an empty density bottle is 20 g. Its mass when filled with water is 40.0 g an d 50.0 g when filled with liquid X. Calculate the density of liquid X if the density of water is 1,000 kgm^-3.

Solution

Mass of water = 40 – 20 = 20 g = 0.02 kg.
Volume of water = 0.02 / 1,000 = 0.00002 m3.
Volume of liquid = volume of bottle
Mass of liquid = 50 – 20 = 30 g
= 0.03 kg
Therefore density of liquid = 0.03 / 0.00002 = 1,500 kgm^-3

Relative Density

Liquids tend to form layers when added to water. The sugar alcohol glycerol (1,261 kg/m³) will sink into the water and form a separate layer until it is thoroughly mixed (glycerol is soluble in water). Vegetable oil (approx. 900 kg/m³) will float on water, and no matter how vigorously mixed, will always return as a layer on the water surface (oil is not soluble in water).

Relative density is the density of a substance compared to the density of water.It is symbolized by (d) and has no units since it’s a ratio.

Relative density (d) = density of substance / density of water.

It is measured using a relative density bottle

Example

The relative density of some type of wood is 0.8. Find the density of the wood in kg/m³.

Solution

Density of substance = d × density of water
Density of substance = 0.8 × 1,000 = 800 kgm^-3

Densities Of Mixtures

We use the following formula to calculate densities of mixtures

Density of the mixture = mass of the mixture / volume of the mixture

Example

100 cm³ of fresh water of density 1,000 kgm^-3 is mixed with 100 cm³ of sea water of density 1030 kgm^-3.

Calculate the density of the mixture.

Solution

Mass = density × volume
Mass of fresh water = 1,000 × 0.0001 = 0.1 kg
Mass of sea water = 1030 × 0.0001 = 0.103 kg
Mass of mixture = 0.1 + 0.103 = 0.203 kg
Volume of mixture = 100 + 100 = 200 cm3 = 0.0002 m³
Therefore density = mass / volume = 0.203 / 0.0002 =1,015 kg/m³.
Archimedes Principle and Law of Floatation

Archimedes’ principle deals with the forces applied to an object by fluids surrounding it. This applied force reduces the net weight of the object submerged in a fluid. When an object is partially or fully immersed in a liquid(as shown in the diagram below), the apparent loss of weight is equal to the weight of the liquid displaced by it

Archimedes’ principle states that:

“The upward buoyant force that is exerted on a body immersed in a fluid, whether partially or fully submerged, is equal to the weight of the fluid that the body displaces and acts in the upward direction at the center of mass of the displaced fluid”.

If you look at the figure above, the weight due to gravity is opposed by the thrust provided by the fluid. The object inside the liquid only feels the total force acting on it as the weight. Because the actual gravitational force is decreased by the liquid’s upthrust, the object feels as though its weight is reduced. The apparent weight is thus given by:

Apparent weight= Weight of object (in the air) – Thrust force (buoyancy)

In simple terms, Archimedes’ principle tells us that the weight loss is equal to the weight of liquid the object displaces. The law states that the buoyant force on an object is equal to the weight of the fluid displaced by the object. Mathematically written as:

Fb = ρ x g x V

Where Fb is the buoyant force, ρ is the density of the fluid, V is the submerged volume, and g is the acceleration due to gravity.

Applications of Archimedes’ Principle

Following are the applications of Archimedes’ principle:

1. Submarine:

The reason why submarines are always underwater is that they have a component called ballast tank which allows the water to enter making the submarine be in its position underwater as the weight of the submarine is greater than the buoyant force.

2. Hot-air balloon:

The reason why hot-air balloons rise and float in mid-air is because the buoyant force of the hot-air balloon is less than the surrounding air. When the buoyant force of the hot-air balloon is more, it starts to descend. This is done by varying the quantity of hot air in the balloon.

3. Hydrometer:

A hydrometer is an instrument used for measuring the relative density of liquids. Hydrometer consists of lead shots which makes them float vertically on the liquid. The lower the hydrometer sinks, the lesser is the density of the liquid.

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