Development of a Mathematical Model


Pollution is the release of environmental contaminants and the major forms of pollution include air pollution, water pollution, radio active pollution, noise pollution, and soil contamination, light pollution and visual pollution. [1]

The leading cause of air pollution is motor vehicle emissions and industrial waste, air pollution leads to acid rain and also causes diseases such as cancer and asthma. The main causes of soil contamination are the release of chlorinated hydrocarbons and heavy metals.

Scenario one

1. Over several months, 1000 litres of unleaded gasoline have leaked from an underground

Development of A Mathematical Model

tank. This has occurred in a residential city area. The city is a hilly area, underlain by heavy fissured Carboniferous limestone, shale beds, clay and volcanic rocks. Major immediate concerns are the possibility of underground movement of the gasoline, pollution of ground water and sewage systems and contamination of housing by toxic and flammable fumes.

The effects are shown in diagram one below.

Diagram one show the effects of the linkages of the underground gasoline on underground water, flammable fumes to households and on the sewage system.

The effects of this spillage is mainly on water pollution and air pollution, the sewage pollution is not likely to have any effect on the people, the pollution of underground water as a result of this spillage and air pollution due to the effects of flammable flames are likely to affect the people and therefore we will develop a mathematical model as follows that explains this effects.

The effect of this spillage will be

The pollution of land

Air pollution due to the flammable fumes

Natural resource pollution and in this case it is the pollution of Carboniferous limestone

Water pollution which in this case is the pollution of underground water

Development of A Mathematical Model

Petroleum products are hydrocarbons and have the chemical formula CnH2n+2; Trimethylpentane is used as gasoline and has the chemical formula C




in this scenario therefore the contact of petroleum with oxygen takes the following reaction:

C8H18 (L) + 12.5O2 (g) → 8CO2 (g) + 9H2O (g) + HEAT

This reaction with oxygen produces carbon dioxide, water and heat; however the incomplete combustion of gasoline is harmful in that it causes the formation of carbon monoxide, and nitric oxide as shown below in the chemical reaction:

C8H18 (L) + 12.5O2 (g) + N2 (g) → 6CO2 (g) + 2CO (g) + 2NO (g) + 9H2O (g) + HEAT [2]

Therefore this will result to the formation of gases that once they are inhaled endanger health, carbon monoxide reacts with red blood cells which carry oxygen from the lungs to the other pats of the body to form an irreversible compound therefore a person who inhales this gas could die due to insufficient oxygen in the body.

Development of A Mathematical Model

Petroleum is insoluble in water and therefore does not react with water, it is less dense than water and therefore will float in water, therefore to its causes to underground water will only affect the availability of clean underground water.

We will not consider the effect on land because there are no major direct effects as a result of the spillage, we shall not also consider the effect of the spillage on sewage lines because the contents are not of use to human beings.

Our major concern therefore is the production of carbon monoxide and nitric oxide, we can form the following mathematical model that analysis the effect of one litre of fuel:

One mole of gas occupies 26.5 litres volume at standard temperature and pressure, therefore one mole of C8H18 will produce 2 moles of carbon monoxide and two moles of nitric oxide, therefore if 1,000 litres of petroleum was spilt then this will produce 37.74 litres of carbon

monoxide and 37.74 moles of nitric oxide.


The model can therefore be specified as follows E= 2 litres carbon monoxide + two litres nitric oxide, and this is per litre of petroleum spilt, Where E is the total effect


Scenario two

Development of A Mathematical Model

1. There has been a spillage of gasoline from a road tanker in a town (population, 5500). Most of the gasoline has entered a river that runs through the town, via road drains. Major concerns are the immediate threat to residents by toxic and flammable fumes, and threat to the ecology of the river, from which water is abstracted and treated for drinking.

The diagram below shows the effect of this spillage:

The spillage will cause water pollution causing it difficult for the treatment of drinking water, a loss of the rivers bio diversity example fish will die as a result of water pollution and air pollution due to the presence of flammable fumes. We will not include the effect on the land because only a small portion of the spillage affects it; also we are not going to consider the economic loss of the spillage.

When oil is spilt in water it forms floats at the top and this results to a decline in the amount of oxygen in the water, when this happens the micro organisms and fish in the water will die as a result of lack of oxygen.

There will also a similar resultant with that of the first scenario where

C8H18 (L) + 12.5O2 (g) + N2 (g) → 6CO2 (g) + 2CO (g) + 2NO (g) + 9H2O (g) + HEAT

Therefore there will be the production of carbon monoxide and nitric oxides which are harmful to health, our mathematical model will therefore be estimated as follows

E= 2 litres carbon monoxide + two litres nitric oxide + loss of biodiversity and this is per litre of petroleum spilt.

Development of A Mathematical Model

Scenario three

(c) There has been an over-application of atrazine herbicide on agricultural land bordering a small lowland river. Major concerns are about the possible impact on soil and river ecosystems.

When there is an over application of herbicides in agriculture, herbicide are mainly used to control weeds, therefore when they are used in excess plants will not grow, the chemicals will filtrate into the soil causing a degradation of agricultural land, the chemicals may also find their way into the rivers causing a loss in the bio diversity of the river and also pollute the river in such a way that the water may not be safe for domestic use. [4]

The effects of the herbicides therefore will be

Degradation of agricultural land

Water pollution leading to a loss on the rivers bio diversity

Water pollution leading to lack of drinking water

Development of A Mathematical Model

The effects to consider in our model will only be water pollution and the degradation of agricultural land, we won’t consider other effects such as air pollution because these effects are so negligible. We therefore specify our model as

E = loss of agricultural land + loss of the rivers biodiversity.


Beychok Milton R. (1967) Aqueous Wastes from Petroleum and Petrochemical Plants, John Wiley and Sons publishers, USA

Pollution and its effects (2006) Pollution, retrieved from

Roy M. Harrison (2001) Pollution: Causes, Effects and Control, published by Royal Society of

Chemistry Environmental Studies, USA

Louis McCabe (1952) Air pollution: proceedings, McGraw-Hill publishers, USA

Miranda Alice (1997) The Internationalization of Environmental Protection, Cambridge

University Press, Ca mbridge

Morris Herbert Goran (1981) Conquest of Pollution, Environmental Research publishers, UK


[2] Harrison (2001)

[3] McCabe (1952)

Development of A Mathematical Model

[4] Milton R. (1967)