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1.1b Domain and Codomain

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Domain and Codomain

  1. In the relation between one set and another, the first set is known as the domain and the second set is known as the codomain.
  2. Elements in the domain is called objects, whereas elements in the codomain mapped to the objects is called the image.
  3. Elements in the codomain not mapped to the objects are not the image.
  4. All images in codomain can be written as a set known as range.
Example:

Domain = {3, 4, 5}
Codomain = {7, 9, 12, 15}
Range = {9, 12, 15} [Note: 7 is not an image because it is not mapped to any object.]

3 is the object of 9, 12 and 15.
4 is the object of 12.
5 is the object of 15.

9, 12 and 15 are the images of 3.
12 is the image of 4.
15 is the image of 5.


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SPM Add Math - Function: Finding Domain, Codomain and Range

10.6.5 Generation of Electricity from Nuclear Fission

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Nuclear Reactor

  1. A nuclear reactor produces tremendous amount of energy through nuclear fission.
  2. The energy liberated from the fusion of nuclear fuel heats the surrounding water.
  3. As a result, steam is generated to drive turbines, which in turn drive the electrical generators.
  4. The main components of a nuclear reactor:
    1. Graphite moderator
      Fast moving neutrons are slowed down by collisions with nuclei in the moderator so that they can cause further fissions. In some nuclear power plant, the moderator is water.
    2. Uranium rod (Fuel)
      Fission reactions occur in the uranium rod to produce nuclear energy. The uranium used is often 'enriched' by increasing the proportion of the isotope uranium-235 above the natural value of 0.7% to 3%.
    3. Control rod
      The rate of the fission reaction is controlled by inserting or withdrawing these rods. The nuclei in the rods absorb neutrons without undergoing any reaction. Sometimes the rod is made of cadmium.
    4. Coolant
      To take away the heat from the nuclear reactor. Substances with high specific heat capacity such as 'heavy' water and carbon dioxide are used.
    5. Thick concrete wall
      To prevent the escape of harmful radiations.
    6. Steam generator
      Water in the generator is heated and changed into steam. The steam then drives the turbines.
    7. Turbine
      To turn the dynamo in the electrical generator to produce electricity.

 

10.6.4 Nuclear Fusion

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  1. In nuclear fusion, two or more small and light nuclei come together to form a heavier nucleus.
  2. This process is accompanied by the release of a huge amount of energy.
  3. Below are two examples of fusion reactions:
  5. Fusion is much more difficult to achieve than fission because the hydrogen nuclei repel each other. Therefore, the nuclei must be heated to 108 K or more so that the nuclei will have enough of kinetic energy to overcome the electrical repulsion between the nuclei.
  6. Example
    1. The Sun get its energy from the fusion of hydrogen nuclei.
    2. A hydrogen bomb uses the principle of nuclear fusion for its design.

 

 

10.6.3 Chain Reaction

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  1. If neutrons from the fission of uranium-235 continue to split other nuclei causing further fission, a chain reaction has occurred.
  2. The number of nuclei which undergo fission multiplies rapidly.
  3. In order for a chain reaction to take place, a minimum of one neutron from each fission must trigger further fission.
  4. At the same time, the mass of fission material must exceed a certain minimal mass known as the critical mass. If the material is less than this value, too many neutrons escape without hitting nuclei, preventing a chain reaction from happening.
(Chain Reaction)

 

 

10.6.2 Nuclear Fission

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Nuclear Reaction

  1. In a nuclear reaction, the mass of the parent particles will become less (know as mass defect). The defected mass is then converted into energy called the nuclear energy.
  2. In short, nuclear energy is the energy released owing to the defect of mass in a nuclear reaction.
  3. There are 2 types of nuclear reaction
    1. nuclear fission
    2. nuclear fusion
  4. Nuclear fission is the process of splitting nucleus into 2 smaller nuclei whereas nuclear fusion is the process which 2 small nuclei combine to form a larger nucleus.

Nuclear Fission

  1. Nuclear fission is a process involving the splitting of a heavy nucleus into two nuclei of roughly equal mass and shooting out several neutrons at the same time.
  2. Nuclear fission seldom occurs spontaneously. Usually, it occurs when the heavy nucleus is bombarded by a neutron.
  3. Fission reaction resulting from neutron absorption is called induced fission. Nuclei that undergo fission without initial neutron absorption are undergoing spontaneous fission.
  4. Two typical examples of fission reactions:


Example 1
In a nuclear reaction, the mass difference in the reaction is 1.5 x 10-8kg. Find the heat released in this reaction. [Speed of light = 3.0 x 108 ms-1]

Answer:
Mass defect, m = 1.5 x 10-8kg

Heat released,
E = mc²
E = (1.5 x 10-8)(3 x 108)
E = 1.35 x 109 J


Example 2
A nuclear explosion released 8.2 x 1013 J of energy. What is the mass defect of uranium-235 in this reaction?
[Speed of light = 3.0 x 108 ms-1]

Answer

 

10.6.1 Nuclear Energy

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  1. In a radioactive decay, one element changes into another in a process called transmutation.
  2. The mass of the daughter particles are less than that of the parent particle. This difference in mass is called mass defect or mass loss.
  3. Mass defect is the missing mass in a nuclear reaction and the missing mass will turn into thermal energy and kinetic energy of the product particles.
  4. The mass lost is converted into energy.
  5. Einstein's formula can be used to calculate the amount of nuclear energy released from the defect of the mass in a radioactive decay or nuclear reaction.


    Einstein Formula

    E = mc²

    where 
    m = mass change, in kg
    c = speed of light, in m s-1
    E = energy changed, in J

 

10.5.1.4 Applications of Radioisotopes in Archeology

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  1. In archaeology radioisotope carbon-14 is used to study and estimate the age of ancient artifacts. This method is named as the radiocarbon dating. 
  2. Radiocarbon dating can be used to estimate the age of organic materials, such as wood and leather, up to about 58,000 to 62,000 years.




Example:
A piece of wood found in a cave of an archaeology site has a carbon-14 activity 25% of the activity from a live plant. Estimate the age of the wood. [Half-life of carbon-14 = 5730 years]

Answer:
100% → 50% → 25%

Carbon-14 take 2 half-life to decay from 100% to 25%, hence the age of the wood
= 2(5730)
= 11460 years