(a) One such reaction fuses two nuclei of carbon-12 ( 6
12C ) to form an atom
of neon ( 10
20Ne ) with the emission of an α-particle. Write an equation for
this process, indicating the atomic number and mass number of each of
the products in the usual way.
(b) Another reaction involving carbon-12 in stars is one in which a
carbon-12 nucleus fuses with a hydrogen nucleus (atomic mass of 1) to
form a nucleus of nitrogen, releasing 1.95 MeV in the process.
Write down the equation for this process, and using values for the
binding energy of the hydrogen and carbon-12 nuclei of 0 MeV and
–92.2 MeV respectively, calculate the binding energy of the nitrogen
nucleus. (Show all your working.)
(c) Convert the energy liberated in the reaction in part (b) into the
equivalent amount of energy in joules, and then calculate the amount of
mass that has been converted into energy as a result of this process.
Using SI units throughout show that the units for the mass are
kilograms.
(Express your answers in scientific notation to an appropriate number of
significant figures.)
(a) The equation for the fusion reaction of two carbon-12 nuclei to form neon-20 with the emission of an alpha particle (helium-4) can be written as:
6
12C + 6
12C → 10
20Ne + 2
4He
(b) The equation for the fusion reaction of carbon-12 with a hydrogen nucleus (proton) to form nitrogen can be written as:
6
12C + 1
1H → 7
13N + energy
From the given binding energy values:
Binding Energy of Hydrogen nucleus (1H) = 0 MeV
Binding Energy of Carbon-12 nucleus (12C) = -92.2 MeV (negative sign indicates the energy release during the reaction)
To calculate the binding energy of the nitrogen nucleus (13N), we need to consider the energy released during the process. The energy released is given as 1.95 MeV.
Total Binding Energy of products = Binding Energy of Nitrogen nucleus (13N) + Energy released
E(total) = E(N) + 1.95 MeV
E(total) = E(N) + (1.95 * 1.6 * 10^-13 J) [1 eV = 1.6 * 10^-19 J]
Since binding energy is negative for nuclei, we can consider the magnitude and take the negative sign later.
E(N) = - (E(total) - 1.95 MeV)
E(N) = - (E(total) - (1.95 * 1.6 * 10^-13 J))
(c) To convert the energy liberated in the reaction to joules and calculate the mass converted into energy, we can use Einstein's famous equation, E = mc^2. Rearranging the equation, we get m = E / c^2.
Energy liberated in the reaction = 1.95 MeV = 1.95 * 1.6 * 10^-13 J
Mass converted into energy (m) = (Energy liberated) / (speed of light)^2
m = (1.95 * 1.6 * 10^-13 J) / (3 * 10^8 m/s)^2
To express the mass in kilograms, let's multiply the numerator and denominator by (1 kg) / (1 J*s):
m = (1.95 * 1.6 * 10^-13 J) / (3 * 10^8 m/s)^2 * (1 kg) / (1 J*s)
m = (1.95 * 1.6 * 10^-13 kg*m^2/s^2) / (3 * 10^8)^2
m = (1.95 * 1.6 * 10^-13 kg*m^2/s^2) / (9 * 10^16)
(a) The equation for the fusion reaction between two carbon-12 nuclei to form an atom of neon-20 with the emission of an alpha particle can be written as:
6
12C + 6
12C → 10
20Ne + 2
4He
In this equation, the atomic number and mass number of each product are indicated.
(b) The equation for the fusion reaction between a carbon-12 nucleus and a hydrogen nucleus to form a nucleus of nitrogen, releasing 1.95 MeV in the process can be written as:
6
12C + 1
1H → 7
13N + 1
0n + 1.95 MeV
To calculate the binding energy of the nitrogen nucleus, we need to use the concept of mass defect. The mass defect (Δm) is the difference between the total mass of the reactants and the total mass of the products.
Mass defect (Δm) = (mass of reactants) - (mass of products)
For the reactants:
Mass of carbon-12 nucleus (12
C): 12.0000 amu
Mass of hydrogen nucleus (1
H): 1.0078 amu
For the products:
Mass of nitrogen nucleus (13
N): x amu (to be determined)
The total mass of the reactants is 12.0000 amu + 1.0078 amu = 13.0078 amu.
The total mass of the products is x amu.
Δm = (mass of reactants) - (mass of products)
Δm = 13.0078 amu - x amu
Δm = -x amu (since the mass defect is negative)
According to Einstein's mass-energy equivalence equation (E = mc^2), the mass defect is converted into energy.
Energy released (E) = Δm * c^2
c is the speed of light in a vacuum, approximately 3.00 x 10^8 m/s.
Inserting the values:
E = (-x amu) * (3.00 x 10^8 m/s)^2
E = (-x amu) * (9.00 x 10^16 m^2/s^2)
E = -9.00 x 10^16 x amu
To calculate the binding energy of the nitrogen nucleus, we can use the given binding energies:
Binding energy of hydrogen nucleus: 0 MeV
Binding energy of carbon-12 nucleus: -92.2 MeV
In this reaction, the binding energy is released, so it is negative.
ΔBE = (binding energy of reactants) - (binding energy of products)
ΔBE = (0 MeV) + (-92.2 MeV)
ΔBE = -92.2 MeV
Since energy is released in the reaction, the absolute value of the binding energy is equal to the energy released:
|ΔBE| = 1.95 MeV
|-92.2 MeV| = 1.95 MeV
Now we can set up a proportion to find x:
|-92.2 MeV| 1.95 MeV
___________ = _________
x y
Solving for x:
x = |-92.2 MeV| * (y / 1.95 MeV)
x = |-92.2 MeV| * (1.95 MeV / 1.95 MeV)
x = |-92. 2 MeV| * (1.00)
x = 92.2 MeV
Therefore, the binding energy of the nitrogen nucleus is 92.2 MeV.
(c) To convert the energy liberated in the reaction in part (b) into the equivalent amount of energy in joules, we need to use the conversion factor 1 MeV = 1.602 x 10^-13 Joules.
Energy in joules = Energy in MeV * (1.602 x 10^-13 J/1 MeV)
Energy in joules = 1.95 MeV * (1.602 x 10^-13 J/1 MeV)
Energy in joules = 3.1209 x 10^-13 J
Next, we can use Einstein's mass-energy equivalence equation (E = mc^2) to calculate the amount of mass that has been converted into energy. Rearranging the equation, we have:
m = E / c^2
Using c = 3.00 x 10^8 m/s:
mass (m) = (3.1209 x 10^-13 J) / (3.00 x 10^8 m/s)^2
mass (m) = (3.1209 x 10^-13 J) / (9.00 x 10^16 m^2/s^2)
mass (m) = 3.47 x 10^-31 kg
Therefore, the amount of mass that has been converted into energy is approximately 3.47 x 10^-31 kg.