1. (a)
ACJCP2Q6 Three alpha particles with equal energies are fired towards the nucleus of a gold atom.
Complete the diagram, showing the possible paths of the alpha particles (labeled 1, 2 and 3) as they passes by, and then moves away from the nucleus. [3] Consider a fourth alpha particle approaching the nucleus head-on. State the energy changes that take place during approach. [1]
1 2 3
Gold Nucleus
(b) In an α-particle scattering experiment, an α-particle and a gold nucleus (in a piece of gold foil) collide head on and the α-particle rebounds. (i) Using the data given, write down a numerical expression for the electrostatic force of repulsion, F , acting on the α-particle at the instant of collision when the distance between α-particle and gold nucleus is x . x2 ] [3.64×10 -26 / x [For α-particle, A = 4, Z = 2; for the gold nucleus, A = 197, Z = 79.] (ii) The electric potential energy of the α-particle and the gold nucleus at the point of impact is Fx . If the initial kinetic energy of the α-particle is 1.8 MeV, calculate a value for x , assuming the gold nucleus has no kinetic energy initially. [1.26×10-13 m] What indication does the value of x give about the size of the two particles involved?
1
238
(c)
A 92 U nucleus undergoes alpha decay. The resulting daughter nucleus is Thorium (Th). Write the equation for this reaction. [1]
(d)
Ionizing radiation is a ‘two-edged sword’. Briefly describe how gamma-ray can be both beneficial to medical science and yet harmful to the human body if not properly handled. [2]
2.
AJCP3Q7/part
(c)
The isotope Iron-59 is a β-emitter with a half-life of 45 days. In order to estimate engine wear, an engine component is manufactured from nonradioactive iron throughout which the isotope Iron-59 has been uniformly distributed. The mass of the component is 2.4 kg and its initial activity is 8.5 x 10 7 Bq. The component is installed in the engine 60 days after manufacture of the component, and then the engine is tested for 30 days. During the testing period, any metal worn off the component is retained in the surrounding oil. Immediately after the test, the oil is found to have a total activity of 880 Bq. Calculate (i) the decay constant for the isotope Iron-59, [1] (ii) the total activity of the component when it was installed, [1] (iii) the mass of iron worn off the component during the test. [2]
(d)
Two deuterium nuclei fuse together to form a Helium-3 nucleus, with the release of a neutron. The reaction is represented by 2 1
2
3
1
H + 1 H → 2 He + 0 n + energy 2
= 2.013 66 u
3
= 3.014 93 u
rest mass of 1 H nucleus rest mass of 2 He nucleus rest mass of neutron rest mass of proton rest mass of electron (i)
(ii) (iii)
= 1.008 67 u = 1.007 28 u = 0.000 55 u
Calculate the binding energy per nucleon for 1. the deuterium nucleus, 2. the Helium-3 nucleus. How much energy is released in this reaction? Estimate the energy available from conversion by fusion of 1 kg of deuterium into Helium-3.
[1] [1] [2] [1]
(e)
Explain whether nuclear fusion or nuclear fission releases more energy per unit mass. [2] 7 -5 (c) 0.0154 / day; 3.37 x 10 Bq; 9.94 x 10 kg; (d) 1.79 x10 -13 J; 4.13 x10 -13 J; 7.82x10 13J (e) There is a greater change in binding energy per nucleon during fusion.
2
3. (ai) (aii)
CJCP3Q8 State what is meant by the term nuclear binding energy . [1] Figure 8.1 shows the variation of binding energy per nucleon number.
A uranium-235 nucleus undergoes fission and produce two fission products of approximately equal nucleon number. Using the data from Figure 8.1, estimate the energy released from the fission of one uranium-235 nucleus. [3] (aiii)
One other possible reaction for uranium-235 is 235 92
1
236
144
90
1
56 Ba + 36 Kr + 2 0 n U + 0 n → 92 U → The masses of particles are given below in atomic mass units, u, where 1 u is a mass of 1.66 x 10 -27 kg.
1
Mass of 0 n
=
1.009 u
Mass of
235 92
U
=
235.124 u
Mass of
144 56
Ba
=
143.923 u
90
Mass of 36 Kr = 89.920 u Calculate the energy released by this reaction. Suggest with a reason which one of the two fission reactions for uranium-235 is more likely to occur. [1] 3
[165 MeV; 4.06 x 10 -11J] Why is radioactive decay described as a ‘random’ process’? [1] (bi) (bII) Describe how you would demonstrate that radioactive decay is a random process. [1] (biii) Explain why it is not advisable advisable for pregnant ladies to be exposed to a radioactive source. [3] (c)
The nuclide
90 38
Sr is a beta-emitter of half life 28 years but the nuclide
238 34
Pu emits two groups of alpha-particles which differ in energy by 0.045 MeV. 90
If a 38 Sr source emits many beta particles in one second today, how long will it take it to emit the same number of beta particles in the year 2064 ? 4s [4] Discuss how the energy is conserved in the two types of plutonium (Pu) disintegration, despite there being this difference in the energies of the alpha particles produced. [1] 4. MJCP3Q4 During one particular fission process, s slow moving neutron bombards a Uranium-236 nucleus giving, among its fission products, s Strontium-90 nucleus and a Xenon-143 nucleus.
Figure 4.1 shows the variation with mass number of the binding energy per nucleon for various nuclides.
4
(a) What is meant be ‘ binding energy per nucleon of a nucleus’? Show that the energy released in the fission process is 247Mev. What other fission products are produced in this process? Why are these particles not taken into account in the calculation? (b) Why does a release of energy occur when there is an increase in binding energy? 90
(c) One of the fission products, Strontium 38 Sr is radioactive and decays with a half life of 28.8 years via beta decay into an isotope of Yttrium(Y). Write down an equation for the decay of Strontium-90. A short while after the fission reaction ended, the activity of Sr was found to be 40 Bq. Find the number of Sr nuclei present in the sample at this time. [5.25 x 10 10] 5. NYJCP2Q6 A student is provided with a freshly prepared sample of a radioactive material and the count rate C from the source is found to vary with time t as shown in Fig. 6(a). A second similar sample of the radioactive material is then prepared and the student repeats the experiment, but with the sample at a higher temperature. The variation with time of the count rate for the second sample is shown in Fig. 6(b). The trend lines are also shown in both graphs.
600
600
Fig. 6 (a)
500
Fig. 6 (b)
500
400
400
s / t n 300 u o c
s / t n 300 u o c
100
100
200
200
0
0 0
10 0
2 00
3 00
4 00
50 0
60 0
7 00
8 00
0
10 0
200
3 00
time/s
(a)
50 0
60 0
700
8 00
time/s
State the evidence that is provided by these two experiments for (i) the random nature of radioactive decay, (ii)
(b)
4 00
(i) (ii)
the spontaneous nature of radioactivity decay.
[4]
Estimate the background count of the second sample (Fig.6(b)). Find from the graph the half-life of either sample.
[3]
5
ai) aii)
bi) bii)
The rate of every single decay is unpredictable. The short term fluctuations for the two graphs are different. Both long term trends are exponential graphs. The decay is spontaneous because its rate of decay is independent of temperature (external condition). The graph comes to a constant rate of 100 counts per s. Back ground count = 100 per s Re-number the count rate values of y-axis by deducting 100 count per s. That is 600 becomes 500 and 100 becomes zero. Then at 0 second, count rate = 450 count per s. Half of it = 225 count per s. Time for count rate to drop by half = 100 s from graph
6. NYJCP3Q7 (a) What is meant by the term binding energy for an atomic nucleus and how can this be calculated? [2] (b) The relationship between the binding energy per nucleon and nucleon number is of the form shown in the graph. Explain from this why energy is released in a fission process. [2] Binding energy / MeV per nucleon
Nucleon number
(c) A possible fusion process which has been suggested for the generation of energy is represented by the following equation: 2 1
2
3
1
H + 1 H → 2 He + 0 n + 5.2 x 10
Rest mass of proton Rest mass of neutron Rest mass of H-2 Rest mass of He-3
-13
J
= 1.673 57 x10 -27 kg = 1.674 96 x10 -27 kg = 3.344 54 x10 -27 kg = 5.008 35 x 10 -27 kg
Using the above data, (i)
explain why H-2 is stable,
[2]
(ii)
prove that the energy released in the nuclear fusion is 5.24 x 10 -13 J. [3]
(iii)
Comment on the following statement on the nuclear reaction:
6
‘The total mass of the two hydrogen nuclei before the reaction is more than the total mass of the helium and neutron because part of it is converted to energy.’ [3] The total relativistic mass has not change. The product particles are moving with high speeds after the collision. They have more mass than when they are at rest (which is called the rest mass) because of their kinetic energy. When they are brought to rest after that, their k.e. and hence some mass will be passed to the particles of the medium which slows them down. So there is no mass converted to energy!
(d) In order to accomplish nuclear fusion, fusion , the particles involved must first overcome the electric repulsion to get close enough for the attractive nuclear strong force to fuse the particles. This requires extremely high temperatures temperatures,, if temperature alone is considered in the process. Consider a proton gas at high temperature. Assume a proton is a sphere of radius r 1 x 10-15 m. Two protons are moving at each other with the same kinetic energy K. ≈
r
(i) Calculate the value of K which is enough to bring the two protons to ‘touch’ one another so that the nuclear force can come into action. Explain briefly the number of significant figure given in your answer. [3] -14 [6 x 10 J] (ii)
If the proton gas behaves like an ideal gas, then the average kinetic 3 energy of the protons in the gas kT where k is the Boltzmann constant. What 2 temperature is required for the protons in the gas to have an average kinetic energy as that calculated in d(i)? [2] 9 [3 x 10 K] (iii) Such nuclear fusion process is happening in the core of the Sun. However, the temperature of the core of the Sun is only about 1.5 x 10 7 K, which is well below the calculated value in (d) (ii). Give two reasons, one in terms of classical mechanics and one in terms of quantum theory, why such process is able to sustain? [3]
7
Nuclear Physics
2011
1. Explain how and why the masses of atoms differ from the sum of the masses of their constituent particles. Radium (Ra) decays to radon (Rn) by the reaction 226 88
Ra →
222 86
4
Rn + 2 He + γ
The atomic masses are: Ra: 3.753×10 -25 kg, Rn: 3.686×10 -25 kg, He: 0.066×10 -25 kg. (a) Estimate the energy released when an atom of 226Ra decays. [9.0×10 -12 J] (b) Estimate the wavelength of the γ photon emitted during this decay given that 4% of the energy released turns to γ radiation. [5.53×10 -13 m] (c) What happens to the remaining 96% of the energy? 2. Sketch a graph of the binding energy per nucleon E against the mass number A for naturally occurring isotopes, indicating an approximate scale for A. Use your graph to explain how energy may be obtained from fission of heavy nuclei. From the equation 3 1
2
1
H + 1 H → X + 0 n + 17.6Mev
(a) (b)
Name the nucleus X. Given that mass of
1 0
2
3
n =1.0087 u, 1 H = 2.0141 u, 1 H = 3.0161 u, Calculate the atomic mass of X in u. [4.0027 u] 3. The first nuclear reaction induced in a laboratory (performed by Rutherford in 1919) can be represented by 14 7
4
17
1
N + 2 He→ 8 O + 1 H, in which a collision takes place between an α-particle travelling at 3.0×10 7 m s-1 and a stationary nitrogen nucleus. (a) The total rest masses are as follows: 14 7 17 8
4
N + 2 He
= 18.005 68 u,
1
O+1H = 18.006 96 u. Calculate the small change in mass, in kilogram, which takes place in this nuclear reaction, and the minimum kinetic energy needed by the α-particle to cause the nuclear reaction. [2.125×10 -30 kg, 1.91×10 -13 J]
(b) The particles move in a straight line. The speed of the proton after the collision is 6.0 × 10 7 m s-1. Use the principle of conservation of momentum to calculate the velocity of the oxygen nucleus after the collision (magnitude and direction). [3.53×10 6 ms-1]
8
4. Radioactive iron, 59Fe, is a radioactive nuclide with a half-life of 46 days. It is used medically in the diagnosis of blood disorders. Measurements are complicated by the fact that iron is excreted, i.e. removed, from the body at a rate such that 69 days after administering a dose, half of the iron atoms in the dose have been excreted. That is to say, iron has a biological half-life of 69 days. (a) If the count rate from a blood sample is 960 counts per minute, what what will it be from a similar blood sample taken 138 days later? [30 counts per min] (b) How long after the first sample was taken would a further similar sample give a count rate of 480 counts per minute? [27.6 days] (c) Derive the relation between T, the effective half-life of the radioactive nuclide in the body, T b, the biological half-life, and T r, the radioactive half-life. 5. An α source with an activity of 150 kBq is placed in a metal can as shown in the figure below. A 100 V d.c. source and a 10 9 resistor are connected in series to the can and the source. This arrangement is sometimes called and ionization chamber.
(a) (b)
(c)
(d)
What is meant in this case by an activity of 150 kBq? A potential difference of 3.4V is registered on the voltmeter. i. Calculate the current in the wire at P. State any assumptions you have made.[3.4 made.[3.4 x 10 -9 A] ii. Calculate the corresponding number of ionizations occurring in the metal can per second. Assume that every ion formed in the metal can is singly charged .[2.125 x 10 10] With the α source removed from the metal can, the voltmeter still registers a potential difference of 0.2V. Suggest 2 reasons why the current is not zero. The half life of the α source is known to be 1600 years. Calculate the decay constant and hence deduce the number of radioactive atoms in the source. [1.37 x 10 -11 s-1; 1.09 x 10 16]
9
6. Nuclei of 238Pu decay with a half-life of 90 years, emitting α particles of energy 5.1 MeV. This isotope is to be used to power a heart pace-maker. (a) Calculate the decay constant of 238Pu. [2.44 x 10 -10 s-1] (b) Determine the minimum number of atoms of 238Pu which must be present to give an initial power of 10mW. Assume that the decay of each 238Pu yields only a single α particle. [5.02 x 10 19] (c) Suggest one reason why answer.
238
Pu is chosen for this application. Justify your
7a)
Define half life
(b)
Show that the half life of a radioactive nuclide is equivalent to
ln 2 where λ λ
is the radioactive decay constant. (c) A fresh sample of 5.00g of sodium-24 (half life 15hrs) is prepared in the laboratory. After 2 hours, another fresh sample of sodium 24 of mass m is added to the original sample. Another 24 hours later, the activity of the sample is measured and found to be 7.00×10 17 Bq. Calculate the mass m added to the sample. [2.03 g] (d) In order to find the volume of water in a large tank, the sample of sodium24 with activity 7.00×10 17 Bq is dissolved completely in water. After 3 days, 100ml of the water is extracted and tested for radioactivity. The activity is found to be 5.00×10 11 Bq. Calculate the volume of water in the tank. [5.02 x 10 6 ml ] 226
222
8. Alpha particles are emitted when stationary 91 Pa decay into 89 Ac. If the kinetic energy K α of the α -particles and its range R in the air are related by the equation R = 0.138 K α1.5 where Kα is in MeV and R is in centimeters, compute the range in air of the α-particles emitted. [5.8313 cm] 226 222 [Data: Pa =226.0280u, Ac =222.0178u, 4He =4.0026u, 1 u = 931.48 MeV] 9. Uranium 238U decays to lead 206Pb with a half life of 4.47 ×10 9 years. The equation representing the decay is as such: 238U → 206Pb + various decay products. A rock is found to contain 4.20 mg of 238U and 2.135 mg of 206Pb. Assuming that the rock contained no lead at formation, and all of the lead present now arises from the decay of uranium, (a) How many atoms of 238U and 206Pb does the rock now contain? (b)
How many atoms of
238
U were there in the rock at formation?
(c) What is the age of the rock? [1.062353 x 10 19; 6.239175 x 10 18; 1.686270 x 10 19; 2.9796 x 10 9 yrs]
10
10. A cardiac pacemaker is a device which is used to ensure that a faulty heart beats at a suitable rate. In one pacemaker, pacemaker, the required electrical energy is provided by converting the energy of radioactive plutonium-238. The atoms of plutonium decay by emitting alpha particles of average energy 8.8 x 10 -13 J. The daughter nucleus is an isotope of uranium. The plutonium has a decay constant of 2.4 x 10 -10 s-1. 238 94
(a)
How many neutrons are in the nucleus of the radioactive plutonium
Pu?
(b)
Suggest why alpha particles are more suitable than either beta particles or gamma radiation for use as the power source.
(c)
A new pacemaker contains 180 mg of plutonium
(i) Determine the number of radioactive atoms in the new pacemaker. [4.55 x 10 20] (ii) the
Show that approximately 1 x 10 11 disintegrations occur each second when power supply is new.
(iii) Determine the initial power, in W, of the source when it is new. [0.088 W] 11. Living tissue contains a small amount of radio carbon-14. Upon death, the activity of the carbon-14 decreases due to radioactive decay with a half-life of 5.7 x 103 years. The radioactivity reading obtained from a certain mass of living tissue is (R + 24) counts per minute where R is the number of counts due to background radiation. Determine the reading that a sample of dead tissue of identical mass would give if it came from the Middle Ages (about 1000 A.D.), from the same meter in counts per minute. State one essential assumption made during the calculations. [R + 21] 12. The half-life of a certain radioactive element is such that 7/8 of a given quantity decays in 12 days. What fraction remains undecayed after 24 days? [1/64] 13. In a cancer therapy unit, patients are given treatment from a certain radioactive source. This source has a half-life of 4 years. years. A particular treatment requires 10 minutes of irradiation when the source is first used. How much time is required is required for this treatment, using the same source, 2 years later? [14.1 minutes]
11
187
14. A radioactive source contains the nuclide 74 W which has a half-life of 24 hours. In the absence of this source, a constant average count-rate of 10 s -1 is recorded. Immediately after the source is placed in a fixed position near the counter, the average count-rate rises to 90 s -1. What average count-rate is expected with the source still in place 24 hours later? [50] 15. A stationary radioactive isotope plutonium-238 has a half-life of 86 years and decays by emitting only an alpha particle. The daughter nucleus is an isotope of uranium. The total kinetic energy of the products is 5.649 MeV. 238
The mass of 94 Pu is 238.0496 u. The mass of an alpha particle is 4.0026 u. (a)
Write down an equation representing the decay, indicating clearly the atomic number and mass number of each nucleus. (b) Calculate the mass of the uranium nucleus formed in the reaction, giving your answer in atomic mass unit to 4 decimal places. (c) Calculate the ratio of the kinetic energy of the alpha particle to that of the uranium nucleus. [234.0410 u; 58.47] (d) Satellites can can be powered by the energy produced by the decay of Pu238. Devices known as radioisotope thermoelectric generators (RTGs) on board the satellites convert energy (in the form of kinetic energy of the products) generated from the decay into electrical energy. A particular satellite requires an electrical power of 25 W. The efficiency of the energy conversion process is 15%. (i) (ii)
Explain what is meant by the activity of a radioactive source. Show that the rate of energy production from the radioactive source is 1.04 x 10 21 eVs-1. (iii) What must the minimum activity of plutonium-238 be in order to provide the required energy? [1.84 x 10 14 s-1] (iv)
Calculate the minimum mass of Pu-238 required to provide this level of activity. [0.284 kg]
(v)
What is the minimum mass of Pu-238 that the satellite should carry before launching if the satellite is intended to function properly at a minimum power of 25 W for 20 years? [0.336 kg]
12
(e) Plutonium is not placed in its pure form in RTGs but is installed as bricks of plutonium dioxide, PuO 2, a ceramic which if shattered, breaks into large pieces, rather than smaller more dangerous dust. (i)
Explain why there is no difference in the probability of decay of Pu-238 whether it exists as pure plutonium or in the form of PuO 2.
(ii)
Suggest a reason why large pieces of PuO 2 pose less of a health risk than plutonium dust, which can be inhaled into the body.
16. When iron is irradiated with neutrons an isotope of iron is formed. This isotope is radioactive with a half-life of 45 days. An iron piston ring of mass 16g was irradiated with neutrons until its activity due to the isotopes is 3.7 ×10 5Bq. Ten days after the irradiation the ring was installed in an engine and after 80 days of continuous use the crankcase oil was found to have a total activity of 1.85×10 3 Bq. Determine the average mass of iron worn off the ring per day, assuming that all the metal removed from the ring accumulated in the oil . [4 x 10-3 g] 17. Nuclide
Half-life/s
Decay Constant /s -1
Americium 241Am 1.48 × 1010 4.68 x 10 -11 Cobalt 60Co 4.18 x 10 -9 Phosphorus 32P 1.24 × 10 6 5.59 x 10 -7 Sodium 24Na 5.42 × 10 4 1.28 x 10 -5 (a) Calculate the half life of 60Co. [1.66 x 10 8s]
λ /kg-1 s-1 Mm 1.94 x 10 -10 6.97 x 10 -8
5.33 x 10 -4
λ for 32P where λ is the decay constant and M m the Mm molar mass of the nuclide. [1.75 x 10 -5 kg-1 s-1]
(b)
Calculate the ratio
(c)
Which of the nuclides have the greatest activity per unit mass? Explain your answer. answer . [Sodium]
(d)
Calculate the activity of a mass of 2.0 x 10 have named in (c). [6.42 x 10 8Bq]
(e)
Why must radioactive waste normally be stored for a period of time before disposal into the environment?
(f)
A laboratory has facilities suitable for the storage of waste radioactive material for periods not exceeding 3 months (7.8 × 10 6 s). For which of the nuclides would storage for 3 months before disposal be worthwhile? Give your reasons. [32P and 24Na]
-12
kg of the nuclide which you
13
Data Analysis : Radioactivity – Nuclear Reactor
A nuclear reactor throughout its life generates power from fission of uranium-235 by neutrons in a reaction of this type: 235 92
1
143
90
1
U + 0 n → 54 Xe + 38 Sr + 3 0 n + 200 MeV The fission products, strontium and xenon, have very small mass and are mostly contained within the fuel cells. The neutrons, on the other hand, have high speeds and are uncharged. These neutrons are able to from the fuel cells into the graphite (carbon) moderator where where they are slowed down. One of the three neutrons is needed to sustain the chain reaction and the others are absorbed by the moderator, control rods and concrete sides. The plan in decommissioning nuclear reactors has to take into account the following information: (i) A reactor has a total mass of 300,000 tonnes and 5% of this may be radioactive. Concrete, graphite and steel take up 8000, 2200 and 3400 3400 tonnes respectively. (ii) The fuel elements can be removed in a normal way. This removes 99.99% of the radioactivity which was on site when the reactor was working. (iii) The remaining radioactivity is mostly neutron-induced in the steel, concrete and graphite structures. (iv) Radioactivity is expressed in Becquerels (Bq). 1 Bq is 1 disintegration per second. (v) British law defines a radioactive substance as one with an activity greater than 400 Bq kg -1. (vi) There are about 2500 known nuclides. Of these, 79 which have a half-life longer than 1 year may be present in a reactor. Most of these 79 nuclides do not reach the activity of 400 Bq kg -1 to necessitate them being called radioactive but some do. (vii) The table below shows the activities of 13 such nuclides in the 3 main sections, namely concrete, graphite and steel of the reactor 10 years after its shutdown. A dash indicates an insignificant amount of nuclide.
14
Nuclide H
Half-life (years) 12
Concrete (1012 Bq) -
Graphite (1012 Bq) 110
Steel (1012 Bq) -
Total (1012 Bq) 110
14 6
C
5700
-
42
1.7
44
36 17
Cl
310 000
-
1.3
-
1.3
41 20
Ca
130 000
0.10
0.90
-
1.0
56 26
Fe
2.7
0.82
4.0
2400
2400
60 27
Co
5.2
0.28
11
680
690
59 28
Ni
80 000
-
0.096
2.1
2.2
63 28
Ni
92
-
17
230
250
63 41
Nb
20 000
-
-
0.013
0.013
108 47
Ag
130
-
0.0017
0.017
0.019
151 62
Sm
90
0.15
0.051
-
0.020
152 63
Eu
12
0.82
0.11
-
0.93
154 63
Eu
16
0.10
0.61
-
0.71
2.3
190
3300
3500
3 1
Total
All figures are given to 2 significant figures. κ represents κ capture. This is a mode of decay in which the nucleus captures one of the atom’s orbiting electrons. Answer the following questions using the above data: (a) What is the activity in terms Bq kg -1 of each of the three main sections of the reactor? [2.88 x 10 6; 8.64 x 10 7; 9.71 x 10 8] (b) Why is it that over a period of 10 years, the radioactive iron-55 in the reactor is less of a problem than the radioactive nickel? (c) Why is it likely that the activity of the long half-life nuclides would be low? (d) How long after shutdown, will the activity of tritium in the reactor fall below 0.43 x 10 12 Ba? [106 years] (e) Identify the nuclide(s) that would determine the time required to isolate the reactor before it is deemed safe to demolish. Explain your reasons. (f) By making appropriate estimations, evaluate numerically, the number of years required to isolate the reactor. [3.3 x 106 years]
15
