Compare the amount of energy released when 1.0 kg of the uranium isotope ^235 U undergoes the f

Compare the amount of energy released when 1.0 kg of the...

Key Concepts

Nuclear Fission

Nuclear fission is a process in which a heavy atomic nucleus splits into two or more smaller nuclei, along with the release of a significant amount of energy. This energy is primarily derived from the differences in nuclear binding energy between the original nucleus and its fission products. In a typical fission reaction, as seen in uranium-235, the mass of the resultant products is slightly less than the original mass, with the missing mass converted into energy according to Einstein’s mass?energy equivalence principle.

Nuclear Fusion

Nuclear fusion is the process by which two light atomic nuclei combine to form a heavier nucleus, releasing energy in the process. The energy released comes from the fact that the binding energy per nucleon of the resulting nucleus is higher than that of the original two nuclei. Fusion reactions, such as those involving hydrogen isotopes, power the stars and offer the potential for a very high yield of energy, though the conditions necessary to initiate and sustain the reaction are typically extreme.

Mass–Energy Equivalence

Mass–energy equivalence is a fundamental principle in physics encapsulated by Einstein’s famous equation E=mc². This principle explains that mass can be converted into energy and vice versa. In both fission and fusion reactions, small amounts of matter are transformed into a large amount of energy, making it possible to compare the energy output from reactions involving different nuclei on a per mass basis.

Binding Energy

Binding energy is the energy required to disassemble a nucleus into its individual protons and neutrons. It reflects the stability of the nucleus and varies across different elements and isotopes. High binding energy per nucleon typically indicates a more stable nucleus. The differences in binding energy before and after a nuclear reaction determine the amount of energy released, allowing for comparisons between the energy outputs of fission and fusion processes.

Transcript

00:01 Hello, and in this question here, we want to compare the nuclear fissions of the uranium isotope 235 with the nuclear fissions of the hydrogen of hydrogen isotopes.

00:17 So we're comparing one kilo of uranium versus one kilo of hydrogen.

00:24 So to do this, we're going to first calculate the energy released in one of these nuclear.

00:32 Reactions and then determine how many nuclear reactions there would be in one kilo of the substance.

00:39 So to begin with, that is determined the energy released in the equation in the first reaction.

00:47 So the energy released will be equal to e, which will be equal to the change in mass of the initial state minus the mass of the final state.

00:58 Okay.

01:00 So this change in mass, this mass, lost by the state system will be converted into energy, and the relation between mass and energy is e equals nc squared from einstein's equation.

01:14 So if we multiply this change in mass by c squared, we can determine energy.

01:20 This change of mass, well, like i said, this is the mass of the initial state, so the mass of the uranium plus the mass of the neutron, minus the mass of the of the final state which is the mass of the barium plus what if we have a minus side out here so we're minusing all of this all of these three masses plus the mass of the crontium plus the mass of three neutrons which is three times the mass of a single neutron okay and this here is equal to delta m so if we multiply by c squared we'll get the energy so from the appendices in the back of the book the mass of a neutron is equal to 1 .0086 -649 u.

02:14 The mass of a barium -barium -141 is equal to 140 .9145 u.

02:28 The mass of a crontium 92 is equal to 91 .440 .5 u.

02:38 92261731 u and the mass of uranium 235 is equal to 235.

02:53 5999 u okay now u is defined as 112 of the mass of a carbon 12 atom and has a value of 1 .66 times 10 to the minus 27 kg and c is equal to 3 times 10 to the 8 meters per second.

03:13 So we have all the variables in this equation here.

03:19 And something they're saying, this allows us to determine that the energy is equal to 1 .18 ,569, u times c squared.

03:32 And something in the values for u and c, we get the energy released in this fishing reaction is 1 .77 times 10 to the minus 10 jubles.

03:43 So we have energy released in one reaction.

03:47 We need to determine how many reactions would take in one kilogram of uranium.

03:52 So to do this, we determine the number of reactions.

03:56 We take one kilogram and we divide by the mass of a single uranium atom.

04:02 Well, from above we have this equal to one kg.

04:06 Times 235 .043 .599 times 1 .66 times 10 to the minus 27 kilograms and this is equal to, well this is equal to 2 .56 times 10 to the 24 and this is the number of a uranium two, three, five in one kg.

04:48 So if we assume each of these uranium will decay, the amount of energy that is released will be the number of decays multiplied by the energy of each individual decay.

05:00 So the total energy is equal to 1 .77 times 10 to the minus 10 joules...

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