How much energy would be required to break a helium nucleus in...

Question

How much energy would be required to break a helium nucleus into its constituents, two protons and two neutrons? The rest masses of a proton (including an electron), a neutron, and neutral helium are, respectively, 1.00783 u, 1.00867 u, and 4.00260 u. (This energy difference is called the total binding energy of the $sub 2 to the fourth power He$ nucleus.)

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice prom together. So first off, let us read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem, determine the energy required to separate a carbon 12 carbon, 12, 6 nucleus into its constituent particles, six protons and six neutrons. This energy difference is known as the total binding energy of the nucleus. The rest masses for the particles are given as follows proton with an electron included MP equals 1.00728 U neutron mn equals 1.00866 U neutral carbon 12 atom MC equals 12.00000 U. Awesome. So whenever I say U what U means is the atomic mass units. So it's an atomic mass units. Awesome. So it appears for this particular problem. The final answer that we're ultimately trying to solve for our end goal is we're trying to figure out the energy that is required in order to separate this carbon 12 nucleus into its respected particles into its constituent particles, which is six protons and six neutrons in this particular case. So now that we know that we're trying to figure out how much energy is required to separate this carbon 12 nucleus. Let's read off multiple choice answers to see what our final answer might be noting that they're all in the same units of mega electron volts. So A is 81 B is 85 C is 89 and D is 92. OK. So quickly recapping here writing down all of our known variables, we know that the mass of the proton including an electron is equal to 1.00728 U. So I'm just gonna say you from now on, but just know whenever I say U I'm saying atomic mass units, we also know the mass of the neutron and the mass of the neutron is given to us by the problem itself as 1.00866 U. And we're also told that the mass of the neutral carbon 12 atom, which is MC is equal to 12. And then we need to make sure we have 12.00000 U. So there's five zeros. Awesome. So now that we have our given variables written down, our next step is we need to calculate the total mass of the free nucleons. So first off, as we should recall, a note, we need to calculate the combined mass of the six protons and the six neutrons when they are separate and not bound within the nucleus. So let's call the mass of the nucleons to be NL. So nucleons, NL. So MNL stands for the mass of the nucleons and the mass of the nucleons is gonna be equal to six multiplied by the mass of the proton with an electron included plus six multiplied by the mass of the neutron, which when we plug in our known variables, we'll find that the mass of the nucleons is equal to six multiplied by our MP value which is 1.00728 U plus six, multiplied by our mass of our neutron, which is 1.00866 U. So let me plug that into our calculator. We will find that the mass of the nucleons is equal to 12.0956 for you. Fantastic. So now at our next step is we need to calculate the mass deficit. So as we should recall and note the mass deficit, which is denoted as delta M delta M is the difference between the total mass of the separate nucleons and the actual mass of the carbon 12 nucleus. So as we should recall, delta M, which is our mass deficit is equal to our mass of our nucleons minus MC. Awesome. So, plugging in our known variables, we know that our mass of our nucleons is 12.09564 U minus 12.00000 U. So that will mean let me plug that into our calculator. That delta M, our mass deficit is equal to 0.09564 U. Awesome. But we have a bit of a problem here, even though this is technically our final answer, we need to convert our units, which right now we have our mass deficit in units of atomic mass units. But we need it to be in units of mega electron volts. So we now need to convert the mass deficit to an energy unit of measurement. So in order to do that, we need to recall and find the binding energy by converting the mass deficit into energy by using Einstein's relation, which as we should recall, Einstein's relation states that E is equal to delta M multiplied by C squared. And let us also recall. So this is a big important one, the conversion factor that one atomic mass unit is equal to 931.5 mega electron volts per the speed of light squared. Awesome. So when we plug this all of our known information into our equation, we will find that E is equal to delta M multiplied by 930. So 931.5 mega electron volts mean as you could see from using our Einstein relation that the speed of light squared value will cancel out. So when we plug in our known variables, so that means we'll take our delta M value which is 0.09564. And then we will multiply it by 931.5 mega electron volts. So when we plug that into our calculator, we will find that E when we round to the nearest whole number or to two significant figures, like all of our multiple choice answers, we will find that E is approximately equal to 89 mega electron volts. And that is our final answer. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be the letter C which once again is 89 mega electron volts. Thank you so much for watching.

Key Concepts

Binding Energy

Mass-Energy Equivalence

Nuclear Mass Defect

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