The nucleus in a 7^12 N atom captures one of the atom's electrons, changing the nucleus to 6^

The nucleus in a 7^12 N atom captures one of the atom's...

Key Concepts

Electron Capture

Electron capture is a process in nuclear physics where a proton-rich nucleus absorbs one of its inner electrons, resulting in the conversion of a proton into a neutron and the emission of a neutrino. This reaction decreases the atomic number of the nucleus while keeping the mass number nearly constant, and it is an important mechanism in the study of radioactive decay and nuclear transmutation.

Mass-Energy Equivalence

Mass-energy equivalence, encapsulated in Einstein's equation E = mc², is a fundamental concept in physics that explains how a difference in mass between the initial and final states of a nuclear reaction is converted into kinetic energy. In nuclear processes like electron capture, this principle is used to determine the energy available for the kinetic energies of the products, including emitted particles such as neutrinos.

Conservation of Energy in Nuclear Reactions

The conservation of energy principle states that the total energy in an isolated system remains constant. In nuclear reactions, the sum of the rest energies and kinetic energies of the products must equal the rest energy of the reactants. This allows physicists to calculate the energy released in reactions by comparing the mass differences before and after the reaction.

Classical and Relativistic Kinetic Energy

In the analysis of nuclear reactions, different expressions for kinetic energy are used depending on the mass and speed of the particles involved. Heavier particles, like nuclei, are often treated using classical kinetic energy formulas because their speeds are relatively low compared to the speed of light. In contrast, nearly massless particles such as neutrinos require a relativistic treatment, where their kinetic energy is related to their momentum by extremely relativistic equations.

Transcript

00:01 For this question, we want to find the energy of the neutrino in a decay of nitrogen 12.

00:11 So let me just write down the decay equation.

00:13 Nitrogen has 7 protons.

00:19 So when it captures one of the electron, right, so this is a beta minus, it actually transformed into a carbon.

00:37 Carbon atom with a neutrino being emitted.

00:43 So we want to find the energy of this neutrino.

00:48 So this is an electron capturing decay or event.

00:54 Now the first thing you want to do is to find what is the change in the masses, between the products and the initial constituents, initial elements that we have.

01:09 And this will tell us how much of the mass is actually converted into energy.

01:16 And this will give us the kinetic energy, the total kinetic energies of the final products.

01:25 So to find the changing mass, we take the mass of the carbon nucleus, right, minus away the six electrons that the carbon atom has.

01:40 Right, we had ignored the mass of the neutrino.

01:44 We assume that neutrino is negligible, has negligible mass.

01:49 Right, then we subtract it with the mass of nitrogen.

01:56 But nitrogen has seven electrons, right? so we gotta take away seven electrons.

02:03 And then we add back the electron that was captured, right? which is over here.

02:10 And we can see that the electrons actually cancel out.

02:14 Right because here is a minus minus so it becomes a plus so it's a plus 6m e over here and then it cancels out so in the end we just get mass of the carbon atom subtracted by mass of the nitrogen atom right this by definition is 12 .0 u this nitrogen over here is 12 .0186132 finally we get our mass difference to be negative 0 .018 6132 u.

03:01 From here then we can get the energy that was released during this decay given by the absolute value of the change in mass multiplied by c squared.

03:16 From here we should get 17 .3381 m e v...

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