Question

The Sun \(_{1}^{1} \mathrm{H}+_{1}^{2} \mathrm{H} \longrightarrow_{2}^{3} \mathrm{He}+\gamma\) is one of the fusion reactions in the Sun. The mass of \(_{1}^{1} \mathrm{H}\) is 1.007825 amu, the mass of \(_{1}^{2} \mathrm{H}\) is \(2.014102 \mathrm{amu},\) and the mass of \(_{2}^{3} \mathrm{He}\) is 3.016029 \(\mathrm{amu}\) a. What is the mass deffect of \(_{2}^{3}\) He? b. What energy is released by the process?

Short answer

The mass defect of \(_{2}^{3} \mathrm{He}\) is 0.005898 amu, which is equivalent to \(9.789708 \times 10^{-30}\) kg. The energy released by the fusion reaction is \(8.810737 \times 10^{-13}\) Joules.

Step-by-step solution

Identify the masses of the reactants and product

The mass of \(_{1}^{1} \mathrm{H}\) is 1.007825 amu, the mass of \(_{1}^{2} \mathrm{H}\) is 2.014102 amu and the mass of \(_{2}^{3} \mathrm{He}\) is 3.016029 amu.

Calculate the total mass of the reactants

Add the masses of \(_{1}^{1} \mathrm{H}\) and \(_{1}^{2} \mathrm{H}\) to obtain the total mass of the reactants. Total mass of reactants = \(1.007825 + 2.014102 = 3.021927\) amu

Calculate the mass defect

Subtract the mass of \(_{2}^{3} \mathrm{He}\) from the total mass of the reactants to find the mass defect. Mass defect = Total mass of reactants - Mass of \(_{2}^{3} \mathrm{He}\) Mass defect = \(3.021927 - 3.016029 = 0.005898\) amu So, the mass defect of \(_{2}^{3} \mathrm{He}\) is 0.005898 amu. #b. Calculate the Energy Released by the Process#

Convert the mass defect to kilograms

We need to convert the mass defect from atomic mass units (amu) to kilograms (kg) to be consistent with the units used in Einstein's mass-energy equivalence formula. Use the conversion factor 1 amu = \(1.660539 \times 10^{-27}\) kg. Mass defect in kg = \(0.005898 \times 1.660539 \times 10^{-27} = 9.789708 \times 10^{-30}\) kg

Apply Einstein's mass-energy equivalence formula

Einstein's mass-energy equivalence formula states that the energy (E) released is equal to the product of the mass defect (∆m) and the speed of light (c) squared. \(E = \Delta m \cdot c^2\) The speed of light, c = \(3 \times 10^{8}\) m/s. Substitute the values of mass defect and the speed of light into the equation.

Calculate the energy released

Calculate the energy released by the process using the mass defect and the speed of light. \(E = (9.789708 \times 10^{-30}\text{ kg}) \times (3\times 10^{8}\text{ m/s})^2\) \(E = 8.810737 \times 10^{-13}\) Joules The energy released by the process is \(8.810737 \times 10^{-13}\) Joules.

Key concepts

Mass Defect Calculation

In nuclear physics, the term mass defect refers to the difference between the combined mass of a nucleus's constituent protons and neutrons and the actual mass of the nucleus. It may sound perplexing, but it's all about the intricate balance of energy and mass.

Let's relate this to the fusion process occurring in the Sun. The hydrogen nuclei undergo fusion to form helium, but if you weigh the hydrogen nuclei and then the resultant helium nucleus, you'll notice the helium weighs slightly less. That missing mass? That’s what we call the mass defect. It’s a bit like making a cake where the finished product weighs less than the sum of its ingredients.

To calculate it, we simply tally up the mass of the reactants, those before the reaction, and subtract the mass of the products, the nucleus after the reaction. The difference is the mass defect, which isn't lost but rather converted into energy. This energy is released according to Einstein's equation, which is where things really get interesting – but more on that in a bit.

Nuclear Reaction Energy

The energy released during a nuclear reaction, such as fusion in the Sun, is due to the mass defect we've just discussed. When these tiny amounts of mass are transformed, they produce an enormous amount of energy.

Imagine our Sun as a giant energy factory, where the mass defect works as a tiny but powerful ingredient to produce incredible amounts of energy. To calculate the actual energy released, we use the mass defect in kilograms and apply it to Einstein's famous formula, which tells us that energy is equal to mass times the speed of light squared. The speed of light is an astounding 299,792,458 meters per second - this is a constant that's vital in connecting the dots between mass and energy.

This is how we get the numbers that tell us exactly how much energy is released when those small bits of mass are transformed during fusion in the Sun. The result? An energy amount that is both mind-boggling and sufficient to sustain life on Earth through the light and heat it provides.

Einstein's Mass-Energy Equivalence

Einstein's mass-energy equivalence is one of the most famous concepts in physics, beautifully encapsulated in the equation we all know: E=mc^2. This equation is essentially a universal currency exchange rate between mass and energy. It tells us that mass can be turned into energy, and energy can be turned into mass.

It's like the nuclear version of the conservation of energy principle, where energy doesn't disappear; it just switches forms. When it comes to nuclear fusion in the Sun, this principle explains how a minuscule loss of mass translates into a monumental gain of energy.

This whole concept puts into perspective the immense power locked within the tiniest particles that make up everything in the universe. In our solar furnace, this energy keeps the Sun shining and supports life on Earth. So next time you look up at the Sun, remember that the warmth you feel is thanks to a little 'lost' mass and Einstein's groundbreaking insight into the nature of energy.