Question

At what rate, in megagram per minute, must hydrogen be consumed in a fusion reaction to provide the sun with the energy it radiates? (Take mass defect per reaction to be $0.028706\,a.m.u$)
A. $3.66 \times {10^{14}}\,mega\,gram\,{\min ^{ - 1}}$
B. $3.66 \times {10^{16}}\,mega\,gram\,{\min ^{ - 1}}$
C. $3.66 \times {10^{15}}\,mega\,gram\,{\min ^{ - 1}}$
D. $3.66 \times {10^{10}}\,mega\,gram\,{\min ^{ - 1}}$

Answer 1

Hint: Nuclear fusion occurs when two or more atomic nuclei unite to generate one or more new atomic nuclei and subatomic particles (neutrons or protons). The release or absorption of energy is caused by the mass differential between the reactants and products. The difference in mass between the nuclei before and after the reaction is due to a difference in atomic binding energy.

Complete step by step answer:
Fusion is the mechanism that generates enormous amounts of energy in active or main sequence stars and other high-magnitude stars. Fusion processes that produce nuclei lighter than iron-\[56\] or nickel-\[62\] will generally release energy. These elements have a high binding energy per nucleon and a small mass per nucleon.

Fusion of lighter nuclei releases energy (an exothermic process), whereas fusion of heavier nuclei results in the resultant nucleons retaining energy (an endothermic reaction). Nuclear fission, on the other hand, is the inverse process. This means that lighter elements like hydrogen and helium are more flammable, whereas heavier elements like uranium, thorium, and plutonium are more fissionable.The fusion reaction is $4H_1^1 \to He_2^4$.

The mass defect $(\vartriangle m) = 0.028706\,amu = 0.028706 \times 1.67 \times {10^{ - 24}} = 4.8 \times {10^{ - 27}}g$
Energy from mass defect is $\vartriangle m{c^2} = 4.8 \times {10^{ - 27}} \times {(3 \times {10^8})^2} = 4.32 \times {10^{ - 10}}$
This is the energy per $4$ hydrogen atoms fusion
Radius of sun (R)=$6.96 \times {10^8}m$
Total energy per minute=$2 \times 4.2 \times {10^4} \times 4\pi (6.96 \times {10^8})$
Let the mass of hydrogen required be $4n$ then the energy released due to mass defect is $n \times 4.32 \times {10^{ - 27}}$.
$ \therefore 4n = 3.66 \times {10^{16}}Mg\,{\min ^{ - 1}}$

So, option B is correct.

Note:To solve this problem there are some very important points which we should keep at our fingertips. The unit conversion is very necessary here, so before solving the question we should always check that each data should be in the same unit.