In a simplified model of stellar energy production

PHYSICS · Multi-part · CBSE · Physics · Nuclei · Nuclei

Problem

In a simplified model of stellar energy production, four protons combine to form a helium-4 nucleus with the emission of two positrons and two neutrinos (whose masses may be neglected):

4\,^{1}_{1}\text{H} \rightarrow \,^{4}_{2}\text{He} + 2e^{+} + 2\nu

. The atomic masses are

m(^{1}_{1}\text{H}) = 1.007825\,\text{u}

and

m(^{4}_{2}\text{He}) = 4.002603\,\text{u}

. Take

1\,\text{u}=1.6605\times10^{-27}\,\text{kg}

and

c = 3.0\times10^{8}\,\text{m}\,\text{s}^{-1}

. (i) Derive an expression for the percentage of the initial mass that is converted into energy in terms of the given atomic masses. (ii) Calculate this percentage for the reaction. (iii) If

2.0\times10^{29}\,\text{kg}

of hydrogen in the Sun’s core undergoes the above reaction, estimate the total energy released in joules. (iv) The present luminosity of the Sun is about

3.8\times10^{26}\,\text{W}

. Assuming constant luminosity, estimate the number of years for which the energy calculated in part (iii) could sustain the Sun’s output.

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